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Mathematics Thesis and Dissertations

2019-03-27 Generalizations of Skew L-Almost Distributive Lattices and their Duals

Yeshiwas, Mebrat http://hdl.handle.net/123456789/9313 Downloaded from DSpace Repository, DSpace Institution's institutional repository Generalizations of Skew L-Almost Distributive Lattices and their Duals

By

Yeshiwas Mebrat Gubena

A dissertation submitted to the Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia, in partial fulfillment of the require- ments of the degree of Doctor of Philosophy in Mathematics.

Mar 18, 2019 This dissertation entitled ”Generalizations of Skew L-Almost Distribu- tive Lattices and their Duals” by Mr. Yeshiwas Mebrat is approved for the award of the degree of ”Doctor of Philosophy in Mathematics”.

Principal Supervisor:

Berhanu Assaye Alaba, (PhD, Associate Professor)

Co - Supervisor:

Mihret Alamneh Taye, (PhD, Associate Professor)

External Examiner:

Venkateswarlu, K. (PhD, Professor)

Internal Examiner:

Berhanu Bekele (PhD, Associate Professor)

Dr. Naveen Kumar Kakumanu is also an external examiner of the dissertation who gives his evaluation electronically.

i Declaration

I hereby declare that the work reported in this dissertation is entirely original and was carried out by me independently in the department of Mathematics, Bahir Dar University, Bahir Dar, under the supervision of Dr. Berhanu Assaye (Associate Professor) and Dr. Mihret Alamneh (Associate Professor). Further I declare that no part of this dissertation formed the basis for the award of any Degree, Diploma, associateship or other similar title to me.

Yeshiwas Mebrat, Bahir Dar University, Bahir Dar, Ethiopia. Mar 18, 2019

ii Certificate

This is to certify that the present dissertation entitled ”Generalizations of Skew L-Almost Distributive Lattices and their Duals” is a genuine record of the work done by Mr. Yeshiwas Mebrat in the department of Mathemat- ics, Bahir Dar University, Bahir Dar, Ethiopia during the period from oct; 2016 to March 18; 2019 and that this work has not previously formed the basis for the award of any degree or diploma to him.

Research supervisors:

Berhanu Assaye (PhD, Associate Professor)

Bahir Dar University, Bahir Dar, Ethiopia.

Mihret Alamneh (PhD, Associate Professor) Bahir Dar University, Bahir Dar, Ethiopia.

iii Acknowledgments

First and foremost I would like to thank the Almighty God for his blessing. He give me strength and courage through out my life. For that, I am forever grateful. I thank my supervisors, Dr. Berhanu Assaye (Associate Professor) and Dr. Mihret Alamneh (Associate Professor) for their continuous support, and providing me with proper guidance. They also share me their experience to do a research in algebra. I take this opportunity to express my thanks to my friends (G. Yohannes (PhD.), T. Bekalu (PhD.), G. Teferi, A. Derso, M. Gezahegn, H. Kidus and Wondewosen) and my class mates (M. Tilahun, N. Derbew and T. Gerima) for the friendship we developed and the discussions we had to solve our problems in mathematics and socially as well. Further, I would like to thank staff members of the department of Mathematics, Bahir Dar and Debre Tabor Universities for all dedication and encouragement they offer me during my study. I thank Bahir Dar University, College of Science, Department of Mathematics for providing the necessary facilities to do my research work. Finally, no words can exactly express the unparalleled support and encouragement given to me by my wife Emawayesh Mandefro, my sister Hiwot Adebabay and my brothers (Samuel and Tewbo) to pursue and complete my PhD. study.

iv Publications

From this dissertation the following Four papers are published. 1. Berhanu Assaye Alaba, Mihret Alamneh Taye and Yeshiwas Mebrat Gubena, Skew Semi- Heyting Algebras, International Journal of Computing Sci- ence and Applied Mathematics, Vol.4, No.1(2018), 10 - 14. 2. Berhanu Assaye Alaba, Mihret Alamneh Taye and Yeshiwas Mebrat Gubena, Skew Semi-Heyting Almost Distributive Lattices, International Journal of Mathematics And its Applications, Vol.5, Issue II-C(2017), 359 - 369. 3. Berhanu Assaye Alaba, Mihret Alamneh Taye and Yeshiwas Me- brat Gubena, Dual Skew Semi-Heyting Almost Distributive Lattices, Interna- tional Journal of Advances in Mathematics, Vol 2019, Number 1(2019), 61-71. 4. Berhanu Assaye Alaba, Mihret Alamneh Taye and Yeshiwas Mebrat Gubena, Skew Heyting Almost Distributive Lattices, Bulletin of The International Mathematical Virtual Institute, Vol.9(2019), 85-94. Prints of the above mentioned published papers are appended at the end of this dissertation.

v Abstract

In this dissertation, we introduce the concepts of Skew Heyting Almost Dis- tributive Lattices (skew HADLs) and characterize it in terms of the set PI(L) of all of the principal ideals of a skew HADL L. Considering a relatively complemented almost distributive lattice we define a congruence relation θ on a skew HADL L and show that each congruence class is a maximal rectangular subalgebra of L and L/θ is a maximal lattice image of L. Using the essences of Heyting algebra, Karin Cvetko-vah introduced the con- cept of skew Heyting algebra. In the same way we introduce the concept of skew semi-Heyting algebra and extend the notions of semi-Heyting algebras. We char- acterize a skew semi-Heyting algebra L as a skew Heyting algebra in terms of a unique binary operation b→ on the upset b↑ for each b ∈ L, on which an induced binary operation → is defined on L. Most of the results discussed related to this concept are published on the journal ”International Journal of Computing Science and Applied Mathematics, Vol. 4, 2018, 10 - 14. ” Based on the notions of skew semi-Heyting algebra we introduce the concept of a skew semi-Heyting almost dis- tributive lattices (skew SHADLs). Besides that, we define a relation θ on a skew SHADL L so that each congruence class is the maximal rectangular subalgebra and L/θ is the maximal lattice image of L. Most of the results of our research related to skew SHADL are published on the journal ”International Journal of Mathematics and its Applications, Vol. 5, 2017, 359-369.” In a similar way we introduce the concept of skew L-algebra and extend the

vi notions of L-algebras. We characterize a skew L-algebra as a Stone lattice, and different conditions on which a skew Heyting algebra becomes a skew L-algebra are given. Using the non commutative nature of an ADL and the concept of skew L- algebra that we introduced, we extend the concept of L-almost distributive lattices (L-ADLs) to skew L-ADLs and characterize skew L-ADLs as skew L-algebras in terms of a congruence relation defined on it. We also characterize skew L-ADLs in terms of the set PI(L) of all of the principal ideals of a skew L-ADL L. Motivated by the results on dual HADL, dual L-ADL, dual pseudocomple- mented ADL, etc., we introduce the concepts of dual skew HADLs, dual skew SHADLs and dual skew L-ADLs. We characterize these algebras in terms of the congruence relations defined on the congruence classes of each of the algebras and each of the set of all the principal ideals of those algebras. Further we study dif- ferent algebraic properties of these algebras. Most of the results discussed related to the concept of dual skew SHADLs are accepted for publication in the journal ”International Journal of advances in Mathematics.

vii Contents

Abstract vi

Introduction 1

1 Preliminaries 9 1.1 Lattices and Almost Distributive Lattices ...... 9 1.2 Some generalizations of Heyting algebras ...... 19 1.3 Skew Lattices, Skew Heyting algebras and Dualities ...... 24

2 Skew Heyting Almost Distributive Lattices 29 2.1 Skew Almost Distributive Lattices ...... 30 2.2 Skew Heyting Almost Distributive Lattices ...... 38

3 Skew Semi-Heyting Almost Distributive Lattices 55 3.1 Skew Semi- Heyting Algebras ...... 56 3.2 Skew Semi-Heyting Almost Distributive Lattices ...... 69

4 Skew L-Almost Distributive Lattices 80 4.1 Skew L-Algebras ...... 81 4.2 Skew L-Almost Distributive Lattices ...... 89

5 Dual Skew Heyting Almost Distributive Lattices 97 5.1 Dual Skew Heyting Almost Distributive Lattices ...... 98

viii 5.2 Dual Skew Semi-Heyting Almost Distributive Lattices ...... 111 5.3 Dual Skew L-Almost Distributive Lattices ...... 122

Bibliography 129

ix Introduction

After the Boole’s axiomatization of the two valued propositional calculus into a Boolean algebra [2], many generalizations both ring theoretically and lattice theo- retically of a Boolean algebra have come into being such as: semirings, distributive lattices, Heyting algebras and almost distributive lattices (ADLs). The concept of an ADL was introduced in 1981 by Swammy, U. M. and Rao, G. C. [47] as a common abstraction of almost the existing ring theoretic general- izations of a Boolean algebra like p-rings [22], regular rings [50], biregular rings [2], associate rings [47], p1-rings [45], Baer rings [24] and m-domain rings [46] on one hand and the class of distributive lattices on the other hand. Heyting algebra is a relatively pseudo-complemented distributive lattice. It arises from the study of non-classical logic. While Boolean algebras provide alge- braic models of classical logic, Heyting algebras provide algebraic models of in- tuitionistic logic. It was first investigated by Skolem T. [42]. It is named as Heyting algebra after the Dutch Mathematician Arend Heyting [1]. It was also studied by Birkhoff, G. under a different name Brouwerian lattice and with a different notation [5]. Following this Epstein, G. and Horn, A. in [8] introduced the concept of L-algebra as a Heyting algebra (H, ∨, ∧, →, 0, 1) with the property (a → b) ∨ (b → a) = 1 for all a, b ∈ H. On the other hand Sankappanavar, H.P. [37] in 2007 defined and investigated a new class of algebras which is called semi-Heyting algebras as an abstraction from Heyting algebras. He showed that semi-Heyting algebras share with Heyt-

1 ing algebras some strong properties, like these algebras are: pseudo-complemented, distributive, congruences on them are determined by filters and every interval in a semi-Heyting algebra is also pseudo-complemented. More over in his study he showed that the variety of semi-Heyting algebras is arithmetical, thus extending the corresponding results of Heyting algebras. An ADL is an algebra (L, ∨, ∧, 0) of type (2, 0, 0) which satisfies all the axioms of a distributive lattice with zero except possibly commutativity of ∨ or commuta- tivity of ∧ or right distributivity of ∨ over ∧ or the absorption law (a ∧ b) ∨ a = a. It was observed that, the set of all principal ideals of an ADL becomes a distribu- tive lattice through which one can extend many concepts existing in the class of distributive lattices to the class of ADLs. Also, Rao, G. C., and Nanaji Rao, G. studied pseudo-complemented ADLs and Stone ADLs [27], Rao, G. C. and Kumar, S. R. introduced and studied normal ADLs and relative normal ADLs [33]. Also Rao, G. C. and Sambasiva Rao [45] studied α-ideals in the class of ADLs. Mo- tivated by the above results Rao, G. C., Berhanu Assaye and Ratnamani, M. V. introduced the concept of a Heyting almost distributive lattice (HADL) [34] as an ADL (H, ∨, ∧, 0) with maximal element m such that for each a in H, ([0, a], ∨, ∧) is a Heyting algebra, where [0, a] = {x ∈ H : 0 ≤ x ≤ a}. Further Rao, G. C., Ratnamani, M. V., Shum, K. P. and Berhanu Assaye introduced the concept of semi-Heyting almost distributive lattice (SHADL) [35] as a generalization of a semi-Heyting algebra in the class of ADLs. Likewise Rao, G. C., Berhanu Assaye and Prasad, R. developed the concept of an L-almost dis- tributive lattice(L-ADL) [36] as an ADL (H, ∨, ∧, 0) with maximal elements such that for each a in H, the interval [0, a] is an L-algebra or equivalently, for any a, b ∈ H, (a → b) ∨ (b → a) is maximal. In that paper they gave several equivalent conditions for an HADL to be an L-ADL and also for an L-ADL to become an L-algebra. Karin Cvetko-vah, on her paper [6] mentioned that the study of non commu-

2 tative lattices began in 1949 with Pascual Jordan’s paper ”Uber nichtkommutative Verbande” that was motivated by certain questions in quantum mechanics to rec- oncile the inconsistency of the two complementary measurements, position and momentum of a particle. Further developments in non commutative lattice theory have occurred in a paper on pseudo-lattices by Schein [39], and in the two papers on near lattices by Schweigert ([40] and [41]). Non commutative lattice is an al- gebra (S, ∨, ∧), where both ∨ and ∧ are associative, idempotent binary operations which are connected by laws of absorption. The precise absorption laws chosen may depend upon the underlying motivation, and differing choices generally pro- duce distinct varieties of algebras. Jordan chose for absorption laws the identities, a ∧ (b ∨ a) = a = (a ∧ b) ∨ a. By augmenting the above identities, Jordan and others obtained a number of varieties of non commutative lattices [20]. Pseudo-lattices were introduced by Schein to study double quasi-ordered sets of the form (A, L, R), where L and R are both quasi-orders on a set A whose inter- section is antisymmetric. Given (A, L, R) and a pair x, y ∈ A, the greatest lower bound of {x, y}, denoted x ∧ y, is an element such that x ∧ y ≤ x(L), x ∧ y ≤ y(R), and x ∧ y is maximal among all such minorants of (x, y) in the partial ordering L ∩ R. Dually, one may define the least upper bound of {x, y}, denoted x ∨ y. Double quasi-ordered sets possessing both greatest lower bounds and least upper bounds for arbitrary pairs of elements are called pseudo-lattices. In general, neither binary operation, ∨ or ∧, is even associative. However, if either is commutative, then so is the other and the algebra reduces to a lattice with both quasi-orderings coinciding with the common lattice ordering. In 1989 Jonathan Leech laid the foundation of modern theory of skew lattices [20]. He define a skew lattice is an algebra (S, ∨,, ∧) in which both ∨ and ∧ satis- fies the idempotent laws, associative laws and absorption laws. For a skew lattice (S, ∨, ∧) and for x, y ∈ S. Two elements of a skew lattice x and y are said to be equivalent, denoted x ≡ y, whenever x∨y ∨x = x and y ∨x∨y = y. In his study he

3 showed that the equivalence as defined above forms a congruence relation. Leech [18] and [19] showed that each right handed skew Boolean algebra can be embedded into a skew Boolean algebra of partial functions from a given set to the co-domain {0, 1}. The notion of skew Heyting algebra was introduced by Karin Cvetko-vah [6]. In that paper it is proved that a Heyting algebra form a variety and that the maximal lattice image of a skew Heyting algebra is a generalized Heyting algebra. Let (P, ≤) be a poset. Thus (P, ≥) is also a poset called the dual of (P, ≤). If φ is a statement about posets and in φ we replace all occurrences of ≤ by ≥, we get the dual of φ. The principle of duality states that ”If a statement φ is true in all posets, then its dual is also true in all posets ” [10]. Let φ be a statement about lattices expressed in terms of the binary operations ∧ and ∨, the dual of φ is the statement we get from φ by interchanging ∧ and ∨. In this case the principle of duality is given by ”if φ is true for all lattices, then the dual of φ is also true for all lattices ”. Unlike in lattices, the dual of an ADL is not an ADL in general. For this reason Rao, G. C. and Naveen Kumar K. introduced the concept of a dual Heyting almost distributive lattice (dual HADL) [28] and dual L-almost distributive lattice (dual L-ADL) [29]. They derive a number of important laws and results satisfied by a dual HADL and dual L-ADL. They also characterize a dual HADL in terms of the lattice of all of its principal ideals.

Motivated by the above results we consider the set La = {x ∧ a|x ∈ L} and introduce the concept of skew Heyting almost distributive lattice (skew HADL) as an ADL (H, ∨, ∧) with a maximal element m but with out 0 such that for each a ∈ H,(La, ∨, ∧, →a, m) is a skew Heyting algebra where →a is a binary opera- tion on La. We derive several conditions on which an ADL becomes a skew HADL for if, we show that an ADL H with a maximal element m, 0 ∈/ H, and to each a ∈ H, x, y, z ∈ La satisfying the following conditions:

(1) y ≤ x →a y

(2)( y →a z) ∧ x = x if and only if z ∧ x ∧ y = x ∧ y,

4 is a skew HADL. Characterization of a skew HADL in terms of a congruence relation and the set of all the principal ideals of the given skew HADL are given. Further, we present different conditions on which a skew HADL becomes a skew Heyting algebra. We introduce the concept of skew semi-Heyting algebra and extend the notions of semi- Heyting algebras. We characterize a skew semi-Heyting algebra L as a skew Heyting algebra in terms of a unique binary operation b→ on b↑ for each b ∈ L from which an induced binary operation → is defined on L. Also, we introduce the concept of skew semi-Heyting almost distributive lattices (skew SHADLs) as an ADL (L, ∨, ∧) with out 0 but with a maximal element m such that for each a ∈ L,(La, ∨, ∧, m) is a skew semi-Heyting algebra with respect to the binary operation →a. Besides that, we define a relation θ on a skew SHADL L so that each congruence class is the maximal rectangular subalgebras and L/θ is the maximal lattice image of L. In a similar way we introduce the concept of skew L-algebra and extend the notions of L-algebras. We characterize a skew L-algebra as a Stone lattice, and different conditions on which a skew Heyting algebra becomes a skew L-algebra are given. Using the non commutative nature of an ADL and the concept of skew L-algebra that we introduced, we extend the concept of L-ADLs to skew L-ADLs and characterize skew L-ADLs as skew L-algebras in terms of the set of all the principal ideals of a given skew L-ADL. We give different conditions on which a skew HADL becomes a skew L-ADL. Also we define a congruence relation θ on a skew L-ADL L in such a way that the quotient algebra L/θ is the maximal lattice image of L and each congruence class is a maximal rectangular subalgebra. We also characterize skew L-ADLs in terms of some conditions related to an interval with 0 and an induced binary operation defined on the interval. Motivated by the results on dual HADL, dual L-ADL and dual pseudo-complemented ADL we extend the concepts of dual skew Heyting algebras, dual skew semi-Heyting algebras and dual skew L-algebras in the class of ADLs and introduce the concepts

5 of dual skew HADLs, dual skew SHADLs and dual skew L-ADLs respectively. This dissertation comprises of five chapters. In chapter one, we collect im- portant definitions and results that are already known for ready reference in the sequel. The first section of chapter one deals on posets, lattices and almost dis- tributive lattices. The second section of this chapter focuses on the results about some generalizations of Heyting algebras like semi-Heyting algebras and L-algebras. The final section of this chapter is on the results about skew lattices, skew Heyting algebras and duality. Chapter two is devoted to the study of skew lattices in the class of ADLs and skew HADLs. In section 2.1, we define a skew ADLs and we give different results on which a skew ADL is a rectangular lattice. Moreover, for a skew ADL S we showed that S/D is a maximal lattice image of S where D is Green’s congruence relation. In section 2.2 we extend the concept of skew Heyting algebras and intro- duce the concept of skew HADLs. We give many conditions for an ADL to become skew HADLs and a skew HADLs to become a skew Heyting algebras, for instance we prove that an ADL L with out 0 but a maximal element m is a skew HADL if and only if the set PI(L) of all principal ideals of L is a skew Heyting algebra. Further we define a congruence relation θ on a skew HADL L so that the congru- ence classes are the maximal rectangular subalgebras and the quotient L/θ is the maximal lattice image of L. Most of the contents of section two of this chapter are communicated for publication in the journal ” Bulletin of the International Math- ematical Virtual Institute” Chapter three is divided into two sections namely, skew semi-Heyting algebras and skew SHADLs. In section 3.1 we introduce the concept of skew semi-Heyting algebra and we characterize it as a skew Heyting algebra in terms of a unique binary operation on which an induced binary operation b→ is defined on b↑. We also give different conditions on which a skew semi-Heyting algebra becomes a skew Heyting algebra. Most of the contents of this section are included in the paper entitled

6 ”Skew Semi-Heyting Algebras” [3] which is published in the journal ”International Journal of Computing Science and Applied Mathematics” Vol.4. 2018. In section 3.2, using the concept of skew semi-Heyting algebras we introduce the concept of a skew SHADL and study its basic properties. We give different condi- tions for an ADL to become skew SHADLs and a skew SHADLs to become a skew semi-Heyting algebras. Most of the results of this section are included in the paper entitled by ”Skew Semi-Heyting Almost Distributive Lattices” [4] published on the journal ”International Journal of Mathematics and its Applications” Vol. 5. 2018. Chapter four contains two sections namely skew L-algebra and skew L-ADL. In section one we introduce the concept of skew L-algebra and extend the notions of L-algebra. We characterize a skew L-algebra as a Stone lattice and different con- ditions on which a skew Heyting algebra becomes a skew L-algebra are derived. Most of the contents of this section are communicated for publication on the jour- nal ”Journal of Algebraic Systems ”. In section 4.2 we define skew L-ADL and as a generalization of skew L-algebra and study its basic properties. We give many equivalent conditions for an ADL to become skew L-ADL and a skew L-ADL to become a skew L-algebra. Chapter five comprises of three sections which are devoted on the study of duals of skew HADL, skew SHADL and skew L-ADL. In the first section we define dual skew Heyting algebra and based on this definition we give the definition of dual skew HADL as: an ADL (L, ∨, ∧, 0) with 0 but with out maximal element is called a dual skew Heyting almost distributive lattice (dual skew HADL) if to each

a a ∈ L the algebra (L , ∨, ∧,a ←, a) is a dual skew Heyting algebra where a← is the binary operation on La = {a ∨ x|x ∈ L}. We give many equivalent conditions for an ADL to become a dual skew HADL and a dual skew HADL to become a dual skew Heyting algebras. In section two of this chapter we define dual skew semi-Heyting algebra and based on this definition we give the definition of a dual skew semi-Heyting almost

7 distributive lattice (dual skew SHADL) as: an ADL (L, ∨, ∧, 0) with 0 but with out maximal element is called a dual skew semi-Heyting almost distributive lattice if to each a ∈ L the algebra (La, ∨, ∧, a) is a dual skew semi-Heyting algebra with respect to the binary operation denoted by a←. We give equivalent conditions for an ADL to become a dual skew SHADL and a dual skew SHADL to become a dual skew semi-Heyting algebras. In the last section of this chapter we give the defini- tion of dual skew L-algebra and then define a dual skew L-ADL. For a relatively complemented ADL L with 0 but with out maximal element such that L is a dual skew L-ADL we define and proved that the relation

θ = {(x, y) ∈ L × L|x ∨ y = x and y ∨ x = y} is an equivalence relation on L. Also we proved that for each equivalence class [x]θ where x ∈ L and for any a ∈ [x]θ, θ is a congruence relation and the quotient [a]θ/θ is the maximal lattice image of [a]θ.

8 Chapter 1

Preliminaries

In this chapter we collect all the necessary preliminaries which will be useful in our discussions in the main text of the dissertation. Even though these pre- liminaries are well known for those working in Lattice Theory and the concept of an Almost Distributive Lattice, it will be convenient for others to have all these elementary notions and results in the beginning of the dissertation for the sake of ready reference. The proofs of most of the results presented in this chapter are either straight forward verifications or wellknown and hence we simply state the results without proofs.

1.1 Lattices and Almost Distributive Lattices

In this section, we give the necessary definitions and results on posets, lattices and almost distributive lattices which will be used in the subsequent sections.

Definition 1.1.1. Let P be a nonempty set. Then a binary relation ”≤” on P satisfying the following properties is called a partial order on P .

(1) Reflexivity: a ≤ a

(2) Antisymmetric: a ≤ b and b ≤ a imply that a = b

9 (3) Transitivity: a ≤ b and b ≤ c imply that a ≤ c for all a, b, c ∈ P .

In this case (P, ≤) is called a partial ordered set or simply a poset. Let (P, ≤) be a poset. If a ≤ b and a 6= b, then we write a < b.

Definition 1.1.2. Let (P, ≤) be a poset and a, b ∈ P . Then we say that a and b are comparable if either a ≤ b or b ≤ a. Otherwise we say that a and b are incomparable. A poset (P, ≤) in which there are no incomparable elements is called a chain or a totally ordered set.

Definition 1.1.3. Let (P, ≤) be a poset, H ⊆ P and a ∈ P . Then

(1) a is called a lower bound of H if a ≤ h for all h ∈ H.

(2) a is called an upper bound of H if h ≤ a for all h ∈ H.

(3) a is called the greatest lower bound or g.l.b or infimum of H if a is a lower bound of H and for any lower bound b of H, we have b ≤ a. In this case we write a = g.l.bH or a = infH.

(4) a is called the least upper bound or l.u.b or supremum of H if a is an upper bound of H and for any upper bound b of H, we have a ≤ b. In this case we write a = l.u.bH or a = supH.

Definition 1.1.4. Let (P, ≤) be a poset and a ∈ P . Then

(1) a is called a minimal element, if x ∈ P and x ≤ a implies that x = a.

(2) a is called a maximal element, if x ∈ P and a ≤ x implies that a = x.

Definition 1.1.5. Let (P, ≤) be a poset and a ∈ P . Then

(1) a is called least element of P if a ≤ x for all x ∈ P . If P has least element, then it is unique and is denoted by 0.

10 (2) a is called greatest element of P if x ≤ a for all x ∈ P . If P has greatest element, then it is unique and is denoted by 1.

Definition 1.1.6. A poset (P, ≤) with 0 and 1 is called a bounded poset.

Theorem 1.1.7. (Zorn’s Lemma): If (P, ≤) is a poset in which every chain has an upper bound, then there exists a maximal element in P.

Definition 1.1.8. A poset (P, ≤) is called a lattice if l.u.b{a, b} and g.l.b{a, b} exist for all a, b ∈ P .

Definition 1.1.9. Let A be a nonempty set. Define A0 = {∅}, a singleton set,

1 n n A = A and A = {(a1, a2, ..., an)|ai ∈ A, 1 ≤ i ≤ n}. A mapping f : A −→ A is called n−ary operation on A and n is called the arity of f. If n = 0, f is called as nullary operation. If n = 1, f is called as unary operation. If n = 2, f is called as binary operation.

We may identify a nullary operation on A with an element of A.

Definition 1.1.10. A language (or type ) of algebras is a set F of function symbols such that a nonnegative integer n is assigned to each member f of F. This integer is called the arity (or rank) of f and f is said to be an n−ary function symbol. The subset of n−ary function symbols in F is denoted by Fn.

Definition 1.1.11. If F is a language of algebras then an algebra A of type F is an ordered pair (A, F ) where A is a nonempty set and F is a family of finitary operations on A indexed by the language F such that corresponding to each n−ary function symbol f in F there is an n−ary operation f A on A. The set A is called the universe (or underline set) of A = (A, F ) and f A’s are called the fundamental operations of A.

If F is finite say F = {f1, f2, ..., fn} and the arity of fi is ni (1 ≤ i ≤ r), then an A A A algebra A of type F is written as an algebra (A, f1 , f2 , ..., fr ) of type (n1, n2, ..., nr).

11 Definition 1.1.12. Let A and B be any two algebras of the same type F.A

A mapping α : A −→ B is called a homomorphism from A to B if αf (a1, a2, ..., an) = B f (αa1, αa2, ..., αan) for each n−ary f in F and each sequence (a1, a2, ..., an) from A.

If α is surjection, then B is said to be a homomorphic image of A and α is called an epimorphism. If α is an injection, then α is called monomorphism. If α is bijection, then α is called an endomorphism. In case A = B, a homomorphism is also called an endomorphism and an isomorphism is referred to as an automorphism. If α : A −→ B is a monomorphism then α is said to be an embedding. We say that A can be embedded in B if there is an embedding α : A −→ B. The definition of a lattice as an algebra is given below.

Definition 1.1.13. An algebra (L, ∨, ∧) of type (2, 2) is called a lattice if it satisfies the following identities:

(1) Commutativity: a ∧ b = b ∧ a and a ∨ b = b ∨ a

(2) Associativity: (a ∧ b) ∧ c = a ∧ (b ∧ c) and (a ∨ b) ∨ c = a ∨ (b ∨ c)

(3) Absorption laws: a ∧ (a ∨ b) = a and a ∨ (a ∧ b) = a.

Now consider the lattice (L, ≤). If we define a ∧ b = g.l.b{a, b} and a ∨ b = l.u.b{a, b}, then we obtain the lattice (L, ∨, ∧). Similarly if we define ≤ by a ≤ b if and only if a ∧ b = a or equivalently a ∨ b = b on the lattice (L, ∨, ∧), then we obtain that (L, ≤) is a lattice. In any lattice (L, ∨, ∧), the following identities are equivalent:

(1) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)

(2)( a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c)

(3) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

12 (4)( a ∧ b) ∨ c = (a ∨ c) ∧ (b ∧ c).

Definition 1.1.14. A lattice (L, ∨, ∧) satisfying any one of the above four identities is called a distributive lattice.

Let (L, ∨, ∧) be a lattice. An element a of L is called zero element or least element of L if a ∧ x = a for all x ∈ L. If L has a least element, then it is unique and it is denoted by 0. Similarly an element a of L is called one element or greatest element of L if a ∧ x = x for all x ∈ L. If L has a greatest element, then it is unique and it is denoted by 1. More over if a lattice (L, ∨, ∧) contains both 0 and 1, then we call it a bounded lattice.

Definition 1.1.15. A bounded lattice (L, ∨, ∧) with 0 and 1 is said to be comple- mented if to each a ∈ L there exists b ∈ L such that a ∨ b = 1 and a ∧ b = 0.

Definition 1.1.16. A bounded distributive and complemented lattice (L, ∨, ∧) is called a Boolean lattice.

Definition 1.1.17. A uniquely bounded complemented distributive lattice (L, ∨, ∧,0 , 0, 1) is called a Boolean algebra.

In other words a Boolean algebra is a Boolean lattice in which 0, 1 and 0(complementation) are also considered to be fundamental operations.

Definition 1.1.18. Let X be a nonempty set and θ be a binary relation on X (that is θ ⊆ X × X). Then θ is said to be an equivalence relation on X if θ satisfies the following conditions:

(1) Reflexive: (x, x) ∈ θ for all x ∈ X

(2) Symmetric: (x, y) ∈ θ implies that (y, x) ∈ θ for all x, y ∈ X

(3) Transitive: (x, y) ∈ θ and (y, z) ∈ θ imply that (x, z) ∈ θ for all x, y, z ∈ X.

For (x, y) ∈ θ. We write xθy.

13 Definition 1.1.19. Let A be an algebra of type F and θ be an equivalence relation on set A. Then θ is a congruence relation on A if θ satisfies the following compati- bility property: For each n−ary function symbol f ∈ F and elements ai, bi ∈ A, if A A aiθbi holds for 1 ≤ i ≤ n then f (a1, a2, ..., an)θf (b1, b2, ..., bn) holds.

Swamy, U. M. and Rao, G. C. introduced the concept of an ADL as a com- mon abstraction to most of the existing ring theoretic generalizations of a Boolean algebra (like regular rings, p−rings, bi regular rings, associate rings, p1-rings etc.) on one hand and distributive lattices on the other hand. In this section we recall the definition of an ADL and certain elementary properties of an ADL. These results are taken from [48].

Definition 1.1.20. An algebra (R, ∨, ∧) of type (2, 2) is called an Almost Distribu- tive Lattice if it satisfies the following axioms:

(1)( x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)

(2) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

(3)( x ∨ y) ∧ y = y

(4)( x ∨ y) ∧ x = x

(5) x ∨ (x ∧ y) = x for all x, y, z ∈ R.

Definition 1.1.21. An algebra (R, ∨, ∧, 0) of type (2, 2, 0) is called an Almost Distributive Lattice (ADL) with 0 if it satisfies the following axioms:

(1) x ∨ 0 = x

(2)0 ∧ x = 0

(3)( x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)

14 (4) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

(5) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

(6)( x ∨ y) ∧ y = y for all x, y, z ∈ R.

It can be seen directly that every distributive lattice is an ADL.

Example 1.1.22. Let X be a nonempty set. Fix x0 ∈ X. For any x, y ∈ X, define  x0 if x = x0 x ∧ y =

y if x 6= x0 and  y if x = x0 x ∨ y =

x if x 6= x0.

Then (X, ∨, ∧, x0) is an ADL with x0 as its zero element. This ADL is called a discrete ADL.

Here onwards by R we mean an ADL (R, ∨, ∧). For any a, b ∈ R, we say that a is less than or equal to b and write a ≤ b, if a ∧ b = a. Then ” ≤ ” is a partial ordering on R. The following hold in any ADL R.

Theorem 1.1.23. Let R be an ADL with 0. Then for any w, x, y, z ∈ R, the following conditions hold:.

(1) x ∨ y = x ⇔ x ∧ y = y

(2) x ∨ y = y ⇔ x ∧ y = x

(3) x ∧ y = y ∧ x = x whenever x ≤ y

(4) ∧ is associative

15 (5) x ∧ y ∧ z = y ∧ x ∧ z

(6) (x ∨ y) ∧ z = (y ∨ x) ∧ z

(7) x ∧ y ≤ y and x ≤ x ∨ y

(8) x ∧ (y ∧ x) = y ∧ x and x ∨ (x ∨ y) = x ∨ y = (x ∨ y) ∨ y

(9) x ∧ x = x and x ∨ x = x

(10) x ∧ 0 = 0 and 0 ∨ x = x

(11) {w ∨ (x ∨ y)} ∧ z = {(w ∨ x) ∨ y} ∧ z

(12) If there exist r ∈ R such that x ≤ r and y ≤ r, then x ∧ y = y ∧ x and x ∨ y = y ∨ x.

Note: It can be observed that an ADL R satisfies all the properties of a distributive lattice except possibly the right distributivity of ∨ over ∧, the commu- tativity of ∨, the commutativity of ∧ and the absorption law (a ∧ b) ∨ a = a. Any one of these properties gives a distributive lattice R.

Theorem 1.1.24. Let (R, ∨, ∧) be an ADL with 0. Then the following are equiva- lent.

(1) (R, ∨, ∧) is a distributive lattice

(2) a ∨ b = b ∨ a for all a, b ∈ R

(3) a ∧ b = b ∧ a for all a, b ∈ R

(4) (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c).

Theorem 1.1.25. Let R be an ADL. Then for any a, b, c ∈ R with a ≤ b, we have the following conditions:

(1) a ∧ c ≤ b ∧ c

16 (2) c ∧ a ≤ c ∧ b

(3) c ∨ a ≤ c ∨ b.

Theorem 1.1.26. Let R be an ADL. For any m ∈ R the following are equivalent.

(1) m is maximal element

(2) m ∨ x = m for all x ∈ R

(3) m ∧ x = x for all x ∈ R.

For any a ∈ R and a partial ordering ≤ on R, the set {x ∈ R|a ≤ x ≤ b} is an interval denoted by [a, b].

Definition 1.1.27. An ADL R is said to be relatively complemented if for any a, b ∈ R with a < b, the interval [a, b] is a complemented lattice.

Theorem 1.1.28. Let R be an ADL. Then the following conditions are equivalent.

(1) R is associative

(2) θa := {(x, y) ∈ R × R : a ∨ x = a ∨ y} is a congruence relation for all a ∈ R

(3) R is a subdirect product of ADLs in each of which there are at most two nonzero elements and every nonzero element is maximal.

Note: In an ADL (R, ∨, ∧, 0) the associative property with respect to ∧ holds, that is for any a, b, c ∈ R, a ∧ (b ∧ c) = (a ∧ b) ∧ c. Whereas the associative property with respect to ∨, that is, for any a, b, c ∈ R, a ∨ (b ∨ c) = (a ∨ b) ∨ c is not known so far. However, Swamy, Rao and others developed the theory by taking the associativity of ∨.

Theorem 1.1.29. Let R be an ADL. Then the following conditions are equivalent.

(1) R is a distributive lattice

17 (2) (R, ≤) is a directed above poset

(3) ∧ is commutative

(4) ∨ is commutative

(5) ∨ is right distributive over ∧

(6) θ = {(a, b) ∈ R : b ∧ a = a} is antisymmetric.

Definition 1.1.30. A nonempty subset I of an ADL R is said to be an ideal of R if it satisfies the following conditions:

(1) a, b ∈ I ⇒ a ∨ b ∈ I

(2) a ∈ I, x ∈ R ⇒ a ∧ x, x ∧ a ∈ I.

Definition 1.1.31. A nonempty subset F of an ADL R is said to be a filter of R if it satisfies the following conditions:

(1) a, b ∈ F ⇒ a ∧ b ∈ F

(2) a ∈ F , x ∈ R ⇒ x ∨ a ∈ F.

An ideal I of R is again an ADL with the induced operations. For any nonempty Wn subset S of R,(S] = {( i=1 si) ∧ a|si ∈ S, a ∈ R, n is a positive integer} is the smallest ideal of R containing S. If S = {x} for any x ∈ R we have (x] = {x∧a|a ∈ R}. The ideal (x] is called the principal ideal of R generated by x.

Lemma 1.1.32. Let R be an ADL and I be an ideal of R. Then for any a, b ∈ R, the following conditions hold:

(1) a ∈ (b] if and only if b ∧ a = a

(2) a ∧ b ∈ I if and only if b ∧ a ∈ I

(3) (a ∧ b] = (b ∧ a].

18 Corollary 1.1.33. Let R be an ADL and x, y ∈ R. Then the following conditions hold:

(1) (x] ∨ (y] = (x ∨ y]

(2) (x] ∧ (y] = (x ∧ y].

1.2 Some generalizations of Heyting algebras

In this section we state results that we already got in different reference ma- terials to use them in the sequel. These results are about the concepts of Heyting algebras, Semi-Heyting algebras, L-algebras, Heyting almost distributive lattices, Semi-Heyting Almost Distributive Lattices and L-Almost Distributive Lattices. On a distributive lattice L, x → y denotes the largest z ∈ L (if it exists) such that x ∧ z ≤ y. L is called a Heyting algebra if x → y exists for all x, y ∈ L. The pseudo-complement of x (when it exists) is denoted by x∗ and defined as x∗ = x → 0. A distributive lattice L is called pseudo-complemented if x∗ exists for all x ∈ L.

Definition 1.2.1. An algebra (L, ∨, ∧, →, 0, 1) of type (2, 2, 2, 0, 0) is called a Heyt- ing algebra if it satisfies the following conditions:

(1)( L, ∨, ∧, 0, 1) is a bounded distributive lattice

(2) x → x = 1

(3) y ≤ x → y

(4) x ∧ (x → y) = x ∧ y

(5) x → (y ∧ z) = (x → y) ∧ (x → z)

(6)( x ∨ y) → z = (x → z) ∧ (y → z) for all x, y, z ∈ L.

19 Equivalently one can use the following definition of a Heyting algebra.

Definition 1.2.2. An algebra (L, ∨, ∧, →, 0, 1) of type (2, 2, 2, 0, 0) is called a Heyt- ing algebra if it satisfies the following axioms:

(H1)( L, ∨, ∧, 0, 1) is a lattice with 0 and 1

(H2) x ∧ (x → y) = x ∧ y

(H3) x ∧ (y → z) = x ∧ ((x ∧ y) → (x ∧ z))

(H4)( x ∧ y) → x = 1 for all x, y, z ∈ L.

Heyting algebras are pseudo-complemented(with x∗ = x → 0 as the pseudo- complement of x) in the sense of the following definition.

Definition 1.2.3. An algebra (L, ∨, ∧, ∗, 0, 1) is a pseudo-complemented lattice, where ∗ is unary, if the following conditions hold:

(1)( L, ∨, ∧, 0, 1) is a lattice with 0, 1

(2) x ∧ (x ∧ y)∗ = x ∧ y∗ for all x, y ∈ L

(3)0 ∗ = 1 and 1∗ = 0.

Theorem 1.2.4. Let L be a Heyting algebra. Then for any a, b, c ∈ L the following conditions hold:

(1) a ∧ c ≤ b ⇔ c ≤ a → b

(2) a ≤ b ⇔ a → b = 1.

Lemma 1.2.5. Let L be a Heyting algebra. Then an equivalence relation θ on L is a congruence relation if and only if for any (a, b) ∈ θ and d ∈ L the following conditions hold:

20 (1) (a ∧ d, b ∧ d) ∈ θ

(2) (a ∨ d, b ∨ d) ∈ θ

(3) (a → d, b → d) ∈ θ

(4) (d → a, d → b) ∈ θ.

Definition 1.2.6. An algebra (L, ∨, ∧, →, 0, 1) is called a semi-Heyting algebra if the following conditions hold:

(SH1)( L, ∨, ∧, 0, 1) is a lattice with 0 and 1

(SH2) x ∧ (x → y) = x ∧ y

(SH3) x ∧ (y → z) = x ∧ ((x ∧ y) → (x ∧ z))

(SH4) x → x = 1.

for any x, y, z ∈ L.

Theorem 1.2.7. Let L be a semi-Heyting algebra. Then L satisfies the following conditions:

(a) x ∧ (y → z) = x ∧ ((x ∧ y) → z)

(b) x ∧ (y → z) = x ∧ (y → (x ∧ z))

for any x, y, z ∈ L.

Definition 1.2.8. A pseudo-complemented lattice L satisfying the identity x∗ ∨ x∗∗ = 1 for all x ∈ L is called a Stone lattice.

Definition 1.2.9. A Heyting algebra L that satisfies the identity (x → y) ∨ (y → x) = 1 for all x, y ∈ L is called a L-algebra.

Lemma 1.2.10. Let L be an L-algebra and x, y, z ∈ L. Then x → (y ∨ z) = (x → y) ∨ (x → z).

21 For any lattice L, we denote the set {z : x ≤ z ≤ y} by [x, y] and call it interval. A distributive lattice L is an L-algebra if and only if every interval in L is a Stone lattice(a relatively pseudocomplmented lattice satisfying x∗ ∨x∗∗ = 1 for all x ∈ L).

Definition 1.2.11. Let (H, ∨, ∧, 0, m) be an ADL with 0 and a maximal element m. Suppose → be a binary operation on H satisfying the following conditions:

(1) x → x = m

(2)( x → y) ∧ y = y

(3) x ∧ (x → y) = x ∧ y ∧ m

(4) x → (y ∧ z) = (x → y) ∧ (x → z)

(5)( x ∨ y) → z = (x → z) ∧ (y → z) for all x, y, z ∈ H. Then (H, ∨, ∧, →, 0, m) is called a Heyting Almost Distributive Lattice(HADL).

Remark 1.2.1: Let H be a HADL and → be a binary operation on H. Then y ≤ x → y implies that (x → y) ∧ y = y, but the converse is not always true. The converse becomes true whenever H is a lattice, and therefore an HADL becomes a Heyting algebra.

Theorem 1.2.12. Let (H, ∨, ∧, →, 0, m) be a HADL. Then the following are equiv- alent:

(1) H is a Heyting algebra

(2) For any a, b, c ∈ H, a ∧ c ≤ b ⇔ c ≤ a → b

(3) b ≤ a → b for all a, b ∈ H.

Theorem 1.2.13. Let H be an ADL with 0 and a maximal element m, then the following are equivalent:

22 (1) H is a HADL

(2) [0, a] is a Heyting algebra for all a ∈ H

(3) [0, m] is a Heyting algebra.

Lemma 1.2.14. Let H be a HADL and a, b ∈ H.Then the following are equivalent:

(1) (a] ⊆ (b]

(2) b ∧ a = a

(3) a ∧ x ≤ b ∧ x for all x ∈ H.

Definition 1.2.15. Let (L, ∨, ∧, 0, m) be an ADL with a maximal element m. Suppose there exists a binary operation → on L satisfying the following conditions:

(1)( x → x) ∧ m = m

(2) x ∧ (x → y) = x ∧ y ∧ m

(3) x ∧ (y → z) = x ∧ ((x ∧ y) → (x ∧ z))

(4)( x → y) ∧ m = (x ∧ m) → (y ∧ m) for all x, y, z ∈ L. Then (L, ∨, ∧, →, 0, m) is called a semi-Heyting almost distribu- tive lattice (SHADL).

Theorem 1.2.16. Let L be an ADL with a maximal element m. Then the following are equivalent:

(1) L is a SHADL

(2) [0, a] is a semi-Heyting algebra for all a ∈ L

(3) [0, m] is a semi-Heyting algebra.

23 Definition 1.2.17. A HADL (L, ∨, ∧, →, 0, m) is called an L-ADL if ((x → y) ∨ (y → x)) ∧ m = m.

Theorem 1.2.18. Let (L, ∨, ∧, →, 0, m) be a L-ADL. Then the following are equiv- alent:

(1) L is L-algebra

(2) For any a, b, c ∈ L, a ∧ c ≤ b ⇔ c ≤ a → b

(3) b ≤ a → b for all a, b ∈ L.

Theorem 1.2.19. Let L be an ADL with 0 and a maximal element m, then the following are equivalent:

(1) L is an L-ADL

(2) [0, a] is an L-algebra for all a ∈ L

(3) [0, m] is an L-algebra.

1.3 Skew Lattices, Skew Heyting algebras and Dualities

In this section we collect certain important elementary properties of skew Heyting algebra and the dual of a given algebra. To begin with we have the following definition.

Definition 1.3.1. A skew lattice is an algebra (L, ∧, ∨) of type (2, 2) such that ∧ and ∨ are both associative and satisfy the following absorption laws: x ∧ (x ∨ y) = x = x ∨ (x ∧ y) and (x ∧ y) ∨ y = y = (x ∨ y) ∧ y for all x, y ∈ L.

The natural partial order can be defined on a skew lattice L by stating that x ≤ y if and only if x ∨ y = y = y ∨ x, or equivalently x ∧ y = x = y ∧ x for x, y ∈ L.

24 Also the natural preorder can be defined by x  y if and only if y ∨ x ∨ y = y, or equivalently x ∧ y ∧ x = x for x, y ∈ L so that Green’s equivalence relation D is defined by xDy if and only if x  y and y  x.

Definition 1.3.2. A skew lattice is called strongly distributive if for all x, y, z ∈ L it satisfies the following identities: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z); and it is called co-strongly distributive if it satisfies the identities: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z).

Definition 1.3.3. A skew Lattice L is called normal if w ∧ x ∧ y ∧ z = w ∧ y ∧ x ∧ z and it is called co-normal if w ∨ x ∨ y ∨ z = w ∨ y ∨ x ∨ z for all w, x, y, z ∈ L.

Note: A skew lattice which is both normal and co-normal is called bi-normal skew lattice. If a skew lattice is strongly distributive, then it is normal. Dually if a skew lattice is co-strongly distributive, then it is co-normal [18].

Definition 1.3.4. A skew lattice S is called rectangular skew lattice if it satisfies the identity x ∨ y = y ∧ x for all x, y ∈ S.

Definition 1.3.5. Let (S, ∨, ∧) be a skew lattice and let x, y ∈ S. Then x and y are said to be equivalent and denoted by x ≡ y if both x ∧ y ∧ x = x and y ∧ x ∧ y = y.

Theorem 1.3.6. Let (S, ∨, ∧) be a skew lattice. Then S is bi regular. That is, for all x, y, and z in S: x ∧ y ∧ x ∧ z ∧ x = x ∧ y ∧ z ∧ x and x ∨ y ∨ x ∨ z ∨ x = x ∨ y ∨ z ∨ x.

Definition 1.3.7. An algebra L = (L, ∨, ∧, →, 1) of type (2, 2, 2, 0) is said to be a skew Heyting algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 1) is a co-strongly distributive skew lattice with top 1

(2) For any a ∈ L, an operation →a can be defined on a↑= {a ∨ x ∨ a|x ∈ L}

such that (a↑, ∨, ∧, →a, 1, a) is a Heyting algebra with top 1 and bottom a

25 (3) An induced binary operation → is defined on L by

x → y = (y ∨ x ∨ y) →y y.

Definition 1.3.8. Let P be a nonempty set and ≤, ≤0 be two partial orders on P . Then we say that ≤ and ≤0 are dual to each other whenever for any a, b ∈ P , a ≤ b if and only if b ≤0 a.

Note that the dual of any partial order ≤ on a nonempty set P is again a partial order on P and is unique. The poset (P, ≤0) is the dual of the poset (P, ≤). Since the dual Hd of a distributive lattice (H, ∨, ∧) is again a distributive lattice, we give the following definition.

Definition 1.3.9. A distributive lattice (H, ∨, ∧) is called a dual Heyting algebra if its dual Hd of H is a Heyting algebra.

Theorem 1.3.10. A distributive lattice (H, ∨, ∧, 0, 1) is a dual Heyting algebra if and only if there exists a binary operation ← satisfying the following:

(1) x ← x = 0

(2) x ∨ (x ← y) = x ∨ y

(3) (x ← y) ∨ y = y

(4) x ← (y ∨ z) = (x ← y) ∨ (x ← z)

(5) (x ∧ y) ← z = (x ← z) ∨ (y ← z) for all x, y, z ∈ H.

In general, the dual of an ADL is not an ADL. For this reason Rao, G. C. and Naveen Kumar Kakumanu introduced the concept of dual Heyting almost distribu- tive lattice as follows.

Definition 1.3.11. An ADL (H, ∨, ∧, 0) with a maximal element m is called a dual Heyting almost distributive lattice (dual HADL), if to each a ∈ H the distributive

26 lattice ([0, a], ∨, ∧, 0, a) is a dual Heyting algebra with respect to the binary opera- tion denoted by ←a.

Theorem 1.3.12. Let H be an ADL with a maximal element m. Then H is a dual HADL if and only if to each a ∈ H, there exists a binary operation ←a on H satisfying the following conditions:

(1) x ←a x = 0

(2) (x ∨ (x ←a y)) ∧ a = (x ∨ y) ∧ a

(3) ((x ←a y) ∨ y) ∧ a = y ∧ a

(4) z ←a (x ∨ y) = (z ←a x) ∨ (z ←a y)

(5) (x ∧ y) ←a z = (x ←a z) ∨ (y ←a z) for all x, y, z ∈ H.

Theorem 1.3.13. Let H be an ADL with a maximal element m. Then the following are equivalent:

(1) H is a dual HADL

(2) [0, m] is a dual Heyting algebra

(3) There exists a binary operation ← on H satisfying the following conditions:

(i) x ← x = 0

(ii) ((x ← y) ∨ y) ∧ m = y ∧ m

(iii) (x ∨ (x ← y)) ∧ m = (x ∨ y) ∧ m

(iv) z ← (x ∨ y) = (z ← x) ∨ (z ← y)

(v) (x ∧ y) ← z = (x ← z) ∨ (y ← z) for all x, y, z ∈ H.

If (H, ∨, ∧, ←, 0, m) is a dual HADL, then for x, y, z ∈ H, it is observed that (x ∨ z) ∧ m ≥ y ∧ m if and only if z ∧ m ≥ (x ← y) ∧ m.

27 Definition 1.3.14. A dual HADL (H, ∨, ∧, ←, 0, m) is said to be a dual L-almost distributive lattice (dual L-ADL) if for any x, y ∈ H,(x ← y) ∧ (y ← x) = 0.

Theorem 1.3.15. Let L be an ADL with a maximal element m. Then the following are equivalent:

(1) L is a dual L-ADL

(2) [0, a] is a dual L-algebra for all a ∈ L

(3) [0, m] is a dual L-algebra.

28 Chapter 2

Skew Heyting Almost Distributive Lattices

Distributive lattices are well known to be precisely those lattices that possess can- cellation: x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the 5-element lattices M3 or N5 as sublattices. In the first section of this chapter we examine cancellation in skew ADLs, where the involved objects are in many ways lattice-like, but the operations ∨ and ∧ no longer need be commutative. In particular, we find sufficient conditions involving the non occurrence of potential sub-objects similar to M3 or N5 that insure that a skew ADL is left cancellative (satisfying the above implication) or just cancellative. Also for a skew ADL L we define a congruence relation θ such that L/θ is a maximal lattice image of L. The second section deals on the concept of skew HADLs. We characterize a skew HADL L in terms of the set of all its principal ideals and a congruence relation θ defined on it such that the congruence classes are the maximal rectangular subal- gebra of L and L/θ is the maximal lattice image of L. We present important results of skew HADL with their complete proofs, and some basic algebraic properties of this algebra are discussed.

29 2.1 Skew Almost Distributive Lattices

In this section we introduce different properties of skew almost distributive lat- tices.

Definition 2.1.1. An algebra (S, ∨, ∧) where S is a nonempty subset of an ADL L is called a Skew Almost Distributive Lattice (skew ADL) whenever x ∨ (y ∨ z) = (x ∨ y) ∨ z for all x, y, z ∈ S.

Note: A relatively complemented ADL is a skew ADL. From the property of ADL a skew ADL is strongly distributive skew lattice. We call a skew ADL S is co-strongly distributive skew ADL if (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z) for all x, y, z ∈ S and using Theorem 1.1.24 one can simply observe that a co-strongly distributive skew ADL with zero is a distributive lattice. Skew ADLs are skew lattices and hence we can apply all concepts of skew lattices to skew ADLs, for if a skew ADL is biregular.

Definition 2.1.2. A skew ADL S is said to be rectangular skew ADL whenever x ∧ y = y ∨ x for all non zero elements x, y ∈ S.

Lemma 2.1.3. Let S be a skew ADL. Then S is normal.

Proof. Suppose S be a skew ADL. Let w, x, y, z ∈ S. Then from Theorem 1.1.23 (5) we obtain that w ∧ x ∧ y ∧ z = w ∧ y ∧ x ∧ z. This shows that S is normal.

Corollary 2.1.4. A rectangular skew ADL is co-normal.

30 Proof. Let S be a rectangular skew ADL and w, x, y, z ∈ S. Then

x ∨ y ∨ z ∨ w = ((x ∨ y) ∨ z) ∨ w

= w ∧ ((x ∨ y) ∨ z) ..... d since S is rectangularc

= w ∧ (z ∧ (x ∨ y))

= w ∧ z ∧ y ∧ x

= w ∧ y ∧ z ∧ x ..... d since S is normal c

= ((w ∧ y) ∧ z) ∧ x

= x ∨ ((w ∧ y) ∧ z) ..... d since S is rectangularc

= x ∨ (z ∨ (w ∧ y))

= x ∨ z ∨ y ∨ w.

Hence S is co-normal.

Lemma 2.1.5. A rectangular skew ADL with 0 is a discrete ADL.

Proof. Let S be a rectangular skew ADL and x, y ∈ S. Then by definition we obtain that x ∧ y = y ∨ x for all x, y ∈ S such that x 6= 0 and y 6= 0. Since y ∧ x = x ∧ y ∧ x = (y ∨ x) ∧ x = x, using the absorption law it follows that y ∨ x = y ∨ (y ∧ x) = y. Which shows that S is a discrete ADL.

Theorem 2.1.6. Let S be a skew ADL with 0. Then S is a rectangular skew ADL if and only if x ∨ y = x for all x, y ∈ S.

Proof. Suppose S be a rectangular skew ADL. It follows that y ∨ x = x ∧ y for all x, y ∈ S such that x 6= 0 and y 6= 0. From Lemma 2.1.5, S is discrete ADL hence x ∨ y = x for all non zero x, y ∈ S. Conversely suppose x ∨ y = x for all x, y ∈ S. Then it follows that x ∨ y ∨ x = x for all x, y ∈ S. Hence x ∨ y = ((y ∧ x) ∨ x) ∨ (y ∨ (y ∧ x)) = (y ∧ x) ∨ (x ∨ y) ∨ (y ∧ x) = y ∧ x. Therefore S is rectangular skew ADL.

31 Theorem 2.1.7. Let S be a skew ADL. A relation θ defined on S by

θ = {(x, y) ∈ S × S|x ∧ y = y and y ∧ x = x} is a congruence relation.

Proof. Let x, y, z ∈ S. Then

(1) x ∧ x = x. Hence θ is reflexive.

(2) Let xθy. Then x ∧ y = y and y ∧ x = x. Hence yθx so that θ is symmetric.

(3) Let xθy and yθz. Then we have x ∧ y = y, y ∧ x = x, y ∧ z = z and z ∧ y = y. Then

x ∧ z = x ∧ (y ∧ z)

= (x ∧ y) ∧ z

= y ∧ z

= z

and

z ∧ x = z ∧ (y ∧ x)

= (z ∧ y) ∧ x

= y ∧ x

= x.

Hence xθz. Transitivity of θ holds and thus θ is an equivalence relation.

(4) Assume that xθy and wθz. Then

(x ∨ w) ∧ (y ∨ z) = ((x ∨ w) ∧ y) ∨ ((x ∨ w) ∧ z)

= ((x ∧ y) ∨ (w ∧ y)) ∨ ((x ∧ z) ∨ (w ∧ z))

= (y ∨ (w ∧ y)) ∨ ((x ∧ z) ∨ z)

= ((y ∨ w) ∧ y) ∨ z

= y ∨ z.

32 Similarly (y ∨ z) ∧ (x ∨ w) = x ∨ w. Hence we have (x ∨ w)θ(y ∨ z). Also

(x ∧ w) ∧ (y ∧ z) = x ∧ y ∧ w ∧ z

= y ∧ w ∧ z

= y ∧ z

and

(y ∧ z) ∧ (x ∧ w) = z ∧ y ∧ x ∧ w

= z ∧ x ∧ w

= x ∧ z ∧ w

= x ∧ w.

Hence we have (x ∧ w)θ(y ∧ z).

Therefore θ is a congruence relation on S.

Lemma 2.1.8. Let S be a skew ADL and x, y ∈ S. For θ given by Theorem 2.1.7, xθy if and only if x ∨ y = y ∧ x.

Proof. Let xθy. Then

x ∨ y = (y ∨ x ∨ y) ∧ (x ∨ y)

= (y ∨ (y ∨ x)) ∧ (x ∨ y)

= (y ∨ x) ∧ (x ∨ y)

= (y ∨ (y ∧ x)) ∧ (x ∨ y)

= y ∧ (x ∨ y).

On the other hand

y ∧ x = y ∧ x ∧ (x ∨ y)

= y ∧ x ∧ (y ∧ (x ∨ y))

= y ∧ (x ∨ y)

= x ∨ y.

33 Thus x∨y = y∧x. Conversely if x∨y = y∧x, then y∧x = x∧y∧x = x∧(x∨y) = x and x ∧ y = y ∧ x ∧ y = y ∧ (y ∨ x) = y which implies that xθy.

Theorem 2.1.9. Let S be a rectangular skew ADL and x, y, z ∈ S. Then the following conditions hold:

(1) x ∧ z = y ∧ z

(2) (x ∨ z) ∧ y = y and y ∨ (x ∧ z) = y

(3) S is a lattice if and only if it is a singleton.

Proof. Suppose S be a rectangular skew ADL and x, y, z ∈ S. Then

(1) Using regularity we get

x ∧ z = ((y ∨ x) ∧ x) ∧ ((y ∨ z) ∧ z)

= (x ∧ y ∧ x) ∧ (z ∧ y ∧ z)

= x ∧ y ∧ x ∧ z ∧ y ∧ z

= x ∧ y ∧ x ∧ y ∧ z ∧ y ∧ z

= x ∧ (y ∧ x ∧ y) ∧ (z ∧ y ∧ z)

= x ∧ y ∧ z.

By symmetry we also have y ∧ z = y ∧ x ∧ z. Since x ∧ y ∧ z = y ∧ x ∧ z we conclude that x ∧ z = y ∧ z.

(2) Using Theorem 2.1.6 we obtained that x ∨ y ∨ z = (x ∨ y) ∨ z = x ∨ z and hence

(x ∨ z) ∧ y = (x ∨ y ∨ z) ∧ y

= (x ∨ z ∨ y) ∧ y

= y.

34 Since x ∧ z = z ∨ x, we have

y ∨ (x ∧ z) = (x ∧ z) ∧ y

= (z ∨ x) ∧ y

= (x ∨ z) ∧ y

= y

(3) Suppose S be a lattice. Then by Theorem 2.1.6 we obtain that x = x ∨ y = y ∨ x = y for all x, y ∈ S. Therefore S is a singleton. The converse is direct.

Note: A skew ADL is said to be left cancellative if it satisfies the implication x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. It is called right cancellative if it satisfies x ∨ z = y ∨ z and x ∧ z = y ∧ z imply x = y. If a skew ADL is both right and left cancellative then it is said to be cancellative.

Theorem 2.1.10. Let S be a skew ADL. Then S is right cancellative skew ADL.

Proof. Suppose S be a skew ADL and x, y, z ∈ S such that x ∧ z = y ∧ z and x ∨ z = y ∨ z. Then

x = x ∨ (x ∧ z)

= x ∨ (y ∧ z)

= (x ∨ y) ∧ (x ∨ z)

= (x ∨ y) ∧ (y ∨ z)

= (y ∨ x) ∧ (y ∨ z)

= y ∨ (x ∧ z)

= y ∨ (y ∧ z)

= y.

Hence S is right cancellative skew ADL.

35 Corollary 2.1.11. Let S be a rectangular skew ADL. Then S is cancellative.

Proof. Suppose S be a rectangular skew ADL such that z∨x = z∨y and z∧x = z∧y for all x, y, z ∈ S. Then, we have

x = (z ∨ x) ∧ x

= (x ∧ z) ∧ x

= x ∧ (z ∧ x)

= x ∧ (z ∧ y)

= (x ∧ z) ∧ y

= (z ∨ x) ∧ y

= (z ∨ y) ∧ y

= y.

This shows that S is left cancellative and hence it is cancellative.

Remark 2.1.1. A rectangular skew ADL S is a distributive lattice and hence it doesn’t contain potential subalgebras similar to M3 (diamond) or N5 (pentagon).

Lemma 2.1.12. Let S be a skew ADL and θ be a congruence relation on S given by Theorem 2.1.7. Then the congruence class [x]θ of x is rectangular skew ADL.

Proof. Suppose S be a skew ADL and x, y, z ∈ S such that y, z ∈ [x]θ. Since yθz we obtain that z ∧ y = y = y ∨ (y ∧ z) = y ∨ z and hence [x]θ is a rectangular skew ADL.

Next, we show that the quotient algebra S/θ is a maximal lattice image of a skew ADL S, where θ is the congruence relation on S. The elements of S/θ are congruence classes.

Theorem 2.1.13. Let S be a skew ADL and θ be a relation given by Theorem 2.1.7. Then S/θ is a maximal lattice image of S.

36 Proof. Suppose S be a skew ADL. From Theorem 2.1.7, θ is a congruence relation on S. Consider S/θ and for x, y ∈ S let [x]θ, [y]θ ∈ S/θ. Define ∧ and ∨ on S/θ by [x]θ ∧ [y]θ = [x ∧ y]θ and [x]θ ∨ [y]θ = [x ∨ y]θ. Then for any w ∈ S,

w ∈ [x ∨ y]θ ⇔ (x ∨ y) ∧ w = w and w ∧ (x ∨ y) = x ∨ y

⇔ (y ∨ x) ∧ w = w and w ∧ (y ∨ x) = y ∨ x

⇔ w ∈ [y ∨ x]θ

Hence [x ∨ y]θ = [y ∨ x]θ. Thus [x]θ ∨ [y]θ = [y]θ ∨ [x]θ. Similarly

w ∈ [x ∧ y]θ ⇔ x ∧ y ∧ w = w and w ∧ (x ∧ y) = x ∧ y

⇔ y ∧ x ∧ w = w and w ∧ (y ∧ x) = y ∧ x

⇔ w ∈ [y ∧ x]θ.

Hence [x ∧ y]θ = [y ∧ x]θ. Thus [x]θ ∧ [y]θ = [y]θ ∧ [x]θ. Which shows that ∧ and ∨ are commutative and therefore S/θ is a lattice. Now consider a congruence relation β on S such that S/β is a lattice. Suppose xθy. Clearly

x ∧ y = y and y ∧ x = x ⇒ [x ∧ y]β = [y]β and [y ∧ x]β = [x]β

⇒ [x]β ∧ [y]β = [y]β and [y]β ∧ [x]β = [x]β

⇒ [x]β = [y]β, since S/β is a lattice

⇒ xβy.

Therefore θ ⊆ β. Suppose H be a lattice image of S. Then there exist an epimorphism f : S −→ H. Define a relation α on S by

α = {(x, y) ∈ S2|f(x) = f(y)}.

Clearly α is reflexive, symmetric and transitive. Let (x1, y1), (x2, y2) ∈ α. Since f is homomorphism we have

f(x1 ∨ x2) = f(x1) ∨ f(x2)

= f(y1) ∨ f(y2)

= f(y1 ∨ y2),

37 which shows that (x1 ∨ x2)α(y1 ∨ y2) similarly (x1 ∧ x2)α(y1 ∧ y2). Hence α is a congruence relation on S. Since a homomorphic image of S is given by S/φ where φ is a congruence relation on S, S/α is the lattice image of S. Consider the set L of lattices given by L = {S/φ|φ is a congruence on S and θ ⊆ φ }. Now take congruence relations θ and β discussed above and assume that S/θ ⊆ S/β. Let a, b ∈ S such that aβb. Since θ ⊆ β and S/θ ⊆ S/β there exist a0, b0 ∈ S such that a ∈ [a]θ = [a0]β and b ∈ [b]θ = [b0]β. Thus aθa0, aβa0 and bθb0, bβb0. Hence aβb, bβb0 ⇒ aβb0 ⇒ a ∈ [b0]β = [b]θ ⇒ aθb. Therefore β ⊆ θ and hence θ = β. Consequently, S/θ = S/β. Therefore S/θ is the maximal element of L so that it is a maximal lattice image of S.

2.2 Skew Heyting Almost Distributive Lattices

Through out this section and the next two chapters for a non empty set L we use the following notations:

(1) La = {x ∧ a|x ∈ L}, for any a ∈ L

(2) For any a ∈ L, →a is the binary operation defined on La

(3) For any b, c ∈ L and b 6= 0, b→ is the binary operation defined on [b, c].

Consider an ADL L with a maximal element m and 0 ∈/ L. Take a ∈ L such that y, z ∈ La. Which implies that y = w ∧ a and z = x ∧ a for some w, x ∈ L. Since

(La, ∨, ∧) is an ADL, we have: y ∧ z = (w ∧ a) ∧ (x ∧ a) = w ∧ x ∧ a = x ∧ w ∧ a =

(x ∧ a) ∧ (w ∧ a) = z ∧ y. This shows that La is a distributive lattice with top a. Using this fact and the concept of skew Heyting algebra, in this section, we introduce a new class of algebra called skew Heyting almost distributive lattices and characterize it in terms of a congruence relation and the set of all of its principal ideals. Considering the congruence relation θ defined on a skew HADL L we show that each congruence class is a maximal rectangular subalgebra of L and L/θ is a

38 maximal lattice image of L. Further we investigate different conditions on which an ADL with a maximal element m and 0 ∈/ L is a skew HADL and several algebraic properties of skew HADLs are studied.

Definition 2.2.1. An ADL L with a maximal element m is said to be a Skew Heyting Almost Distributive Lattice (skew HADL) whenever to each a ∈ L the algebra (La, ∨, ∧, →a, a) is a skew Heyting algebra.

Example 2.2.2. Let L be an ADL with a maximal element m. Let b ∈ L. For any a ∈ L such that a ≤ b define a binary operation a→ on [a, b] by  b if x ≤ y x a→ y = y otherwise.

Let a binary operation →b on Lb induced from a→ be defined by x →b y = (x∨y) y→ y. Hence (Lb, ∨, ∧, →b, b) is a skew Heyting algebra and therefore L is a skew HADL.

Proof. Suppose L be an ADL with a maximal element m. We claim that for any b ∈ L, Lb is a skew Heyting algebra. Clearly Lb is co-strongly distributive skew lattice. Now take a ∈ Lb and consider [a, b] such that x, y, z ∈ [a, b]. Then

(1) x a→ x = b.

(2) Since b is the top element in [a, b] we have  b ∧ y if x ≤ y (x a→ y ) ∧ y = y ∧ y otherwise = y.

39 (3) Using the fact that x ≤ y implies x = x ∧ y we obtain that  x ∧ b if x ≤ y x ∧ (x a→ y ) = x ∧ y otherwise = x ∧ y ∧ b

= x ∧ y.

(4) Since [a, b] is a lattice, y ∧ z ≤ y, y ∧ z ≤ z, x ≤ x ∨ y and y ≤ x ∨ y. Thus, we obtain that  b if x ≤ y ∧ z x a→(y ∧ z) = y ∧ z otherwise  b if x ≤ y and x ≤ z = y ∧ z otherwise   b if x ≤ y b if x ≤ z = ∧ y otherwise z otherwise = (x → y) ∧ (x → z).

(5) Since [a, b] is a lattice and x, y, z ∈ [a, b] we have

x ∨ y ≤ z ⇒ x ∨ y = (x ∨ y) ∧ z

⇒ x ∧ (x ∨ y) = x ∧ ((x ∨ y) ∧ z)

⇒ x = (x ∧ (x ∨ y)) ∧ z

⇒ x = x ∧ z

⇒ x ≤ z.

40 Similarly x ∨ y ≤ z ⇒ y ≤ z and it follows that  b if x ∨ y ≤ z (x ∨ y) a→ z = z otherwise  b if x ≤ z and y ≤ z = z otherwise   b if x ≤ z b if y ≤ z = ∧ z otherwise z otherwise = (x → z) ∧ (y → z).

Hence ([a, b], ∨, ∧, a→, a, b) is a Heyting algebra. Consequently, it is possible to define a binary operation →b on Lb by x →b y = (x∨y) y→ y. As x, y ∈ [a, b], we have x ∨ y = y ∨ x ∨ y and hence x →b y = (y ∨ x ∨ y) y→ y so that (Lb, ∨, ∧, →b, b) is a skew Heyting algebra. Therefore L is a skew HADL.

Theorem 2.2.3. Let L be an ADL with a maximal element m. Then the following conditions are equivalent.

(1) L is a skew HADL

(2) Lm is a skew Heyting algebra

(3) (i) for any b ∈ L, ([b, m], ∨, ∧, b→, b, m) is a HADL

(ii) there exists a binary operation → on L defined by x → y = ((x ∨ y) ∧ m)

(y∧m)→ (y ∧ m).

Proof. Let L be a skew HADL. Then (2) holds directly from the definition. Assume that (2) holds. Since L has maximal element(s), for each b ∈ L there exist a maximal element m such that b ≤ m and hence b ∈ Lm. As Lm is a skew Heyting algebra b↑ = [b, m] is a Heyting algebra and therefore an HADL. Since Lm is a skew

Heyting algebra, the induced operation →m on Lm from b→ on [b, m] is given by

41 x →m y = (y ∨ x ∨ y) y→ y. Thus it is possible to define a binary operation → on

L by x → y = (x ∧ m) →m (y ∧ m). But

(x ∧ m) →m (y ∧ m) = ((y ∧ m) ∨ (x ∧ m) ∨ (y ∧ m)) (y∧m)→ (y ∧ m)

= ((y ∨ x ∨ y) ∧ m) (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m),

and hence x → y = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m).

Conversely, suppose condition (3) hold and let a ∈ L. Then La is a co- strongly distributive skew lattice. By (i) for any b ∈ La, [b, m] is a HADL. Since L has maximal element for some maximal element m in L we have a ≤ m so that [b, a] ⊆ [b, m]. Theorem 1.2.13 asserts that ([b, a], ∨, ∧, b→, b, a) is a Heyting algebra. The maximal element in La is a, thus by (2) it is possible to define →a on

La by x →a y = ((x ∨ y) ∧ a) (y∧a)→ (y ∧ a). But

((x ∨ y) ∧ a) (y∧a)→ (y ∧ a) = ((y ∨ x ∨ y) ∧ a) (y∧a)→ (y ∧ a)

= (y ∨ x ∨ y) y→ y,

and hence x →a y = (y ∨ x ∨ y) y→ y. Therefore (La, ∨, ∧, →a, a) is a skew Heyting algebra so that L is a skew HADL.

Consequently, by a skew HADL we mean an algebra (L, ∨, ∧, →, m) of type (2, 2, 2, 0) satisfying condition (3) of Theorem 2.2.3.

Corollary 2.2.4. Let L be a skew HADL. Then for any a ∈ L, [a, m] is a Heyting algebra.

Proof. Clear by Theorem 2.2.3(1).

The following lemma is analogous with the statement, any interval on a Heyting algebra is again a Heyting algebra.

Lemma 2.2.5. Let L be a skew HADL. Then for any b ∈ L, [b, m] is a skew HADL.

42 Proof. From Corollary 2.2.4 for any b ∈ L we have [b, m] is a Heyting algebra. Following this for any c ∈ [b, m], [b, c] is a Heyting algebra so that it is a skew Heyting algebra with top element c. Therefore [b, m] is a skew HADL.

Corollary 2.2.6. Let L be a skew HADL. If x, y ∈ L such that x ≤ y and a, b ∈

[y, m], then a x→ b = a y→ b.

Proof. Let x, y ∈ L such that x ≤ y. Then [y, m] ⊆ [x, m]. If a, b ∈ [y, m], then a y→ b ∈ [y, m] and hence a y→ b ∈ [x, m]. By Corollary 2.2.4, [x, m] and [y, m] are

Heyting algebras. Since a, b ∈ [x, m], a x→ b also belongs to [x, m]. The maximal element characterization of a x→ b and a y→ b on the Heyting algebra [x, m] forces the two elements are equal.

Lemma 2.2.7. Let L be a skew HADL. Then the following conditions hold:

(1) x → y = (x (y∧m)→ y) ∧ m

(2) x ∧ (x → y) = x ∧ y ∧ m

(3) (x ∨ y) → z = (x → z) ∧ (y → z). for all x, y, z ∈ L.

Proof. Let L be a skew HADL and x, y, z ∈ L. Clearly [y ∧ m, m] is a HADL and hence we have

(1) (x (y∧m)→ y) ∧ m = (m (y∧m)→ (x (y∧m)→ y)) ∧ m

= ((x ∧ m) (y∧m)→ y) ∧ m ∧ m

= ((x ∧ m) (y∧m)→ y) ∧ ((x ∧ m) (y∧m)→ m) ∧ m

= ((x ∧ m) (y∧m)→ (y ∧ m)) ∧ ((y ∧ m) (y∧m)→ (y ∧ m))

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = x → y.

43 (2) x ∧ (x → y) = x ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= m ∧ x ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= x ∧ m ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= x ∧ (x ∨ y) ∧ m ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= x ∧ {((x ∨ y) ∧ m) ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}} = x ∧ {((x ∨ y) ∧ m) ∧ (y ∧ m)}

= x ∧ (y ∧ m)

= x ∧ y ∧ m.

(3) (x ∨ y) → z = {((x ∨ y) ∨ z) ∧ m} (z∧m)→ (z ∧ m)

= {((x ∨ y) ∨ z ∨ z) ∧ m} (z∧m)→ (z ∧ m)

= {((z ∨ x) ∨ (y ∨ z)) ∧ m} (z∧m)→ (z ∧ m)

= {((x ∨ z) ∧ m) (z∧m)→ (z ∧ m)}

∧{((y ∨ z) ∧ m) (z∧m)→ (z ∧ m)} = (x → z) ∧ (y → z).

Corollary 2.2.8. Let L be a skew HADL. Then x → y = (x → y) ∧ m

Proof. Let L be a skew HADL and x, y ∈ L. Then

(x → y) ∧ m = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)) ∧ m

= {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)} ∧ ((y ∧ m) (y∧m)→ (y ∧ m))

= {((x ∨ y) ∨ y) ∧ m} (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = x → y.

44 The next theorem characterizes a skew HADL in terms of a congruence relation θ defined on it. To prove the theorem we use the following lemma.

Lemma 2.2.9. Let L be a skew HADL and x, y, z ∈ L such that x ∧ m = y ∧ m. Then the following statements hold:

(1) x → y = m

(2) x → z = y → z and z → x = z → y.

Proof. Let L be a skew HADL and x, y, z ∈ L. Suppose x ∧ m = y ∧ m. Then

(1) x → y = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)

= ((x ∧ m) ∨ (y ∧ m)) (y∧m)→ (y ∧ m)

= (y ∧ m) (y∧m)→ (y ∧ m) = m.

(2) x → z = ((x ∨ z) ∧ m) (z∧m)→ (z ∧ m)

= ((x ∧ m) ∨ (z ∧ m)) (z∧m)→ (z ∧ m)

= ((y ∧ m) ∨ (z ∧ m)) (z∧m)→ (z ∧ m)

= ((y ∨ z) ∧ m) (z∧m)→ (z ∧ m) = y → z

By a similar argument, we obtain z → x = z → y.

Theorem 2.2.10. Let L be an ADL with a maximal element m. If L is a skew HADL and θ defined by

θ = {(x, y) ∈ L × L|x ∧ y = y and y ∧ x = x} is a relation on L, then the following conditions hold:

45 (1) θ is a congruence relation on L

(2) L/θ is the maximal lattice image of L

(3) The congruence classes are the maximal rectangular subalgebras of L.

Proof. Suppose L be a skew HADL. Thus by Theorem 2.1.7, θ is a congruence relation on the skew ADL L. To prove that θ is a congruence relation on a skew HADL L, it suffices to show that (x → a)θ(y → a) and (a → x)θ(a → y) hold. From the property of ADLs given by Theorem 1.1.23 (5) and the given condition: (x, y) ∈ θ implies x ∧ y = y and y ∧ x = x, one can simply observe that x ∧ m = y ∧ x ∧ m = x ∧ y ∧ m = y ∧ m. Indeed, (2) of the above lemma assures that (x → a)θ(y → a) and (a → x)θ(a → y). Hence θ is a congruence relation on the skew HADL L. Using Theorem 2.1.13 one can conclude that L/θ is a maximal lattice image of L. Suppose x, y ∈ [z]θ. Then xθy. Hence y ∨ x = y ∨ (y ∧ x) = y = x ∧ y. Therefore each congruence class is a rectangular subalgebra of L. Let R be the set of all rectangular subalgebras of L. Now take an arbitrary congruence class [x]θ for some x ∈ L and a rectangular subalgebra T of L such that [x]θ ⊆ T and let r ∈ T . Since x ∈ T and T is a rectangular subalgebra of L we have r ∨ x = x ∧ r and x ∨ r = r ∧ x. Thus x ∧ r = (r ∧ x) ∧ r = (x ∨ r) ∧ r = r which implies that r ∨x = x∧r = r and x∨r = x∨(x∧r) = x. Hence r ∈ [x]θ. Therefore T ⊆ [x]θ and we conclude that [x]θ = T . Hence [x]θ is a maximal element of R, i.e. each congruence class is a maximal rectangular subalgebra of L.

Remark 2.2.1. A skew Heyting algebra has a top element and need not to contain bottom element. If a skew Heyting algebra L contains a bottom element, then it is a HADL.

Theorem 2.2.11. Let L be an ADL with a maximal element m. Then L is a skew

HADL if and only if for any a, z ∈ L such that a ∈ Lz and w, x, y ∈ [a, z], the following conditions hold:

46 (1) x ≤ w a→ x

(2) (x a→ y) ∧ w = w if and only if y ∧ w ∧ x = w ∧ x.

Proof. Let L be a skew HADL. Indeed for any z ∈ L, Lz is a skew Heyting algebra.

Let a ∈ Lz and w, x, y ∈ [a, z]. Since [a, z] is a Heyting algebra (1) is clear.

Now, assume that (x a→ y) ∧ w = w. Then w ∧ x = ((x a→ y) ∧ w) ∧ x = x ∧ ((x a→ y) ∧ w) = (x ∧ y) ∧ w = y ∧ w ∧ x.

On the other hand given that y ∧ w ∧ x = w ∧ x, we obtain x a→ (w ∧ x) = x a→ (y ∧ w ∧ x). Hence (x a→ w) ∧ z = (x a→ y) ∧ (x a→ w) ∧ z. Therefore,

(x a→ y) ∧ w = (x a→ y) ∧ (x a→ w) ∧ w

= (x a→ y) ∧ (x a→ w) ∧ z ∧ w

= (x a→ w) ∧ z ∧ w

= z ∧ (x a→ w) ∧ w = z ∧ w

= w.

Conversely, let z ∈ L and assume that (1) and (2) hold. Now for any a ∈ Lz, take c, d, e ∈ [a, z]. By (1), c ≤ e a→ c such that e a→ c ∈ [a, z] and then by (2) we get

d ≤ e a→ c ⇔ ( e a→ c ) ∧ d = d ⇔ c ∧ d ∧ e = d ∧ e

⇔ d ∧ e ∧ c = d ∧ e

⇔ d ∧ e ≤ c.

Hence [a, z] is a Heyting algebra. Now define an induced binary operation →z on Lz by x →z y = (y ∨ x ∨ y) y→ y. Hence Lz is a skew Heyting algebra and therefore L is a skew HADL.

47 Corollary 2.2.12. On a skew HADL L, ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = m if and only if y ∧ x = x.

Proof. Assume ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = m. Then

x = m ∧ x

= {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)} ∧ x

= x ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)} ∧ x

= x ∧ ((x ∨ y) ∧ m) ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)} ∧ x = x ∧ (((x ∨ y) ∧ m) ∧ (y ∧ m)) ∧ x

= x ∧ m ∧ y ∧ m ∧ x

= m ∧ y ∧ x

= y ∧ x.

Hence y ∧ x = x. The converse is straight forward.

Lemma 2.2.13. Let L be a skew HADL and x, y ∈ L. Then ((x ∧ m) (y∧m)→ (y ∧ m)) ∧ m = x → y

Proof. Suppose L be a skew HADL. Since [y ∧ m, m] is a HADL, for any x, y ∈ L we obtain that

((x ∧ m) (y∧m)→ (y ∧ m)) ∧ m = ((x ∧ m) (y∧m)→ (y ∧ m))

∧((y ∧ m) (y∧m)→ (y ∧ m))

= ((x ∧ m) ∨ (y ∧ m)) (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = x → y.

Theorem 2.2.14. Let L be an ADL with a maximal element m. Then L is a skew HADL if and only if the set PI(L) of all principal ideals of L is a skew Heyting algebra.

48 Proof. Suppose L be a skew HADL. Clearly PI(L) is a distributive lattice. Define → on PI(L) by (x] → (y] = (x → y].

Then we have (x] → (y] = (((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)]. For a, b, c, d ∈ L, let (a] = (b] and (c] = (d]. Then by Lemma 1.2.14 we have a ∧ b = b, b ∧ a = a, c ∧ d = d, d ∧ c = c. Clearly b ∧ m = a ∧ b ∧ m = b ∧ a ∧ m = a ∧ m and similarly c ∧ m = d ∧ m. Following Lemma 2.2.13 we have b → d = ((b ∧ m) (d∧m)→

(d ∧ m)) ∧ m = ((a ∧ m) (c∧m)→ (c ∧ m)) ∧ m = a → c. Thus (a → c] = (b → d] so that (a] → (c] = (b] → (d]. Therefore the binary operation → is well defined on PI(L). It is obvious that PI(L) is a co-strongly distributive skew lattice with top element (m] = L. We show that an upset (a]↑ = {(b] ∈ PI(L)|(a] ⊆ (b]} where a, b ∈ L and (a], (b] ∈ PI(L) is a Heyting algebra. Let a, b, c, d ∈ L, such that

(b], (c], (d] ∈ (a]↑ and define (a]→ on (a]↑ by (b] (a]→ (c] = (b a→ c]. Clearly (a] ⊆ (c]. Since [a, m] and [c, m] are HADL, for all x, y ∈ [c, m] we obtain that x c→ y = x a→ y. Thus we have

(b] (a]→ (c] = (m ∧ b] (m∧a]→ (m ∧ c]

= (b ∧ m] (a∧m]→ (c ∧ m]

= ((b ∧ m) (a∧m)→ (c ∧ m)]

= (m ∧ ((b ∧ m) (a∧m)→ (c ∧ m))]

= ((b ∧ m) (a∧m)→ (c ∧ m)) ∧ m]

= ((b ∧ m) (a∧m)→ (c ∧ m)) ∧ ((c ∧ m) (a∧m)→ (c ∧ m))]

= (((b ∨ c) ∧ m) (a∧m)→ (c ∧ m)]

= (((b ∨ c) ∧ m) (c∧m)→ (c ∧ m)] = (b] → (c].

Using this fact we have the following:

49 (1) (b] (a]→ (b] = (b] → (b] = (b → b] = (m], since [b, m] is a Heyting algebra. (2) Since [b ∧ m, m] is a Heyting algebra we have

((c] (a]→ (b]) ∧ (b] = ((c]→(b]) ∧ (b]

= (((c ∨ b) ∧ m) (b∧m)→ (b ∧ m)]) ∧ (m ∧ b]

= (((c ∨ b) ∧ m) (b∧m)→ (b ∧ m)]) ∧ (b ∧ m]

= ((((c ∨ b) ∧ m) (b∧m)→ (b ∧ m)) ∧ (b ∧ m)] = (b ∧ m]

= (m ∧ b]

= (b].

(3) By the same reason as (2) we obtain that

(d] ∧ ((d] (a]→ (b]) = (d] ∧ ((d] → (b])

= (d] ∧ (((d ∨ b) ∧ m) (b∧m)→ (b ∧ m)]

= (d ∧ m] ∧ (((d ∨ b) ∧ m) (b∧m)→ (b ∧ m)]

= (d ∧ (d ∨ b) ∧ m] ∧ (((d ∨ b) ∧ m) (b∧m)→ (b ∧ m)]

= (d ∧ m] ∧ ((d ∨ b) ∧ m] ∧ (((d ∨ b) ∧ m) (b∧m)→ (b ∧ m)]

= (d ∧ m] ∧ (((d ∨ b) ∧ m) ∧ (((d ∨ b) ∧ m) (b∧m)→ (b ∧ m))] = (d ∧ m] ∧ (((d ∨ b) ∧ m) ∧ (b ∧ m)]

= ((d ∧ m) ∧ (b ∧ m)]

= (d ∧ b ∧ m]

= (m ∧ d ∧ b]

= (d ∧ b]

= (d] ∧ (b]

50 (4) ((c] ∨ (b]) (a]→ (d]

= (c ∨ b] (a]→ (d] = (c ∨ b]→(d]

= (((c ∨ b ∨ d) ∧ m) (d∧m)→ (d ∧ m)]

= ((((c ∨ d) ∧ m) (d∧m)→ (d ∧ m)) ∧ (((b ∨ d) ∧ m) (d∧m)→ (d ∧ m))]

= (((c ∨ d) ∧ m) (d∧m)→ (d ∧ m)] ∧ (((b ∨ d) ∧ m) (d∧m)→ (d ∧ m)] = ((c]→(d]) ∧ (b]→(d])

= (c] (a]→ (d]) ∧ ((b] (a]→ (d]).

(5) Applying Lemma 2.2.7 (1) we obtain that

(b] (a]→ ((c] ∧ (d]) = (b] (a]→ (c ∧ d] = (b]→(c ∧ d]

= (b → (c ∧ d)]

= ((b (c∧d∧m)→ (c ∧ d)) ∧ m]

= (((b (c∧d∧m)→ c) ∧ (b (c∧d∧m)→ d)) ∧ m]

= (((b (c∧d∧m)→ c) ∧ m) ∧ ((b (c∧d∧m)→ d) ∧ m)] = ((b → c) ∧ (b → d)]

= (b → c] ∧ (b → d]

= ((b] → (c]) ∧ ((b] → (d])

= ((b] (a]→ (c]) ∧ ((b] (a]→ (d])

Hence (a]↑ is a Heyting algebra.

51 Now for any (a], (b] ∈ PI(L) we have

((b] ∨ (a] ∨ (b]) (b]→ (b] = (b ∨ a ∨ b] (b]→ (b]

= (m ∧ (b ∨ a ∨ b)] (m∧b]→ (m ∧ b]

= ((b ∨ a ∨ b) ∧ m] (b∧m]→ (b ∧ m]

= ((a ∨ b) ∧ m] (b∧m]→ (b ∧ m]

= ((a ∨ b) ∧ m) (b∧m)→ (b ∧ m)] = (a] → (b].

Hence an induced binary operation → is defined on PI(L) by (a] → (b] =

((b] ∨ (a] ∨ (b]) (b]→ (b]. Therefore PI(L) is a skew Heyting algebra. Conversely, suppose PI(L) be a skew Heyting algebra. For each z ∈ L we show that Lz is a skew Heyting algebra. For x, y ∈ Lz define x →z y = w ∧ z, for some w ∈ L such that (x] → (y] = (w]. Let (u] = (v] for some u, v ∈ L. Then from Lemma 1.2.14, we have u∧v = v and v ∧u = u. Thus u∧z = v ∧u∧z = u∧v ∧z = v ∧ z. Now take a, b, c, d ∈ Lz such that a = c and b = d. Then we obtain that (a] = (c] and (b] = (d]. Consequently (a] → (b] = (c] → (d]. Since (a] → (b] = (e] and (c] → (d] = (f] for some e, f ∈ L, we have (e] = (f] and hence e ∧ z = f ∧ z so that a →z b = c →z d. Therefore the binary operation →z is well defined on Lz.

Since Lz is a distributive lattice, it is co-strongly distributive skew lattice with top z.

For any y ∈ Lz, we claim that [y, z] is a Heyting algebra. For any w, x ∈ [y, z], define y→ on [y, z] by w y→ x = w →z x. Clearly for any a, b, c, d ∈ [y, z] we have (a], (b], (c], (d] ∈ (y]↑. Then

(i) Since (y]↑ is a Heyting algebra, (a] → (a] = (m]. Therefore

a y→ a = a →z a = m ∧ z = z.

(ii) Let (a] → (b] = (t] for some t ∈ L. Thus a y→ b = a →z b = t ∧ z. Hence

(a y→ b) ∧ b = t ∧ z ∧ b = t ∧ b = b, this is because of (b] ⊆ (a] → (b] ⇒ (b] ⊆

52 (t] ⇒ t ∧ b = b.

(iii) Let (a] → (b] = (t] for some t ∈ L. Then a y→ b = t ∧ z. Now

(a ∧ t] = (a] ∧ (t]

= (a] ∧ ((a] → (b])

= (a] ∧ (b]

= (a ∧ b].

Hence a∧t∧z = a∧b∧z. Therefore a∧(a y→ b) = a∧t∧z = a∧b∧z = a∧b.

(iv) Let (a] → (b] = (u] and (a] → (c] = (v] for some u, v ∈ L. Then a y→

b = u ∧ z and a y→ c = v ∧ z. Since

(a] → (b ∧ c] = (a] → ((b] ∧ (c])

= ((a] → (b]) ∧ ((a] → (c]

= (u] ∧ (v]

= (u ∧ v],

we get a y→ (b ∧ c) = u ∧ v ∧ z = u ∧ z ∧ v ∧ z = (a y→ b) ∧ (a y→ c).

(v) Let (a] → (c] = (u] and (b] → (c] = (v] for some u, v ∈ L. Then we obtain

that a y→ c = u ∧ z and b y→ c = v ∧ z. Further

(a ∨ b] → (c] = ((a] ∨ (b]) → (c]

= ((a] → (c]) ∧ ((b] → (c])

= (u] ∧ (v]

= (u ∧ v].

Therefore (a ∨ b) y→ c = u ∧ v ∧ z = u ∧ z ∧ v ∧ z = (a y→ c) ∧ (b y→ c).

(i)-(v) above shows that [y, z] is a Heyting algebra.

For each (a], (b], (c] ∈ PI(L) define (c]→ on (c]↑ by (a] (c]→ (b] = (a c→ b].

53 Since PI(L) is a skew Heyting algebra, for x, y ∈ Lz we have the following:

(x] → (y] = ((y] ∨ (x] ∨ (y]) (y]→ (y]

= (y ∨ x ∨ y] (y]→ (y]

= ((y ∨ x ∨ y) y→ y].

Which implies that x →z y = ((y ∨ x ∨ y)y→y) ∧ z = (y ∨ x ∨ y)y→y. Hence

Lz is a skew Heyting algebra. Therefore L is a skew HADL.

54 Chapter 3

Skew Semi-Heyting Almost Distributive Lattices

Every Heyting algebra is a skew Heyting algebra but a semi-Heyting algebra is not necessarily a skew semi-Heyting algebra and we discuss this idea through examples. In the first section of this chapter we present different conditions on which a skew semi -Heyting algebra becomes a skew Heyting algebra. Considering a bi-normal skew semi-Heyting algebra L, for each b ∈ L we define a congruence relation on each upset b↑.

Using the fact that La is a distributive lattice with top a and the concepts of skew semi-Heyting algebra, in the second section of this chapter we introduce a new class of algebras called skew semi-Heyting almost distributive lattice(skew SHADL). We characterize it in terms of a congruence relation and the set of all of its principal ideals. Considering the congruence relation θ defined on a skew SHADL L we show that each congruence class is a maximal rectangular subalgebra of L and L/θ is a maximal lattice image of L. Further we investigate different conditions on which an ADL with a maximal element m is a skew SHADL and some algebraic properties of skew SHADLs are studied.

55 3.1 Skew Semi- Heyting Algebras

In this section, we introduce the concept of skew semi-Heyting algebra, charac- terize it as a skew Heyting algebra in terms of a unique binary operation on which an induced binary operation is defined, and we investigate some of its algebraic properties.

Definition 3.1.1. An algebra (L, ∨, ∧, →, 1) of type (2, 2, 2, 0) is said to be a skew semi-Heyting algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 1) is a co-strongly distributive skew lattice with top 1

(2) For any a ∈ L, an operation a→ can be defined on a↑= {x ∈ L|a ≤ x} such

that (a↑, ∨, ∧,a →, a, 1) is a semi-Heyting algebra with top 1 and bottom a

(3) An induced binary operation is defined on L by x → y = (y ∨ x ∨ y) y→ y.

Example 3.1.2. Let L be a co-strongly distributive skew lattice with top 1. Let b ∈ L such that x, y ∈ b↑. Define a binary operation b→ on b↑ by  1 if x ≤ y x b→ y = y otherwise.

One can show that b↑ is a semi-Heyting algebra. Thus defining a binary operation

→ on L induced from b→ by x → y = (y∨x∨y) y→ y makes L a skew semi-Heyting algebra.

Note: A semi-Heyting algebra is not necessarily a skew semi-Heyting algebra. The following example justifies this.

Example 3.1.3. Let L = {0, x, 1} be a chain with 0 < x < 1 which is the lattice reduct of the two semi-Heyting algebras with the binary operation → defined on L by the tables given below. It can be verified that the algebra (L, ∨, ∧, →, 1) where

56 the binary operation → defined by Table 1 is a skew semi-Heyting algebra. Now, if we assume that the algebra (L, ∨, ∧, →, 1) where the binary operation → defined by Table 2 is a skew semi-Heyting algebra, then for any y ∈ L y↑ is a semi-Heyting algebra. But when we apply the definition of → in Table 2 we obtain that 0 → x =

(x ∨ 0 ∨ x) x→ x ⇒ 0 → x = x x→ x ⇒ x = 1, which is impossible. This is a contradiction and hence (L, ∨, ∧, →, 1) is not a skew semi-Heyting algebra.

→ 0 x 1 → 0 x 1 0 1 1 1 0 1 x 1 x 0 1 1 x 0 1 1 1 0 x 1 1 0 x 1

Table 3.1: a skew semi-HA Table 3.2: Not a skew semi-HA

Next, we discuss some useful arithmetical properties of skew semi-Heyting alge- bras.

Lemma 3.1.4. Let L be a skew semi-Heyting algebra. Then for any a ∈ L, a↑ is a distributive lattice.

Proof. Suppose L be a skew semi-Heyting algebra and let a ∈ L. Then a↑ is a semi-Heyting algebra. Hence a↑ is a distributive lattice.

Theorem 3.1.5. Let L be a skew semi-Heyting algebra and x, y, z ∈ L. Then the following conditions hold:

(1) 1 → x = x

(2) x → 1 = 1

(3) x ≤ y ⇒ x → y = 1

(4) y ≤ x → y

57 (5) (x ∨ y) → x = y → x

(6) x ≤ y ≤ z ⇒ y ≤ x → z

(7) (x ∧ y) → x = 1

(8) x → (y → x) = y → (x → y)

Proof. Suppose L be a skew semi-Heyting algebra and x, y, z ∈ L . Then (1) and (2) hold trivially.

(3) Suppose x ≤ y. Then x → y = (y ∨ x ∨ y) y→ y = 1. (4) Since y↑ is a semi-Heyting algebra we have

y ∧ (x → y) = y ∧ ((y ∨ x ∨ y) y→ y)

= y ∧ {(y ∧ (y ∨ x ∨ y)) y→ y}

= y ∧ (y y→ y) = y ∧ 1

= y.

Moreover, since y ∨ x ∨ y and y belongs to y↑ we obtain that ((y ∨ x ∨ y) y→ y) ∧ y = y. Hence (4) follows. (5) For any x ∈ L, x↑ is a semi-Heyting algebra and hence it is a lattice. Thus

(x ∨ y) → x = (x ∨ (x ∨ y) ∨ x) x→ x

= (x ∨ y ∨ x) x→ x = y → x.

(6) Let x ≤ y ≤ z. Then

y ∧ (x → z) = y ∧ ((z ∨ x ∨ z) z→ z)

= y ∧ (z z→ z) = y ∧ 1

= y.

58 Clearly, (x → z) ∧ y = y

(7) (x ∧ y) → x = (x ∨ (x ∧ y) ∨ x) x→ x = x x→ x = 1. (8) It holds by applying (3) and (4).

Lemma 3.1.6. Let L be a skew semi-Heyting algebra and x, y ∈ L. Then x∨y ∈ y↑ if and only if y ∨ x ∈ x↑. Whenever x ∨ y and y ∨ x both belongs to y↑, then they are equal.

Proof. Suppose x ∨ y ∈ y↑. Since y ∈ y↑, we have y ∨ (x ∨ y) = (x ∨ y) ∨ y. Using associative property of skew lattice we obtain y ∨ x ∨ y = x ∨ y. Thus (y ∨ x ∨ y) ∨ x = x ∨ y ∨ x so that y ∨ x = x ∨ y ∨ x. Clearly x ∨ y ∨ x ∈ x↑ for some y ∈ L which implies that y ∨ x ∈ x↑. Suppose both x ∨ y and y ∨ x belongs to a lattice y↑. Since y ∈ y↑ we have y ∨ x ∨ y = (y ∨ x) ∨ y = y ∨ (y ∨ x) = y ∨ x and y ∨ x ∨ y = y ∨ (x ∨ y) = (x ∨ y) ∨ y = x ∨ y. Hence x ∨ y = y ∨ x ∨ y = y ∨ x.

Following the above lemma we have the next theorem.

Theorem 3.1.7. Let L be a skew semi-Heyting algebra and x, y, z ∈ L such that x ∨ y ∈ y↑. Then the following conditions hold:

(1) x → y = 1 ⇒ x ∨ y = y and x ∧ y = x

(2) x ∧ (y → z) = x ∧ ((x ∧ y) → z)

(3) x ∧ ((x → y) → y) = x

(4) (x → y) ∧ (y → x) = 1 ⇒ y ∧ x = y

(5) x ∧ (x → y) = x ∧ y

(6) x ≤ y ⇒ x ∧ (z → y) = x ∧ y

(7) (x ∨ y) → (x ∧ y) = 1 ⇒ x ∧ y = x.

59 Proof. (1) Let x → y = 1. Then (y ∨ x ∨ y) y→ y = 1.

⇒ (x ∨ y) ∧ {(y ∨ (x ∨ y)) y→ y} = x ∨ y

⇒ (x ∨ y) ∧ {{(x ∨ y) ∧ (y ∨ (x ∨ y))} y→ ((x ∨ y) ∧ y)} = x ∨ y

⇒ (x ∨ y) ∧ {{(y ∨ (x ∨ y)) ∧ (x ∨ y)} y→ ((x ∨ y) ∧ y)} = x ∨ y

⇒ (x ∨ y) ∧ ((x ∨ y) y→ y) = x ∨ y ⇒ x ∨ y = y.

Similarly, (y ∨ x ∨ y) y→ y = 1

⇒ x ∧ ((y ∨ x ∨ y) y→ y) = x

⇒ x ∧ (x ∨ y) ∧ ((y ∨ x ∨ y) y→ y) = x

⇒ x ∧ (x ∨ y) ∧ {((x ∨ y) ∧ (y ∨ x ∨ y)) y→ ((x ∨ y) ∧ y)} = x

⇒ x ∧ {(x ∨ y) ∧ ((x ∨ y) y→ y)} = x ⇒ x ∧ ((x ∨ y) ∧ y) = x

⇒ x ∧ y = x.

(2) From the fact that L is co-strongly distributive skew lattice we have

x ∧ (y → z) = x ∧ {(z ∨ y ∨ z) z→ z}

= x ∧ (x ∨ z) ∧ {(z ∨ y ∨ z) z→ z}

= x ∧ (x ∨ z) ∧ {((x ∨ z) ∧ (z ∨ y ∨ z)) z→ ((x ∨ z) ∧ z)}

= x ∧ {{((x ∨ z) ∧ z) ∨ ((x ∨ z) ∧ (y ∨ z))} z→ z}

= x ∧ {{(z ∨ ((x ∧ y) ∨ z))} z→ z} = x ∧ ((x ∧ y) → z).

60 (3) x ∧ ((x → y) → y)

= x ∧ {((y ∨ x ∨ y) y→ y) → y}

= x ∧ {(y ∨ ((y ∨ x ∨ y) y→ y) ∨ y) y→ y}

= x ∧ (x ∨ y) ∧ {(y ∨ ((y ∨ x ∨ y) y→ y) ∨ y) y→ y}

= x ∧ (x ∨ y) ∧ {((x ∨ y) ∧ (y ∨ ((y ∨ x ∨ y) y→ y) ∨ y)) y→ ((x ∨ y) ∧ y)}

= x ∧ {{((x ∨ y) ∧ y) ∨ ((x ∨ y) ∧ ((y ∨ x ∨ y) y→ y)) ∨ ((x ∨ y) ∧ y)} y→ y}

= x ∧ {y ∨ {(x ∨ y) ∧ {((x ∨ y) ∧ (y ∨ x ∨ y)) y→ ((x ∨ y) ∧ y)} ∨ y} y→ y}

= x ∧ {{y ∨ ((x ∨ y) ∧ ((x ∨ y) y→ y)) ∨ y} y→ y}

= x ∧ (y y→ y) = x ∧ 1

= x.

(4) From Lemma 3.1.6 we have the fact that x ∨ y ∈ y↑ implies that y ∨ x ∈ x↑. Thus (x → y) ∧ (y → x) = 1

⇒ ((y ∨ x ∨ y) y→ y) ∧ ((x ∨ y ∨ x) x→ x) = 1

⇒ y ∧ {((y ∨ x ∨ y) y→ y) ∧ ((x ∨ y ∨ x) x→ x)} = y

⇒ (y ∧ ((y ∨ x ∨ y) y→ y)) ∧ ((x ∨ y ∨ x) x→ x) = y

⇒ y ∧ ((x ∨ (y ∨ x)) x→ x) = y

⇒ y ∧ (y ∨ x) ∧ ((x ∨ (y ∨ x)) x→ x) = y

⇒ y ∧ (y ∨ x) ∧ {{(y ∨ x) ∧ (x ∨ (y ∨ x))} x→ (y ∨ x) ∧ x)} = y

⇒ y ∧ (y ∨ x) ∧ {((x ∨ (y ∨ x)) ∧ (y ∨ x)) x→ (y ∨ x) ∧ x)} = y

⇒ y ∧ ((y ∨ x) x→ x) = y

⇒ (y ∧ (y ∨ x)) ∧ ((y ∨ x) x→ x) = y

⇒ y ∧ ((y ∨ x) ∧ ((y ∨ x) x→ x)) = y ⇒ y ∧ ((y ∨ x) ∧ x) = y

⇒ y ∧ x = y.

61 (5) x ∧ (x → y) = x ∧ ((y ∨ x ∨ y) y→ y)

= (x ∧ (x ∨ y)) ∧ {((y ∨ x) ∨ y) y→ y}

= x ∧ {(x ∨ y) ∧ {((y ∨ x) ∨ y) y→ y}}

= x ∧ (x ∨ y) ∧ {((x ∨ y) ∧ ((y ∨ x) ∨ y)) y→ ((x ∨ y) ∧ y)}

= x ∧ (x ∨ y) ∧ {((y ∨ (x ∨ y)) ∧ (x ∨ y)) y→ y}

= x ∧ (x ∨ y) ∧ ((x ∨ y) y→ y) = x ∧ ((x ∨ y) ∧ y)

= x ∧ y.

(6) Assume that x ≤ y. Then x ∧ (z → y) = x ∧ y ∧ ((y ∨ z ∨ y) y→ y) = x ∧ y. (7) Suppose (x ∨ y) → (x ∧ y) = 1. Then x ∧ ((x ∨ y) → (x ∧ y)) = x. From (2) we have x ∧ ((x ∧ (x ∨ y)) → (x ∧ y)) = x. Thus, applying (5) on x ∧ (x → (x ∧ y)) = x yields x ∧ (x ∧ y) = x. Therefore x ∧ y = x.

Next, we give a characterization of skew semi-Heyting algebra.

Theorem 3.1.8. Let L be a skew semi-Heyting algebra and z ∈ L such that w, x, y ∈ z↑. Then the following are equivalent.

(1) L is a skew Heyting algebra

(2) w ≤ x ⇒ w z→ x = 1

(3) w ≤ x ⇒ x z→ y ≤ w z→ y

(4) (w ∨ x) z→ y = (w z→ y) ∧ (x z→ y).

Proof. Suppose L be a skew Heyting algebra. Let w, x, y ∈ z↑. Clearly z↑ is a skew

Heyting algebra. If w ≤ x, then w z→ x = (x ∨ w ∨ x) x→ x = x x→ x = 1. Thus (1)⇒(2) holds. Using the fact that y↑ is a Heyting algebra and L is co-normal,

62 whenever w ≤ x we have

(x z→ y) ∧ (w z→ y) = ((y ∨ x ∨ y) y→ y) ∧ ((y ∨ w ∨ y) y→ y)

= ((y ∨ x ∨ y) ∨ (y ∨ w ∨ y)) y→ y

= (y ∨ x ∨ y ∨ w ∨ y) y→ y

= (y ∨ x ∨ w ∨ y) y→ y

= (y ∨ w ∨ x ∨ y) y→ y

= (y ∨ x ∨ y) y→ y

= x z→ y.

This shows that (1)⇒(3). From the fact that y↑ is a Heyting algebra, we obtain the following result

(w z→ y) ∧ (x z→ y) = ((y ∨ w ∨ y) y→ y) ∧ ((y ∨ x ∨ y) y→ y)

= {(y ∨ w ∨ y) ∨ (y ∨ x ∨ y)} y→ y

= (y ∨ w ∨ x ∨ y) y→ y

= (w ∨ x) z→ y.

Consequently (1)⇒(4). To prove the converse, we show that for any z ∈ L the semi-Heyting algebra z↑ is a Heyting algebra. For this it suffices to show that for any w, x ∈ z↑ each of the conditions (2), (3) and (4) implies that (w ∧ x) z→ w = 1. Suppose (2) holds and let z ∈ L such that w, x ∈ z↑. Then w ∧ x = x ∧ w and (x ∧ w) ∨ w = w ⇒ x ∧ w ≤ w ⇒ w ∧ x ≤ w. From the given assumption we obtain that (w ∧ x) z→ w = 1. Now assume that (3) holds. Clearly w, x ∈ z↑ implies that w ∧ x ≤ w. Then by

(3), we have w z→ y ≤ (w ∧ x) z→ y for any y ∈ L. Taking y = w we get w z→ w ≤ (w ∧ x) z→ w. This in turn implies that 1 ≤ (w ∧ x) z→ w. Hence (w ∧ x) z→

63 w = 1. Finally let (4) holds. Then for any w, x ∈z↑,

(w ∧ x) z→ w = 1 ∧ ((w ∧ x) z→ w)

= (w z→ w) ∧ ((w ∧ x) z→ w)

= (w ∨ (w ∧ x)) z→ w

= w z→ w = 1.

This proves the theorem.

In the following theorem we give another characterization of a skew semi-Heyting algebra.

Theorem 3.1.9. Let L be a skew semi-Heyting algebra. Then L is a skew Heyting algebra if and only if for all z ∈ L, the binary operation z→ on z↑ is the same as the induced binary operation → on L.

Proof. Assume that L is a skew Heyting algebra. Thus for any z ∈ L, z↑ is a Heyting algebra so that it is a skew Heyting algebra. Since the binary operation → on a skew Heyting algebra is unique, x z→ y = x → y for any x, y ∈ z↑. Conversely suppose x z→ y = x → y for any x, y ∈ z↑. Since z↑ is semi-Heyting algebra we need to prove that (x ∧ y) z→ x = 1. Using (SH4) for the semi-Heyting algebra x↑ we get,

(x ∧ y) z→ x = (x ∧ y)→x

= (x ∨ (x ∧ y) ∨ x) x→ x

= x x→ x = 1.

Hence z↑ is a Heyting algebra and therefore L is a skew Heyting algebra.

64 Consider the skew semi-Heyting algebra L given by Example 3.1.3 whose induced binary operation → is defined by Table 1. In that example the induced binary operation → on L is the same as the binary operation z→ defined on z↑ for each z ∈ L. Therefore the skew semi-Heyting algebra is a skew Heyting algebra. In the next theorem we give an axiomatization for a skew semi-Heyting algebra.

Theorem 3.1.10. Let (L, ∨, ∧, →, 0, 1) be an algebra of type (2, 2, 2, 0, 0) such that (L, ∨, ∧, 0, 1) is a co-strongly distributive skew lattice and let z ∈ L. Then (L, ∨, ∧, →, 1) is a skew semi-Heyting algebra if and only if the following conditions hold:

(1) w z→ w = 1 for all w ∈ z↑

(2) w ∧ (w z→ x) = w ∧ x for all w, x ∈ z↑

(3) w ∧ (x z→ y) = w ∧ ((w ∧ x) z→ y) for all w, x, y ∈ z↑

(4) w ∧ (x z→ y) = w ∧ (x z→ (w ∧ y)) for all w, x, y ∈ z↑

(5) x ≤ w → x for all w, x ∈ L

(6) w → x = (x ∨ w ∨ x) x→ x for all w, x ∈ L.

Proof. Let (L, ∨, ∧, →, 1) be a skew semi-Heyting algebra. Then for any z ∈ L, z↑ is a semi-Heyting algebra and hence (1) and (2) hold from the definition of semi- Heyting algebra. From Theorem 1.2.7, (3) and (4) hold directly. But condition (5) follows from Theorem 3.1.5 (4), and (6) is direct from the assumption. Since L is co-strongly distributive skew lattice, for any z ∈ L, z↑ is a lattice. For any w, x, y ∈ z↑, we set w z→ x = w → x. Clearly, (5) implies that w → x ∈ x↑ ⊆ z↑.

Thus the restriction z→ of → to z↑ is well defined. Since z↑ is commutative (1),

(2) and, (3) and (4) for → simplify respectively to (SH4), (SH2) and (SH3) for z→ making z→ is the binary operation on z↑. Now, we prove that z↑ is a semi-Heyting algebra and for this it is enough to

65 show that w ∧ (x z→ y) = w ∧ ((w ∧ x) z→ (w ∧ y)). Then from (3) we have w ∧ (x z→ y) = w ∧ ((w ∧ x) z→ y), and from (4) we get w ∧ ((w ∧ x) z→ y) = w ∧ ((w ∧ x) z→ (w ∧ y). Hence w ∧ (x z→ y) = w ∧ ((w ∧ x) z→ (w ∧ y)). This shows that for each z ∈ L, (z↑, ∨, ∧, z→, z, 1) is a semi-Heyting algebra. Therefore using (6) it is possible to define an induced binary operation → on L by x → y = (y ∨ x ∨ y) y→ y that makes L is a skew semi-Heyting algebra.

Alternatively we can use the following axiomatization of a skew semi-Heyting algebra.

Corollary 3.1.11. Let (L, ∨, ∧, →, 0, 1) be an algebra of type (2, 2, 2, 0, 0) such that (L, ∨, ∧, 0, 1) is a co-strongly distributive skew lattice and let b ∈ L. Then (L, ∨, ∧, →, 1) is a skew semi-Heyting algebra if and only if the following conditions hold:

(1) w ≤ (w z→ x) z→ x for all w, x ∈ z↑

(2) w ∧ (w z→ x) = w ∧ x for all w, x ∈ z↑

(3) w ∧ (x z→ y) = w ∧ ((w ∧ w) z→ y) for all w, x, y ∈ z↑

(4) w ∧ (x z→ y) = w ∧ (x z→ (w ∧ y)) for all w, x, y ∈ z↑

(5) x ≤ (w → x) for all w, x ∈ L

(6) x → y = (y ∨ x ∨ y) y→ y for all x, y ∈ L.

Proof. Suppose (L, ∨, ∧, →, 1) be a skew semi-Heyting algebra. Clearly for any w, x ∈ z↑ we have

w ∧ ((w z→ x) z→ x) = w ∧ ((w ∧ (w z→ x)) z→ (w ∧ x))

= w ∧ ((w ∧ x) z→ (w ∧ x)) = w ∧ 1

= w.

66 Hence w ≤ w ∧ ((w z→ x) z→ x) so that (1) holds. The proof of (2)-(6) follows from the fact that for any z ∈ L, z↑ is a semi-Heyting algebra and the definition of skew semi-Heyting algebra. Conversely from (2) we have 1 ∧ (1 z→ x) = 1 ∧ x = x ⇒ 1 z→ x = x to each x ∈ z↑. Thus using (1) we get 1 ≤ (1 z→ x) z→ x, which implies that 1 ≤ x z→x. Thus x z→ x = 1. Therefore z↑ is a semi-Heyting algebra. The rest of the proof follows from Theorem 3.1.10.

Theorem 3.1.12. Let (L, ∨, ∧, →, 1) be a skew semi-Heyting algebra. Then for any b, y ∈ L, the algebra ([b, y], ∨, ∧, b→, y) is a skew semi-Heyting algebra.

Proof. Suppose L be a skew semi-Heyting algebra and b, y ∈ L. We show that for any d ∈ [b, y] the algebra ([d, y], ∨, ∧, d→, d, y) is a semi-Heyting algebra . Clearly [b, y] is co-strongly distributive skew lattice. Since L is a skew semi-Heyting algebra, for any b ∈ L, b↑ is a semi-Heyting algebra and therefore [b, y] is a semi-Heyting algebra. Again by the same reason [d, y] is a semi-Heyting algebra. Finally if we define b→ on [b, y] by w b→ x = w → x, then we obtain w b→ x = (x ∨ w ∨ x) x→ x so that ([b, y], ∨, ∧, b→, y) is a skew semi-Heyting algebra.

Lemma 3.1.13. Let the algebra ([x, y], ∨, ∧, x→, y) be a skew semi-Heyting algebra.

Then the algebra ([x, y], ∨, ∧, x→, x, y) is a Heyting algebra.

Proof. Let ([x, y], ∨, ∧, x→, y) be a skew semi-Heyting algebra and a, b ∈ [x, y].

Then by definition ([b, y], ∨, ∧, b→, y) is a semi-Heyting algebra. If we take b = x, then we obtain that ([x, y], ∨, ∧, x→, y) is a semi-Heyting algebra. Since

(a ∧ b) x→ a = (a ∨ (a ∧ b) ∨ a) a→ a

= a a→ a = y.

We conclude that ([x, y], ∨, ∧, x→, x, y) is a Heyting algebra.

Note: Using the above lemma we can generalize that a skew semi-Heyting algebra with bottom is a Heyting algebra.

67 Theorem 3.1.14. Let L be a bi-normal skew semi-Heyting algebra and b ∈ L. Let θ be the relation on b↑ defined by

θ = {(x, y) ∈ b↑ × b↑|b0 ∧ x = b0 ∧ y for some b0 ∈ b↑}.

Then θ is a congruence relation on b↑.

Proof. Suppose L be a bi-normal skew semi-Heyting algebra. Clearly θ is reflexive and symmetric. Let b ∈ L and x, y, z ∈ b↑ such that (x, y) ∈ θ and (y, z) ∈ θ. This indicates that there exist b1 and b2 ∈b↑ such that b1 ∧ x = b1 ∧ y and b2 ∧ y = b2 ∧ z. Then

b2 ∧ b1 ∧ x = b2 ∧ b1 ∧ y

= b1 ∧ b2 ∧ y

= b1 ∧ b2 ∧ z

= b2 ∧ b1 ∧ z.

This implies that (x, z) ∈ θ and hence θ is transitive. Therefore θ is an equiv- alence relation. Let (x1, y1) ∈ θ and (x2, y2) ∈ θ. Then there exist b1, b2 ∈b↑ such that b1 ∧ x1 = b1 ∧ y1 and b2 ∧ x2 = b2 ∧ y2. Now we show that (x1 ∧ x2, y1 ∧ y2) ∈ θ, and (x1 ∨x2, y1 ∨y2) ∈ θ. Using the normality of L we obtain that b1 ∧b2 ∧x1 ∧x2 = b1 ∧ x1 ∧ b2 ∧ x2 = (b1 ∧ x1) ∧ (b2 ∧ x2) = (b1 ∧ y1) ∧ (b2 ∧ y2) = b1 ∧ b2 ∧ y1 ∧ y2.

Hence (x1 ∧ x2, y1 ∧ y2) ∈ θ. Similarly, since b↑ is distributive lattice we have

b1 ∧ b2 ∧ (x1 ∨ x2) = (b1 ∧ b2 ∧ x1) ∨ (b1 ∧ b2 ∧ x2)

= (b2 ∧ b1 ∧ x1) ∨ (b1 ∧ b2 ∧ x2)

= (b2 ∧ b1 ∧ y1) ∨ (b1 ∧ b2 ∧ y2)

= b1 ∧ b2 ∧ (y1 ∨ y2).

which shows that (x1 ∨ x2, y1 ∨ y2) ∈ θ.

68 Finally we obtain that

b1 ∧ b2 ∧ (x1 → x2) = b1 ∧ b2 ∧ ((b1 ∧ b2 ∧ x1) → (b1 ∧ b2 ∧ x2))

= b1 ∧ b2 ∧ ((b2 ∧ b1 ∧ x1) → (b1 ∧ b2 ∧ x2))

= b1 ∧ b2 ∧ ((b2 ∧ b1 ∧ y1) → (b1 ∧ b2 ∧ y2))

= b1 ∧ b2 ∧ ((b1 ∧ b2 ∧ y1) → (b1 ∧ b2 ∧ y2))

= b1 ∧ b2 ∧ (y1 → y2).

Since b↑ is a lattice b1 ∧ b2 ∈b↑, thus (x1 → x2, y1 → y2) ∈ θ. Therefore θ is a congruence relation on b↑.

3.2 Skew Semi-Heyting Almost Distributive Lat- tices

In this section we introduce the concept of skew semi-Heyting almost distributive lattices (skew SHADLs). We characterize skew SHADL in terms of skew HADL and the set of all of its principal ideals. From the definition of skew HADL, to each a in a skew HADL L we have La is a skew Heyting algebra. Hence to each b ∈ La, [b, a] is a Heyting algebra so that it is a semi-Heyting algebra. Thus defining an induced binary operation →a on La from the binary operation b→ on the semi-Heyting algebra [b, a] makes La is a skew semi-Heyting algebra. In turn this makes L is a skew SHADL.

Definition 3.2.1. An ADL L with a maximal element m is said to be Skew Semi- Heyting Almost Distributive Lattice (skew SHADL) whenever to each a ∈ L the algebra (La, ∨, ∧, →a, a) is a skew semi-Heyting algebra.

The following theorem proves that a skew SHADL is a generalization of skew semi-Heyting algebra in the class of ADL.

69 Lemma 3.2.2. Let L be an ADL with a maximal element m. If L is a skew semi- Heyting algebra, then it is a skew SHADL.

Proof. Let L be a skew semi-Heyting algebra. For any z ∈ L, consider the set Lz which is co-strongly distributive skew lattice. Take x ∈ Lz so that the subset [x, z] of x↑ is a semi-Heyting algebra. Now define →z on Lz by w →z y = w → y to each w, y ∈ Lz. Hence we obtain that w →z y = (y ∨ w ∨ y) y→ y. This implies that Lz is a skew semi-Heyting algebra. Therefore L is a skew SHADL.

The next theorem gives a justification to our Definition 3.2.1 of skew SHADL.

Theorem 3.2.3.

Theorem 3.2.4. Let L be an ADL with a maximal element m. Then the following conditions are equivalent.

(1) L is a skew SHADL

(2) Lm is a skew semi-Heyting algebra

(3) (i) for any b ∈ L, ([b, m], ∨, ∧, b→, b, m) is a SHADL (ii) there exists a binary operation → on L defined by x → y = ((x ∨ y) ∧ m)

(y∧m)→ (y ∧ m).

Proof. Let L be a skew SHADL. Then (2) holds directly from the definition. Assume that (2) holds. Since L has maximal element(s), for each b ∈ L there exist a maximal element m such that b ≤ m and hence b ∈ Lm. As Lm is a skew semi-Heyting algebra b↑ = [b, m] is a semi-Heyting algebra and therefore an SHADL. Since Lm is a skew semi-Heyting algebra, the induced operation →m on Lm from b→ on [b, m] is given by x →m y = (y ∨x∨y) y→ y. Thus it is possible to define a binary operation

→ on L by x → y = (x ∧ m) →m (y ∧ m). But

(x ∧ m) →m (y ∧ m) = ((y ∧ m) ∨ (x ∧ m) ∨ (y ∧ m)) (y∧m)→ (y ∧ m)

= ((y ∨ x ∨ y) ∧ m) (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m),

70 and hence x → y = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m).

Conversely, suppose condition (3) hold. Let z ∈ L. Then Lz is a co-strongly distributive skew lattice. By (i) for any b ∈ Lz, [b, z] is a SHADL. Since the SHADL

[b, z] is a lattice we have ([b, z], ∨, ∧, b→, b, z) is a semi-Heyting algebra. Now define

→z on Lz by x →z y = x → y. Using (ii) we obtain that x →z y = ((x ∨ y) ∧ z)

(y∧z)→ (y ∧ z). But

((x ∨ y) ∧ z) (y∧z)→ (y ∧ z) = ((y ∨ x ∨ y) ∧ z) (y∧z)→ (y ∧ z)

= (y ∨ x ∨ y) y→ y.

Therefore (Lz, ∨, ∧, →z, z) is a skew semi-Heyting algebra and hence L is a skew SHADL.

Following this, by a skew SHADL we mean an algebra (L, ∨, ∧, →, m) of type (2, 2, 2, 0) satisfying condition (3) of Theorem 3.2.3.

Corollary 3.2.5. Let L be a skew SHADL. Then for any a ∈ L, [a, m] is a semi- Heyting algebra.

Proof. Let L be a skew SHADL and a ∈ L. Then from Theorem 3.2.3 (1) we obtain that [a, m] is a SHADL. Since [a, m] is a lattice it is a semi-Heyting algebra.

Corollary 3.2.6. Let L be a skew SHADL. If x, y ∈ L such that x ≤ y and a, b ∈ [y, m], then a x→ b = a y→ b.

Proof. Assume that L is a skew SHADL and let x, y ∈ L such that x ≤ y. It is obvious that Lm is a skew semi-Heyting algebra. Following this [x, m] and [y, m] are semi-Heyting algebras with [y, m] ⊆ [x, m]. If a, b ∈ [y, m], then a y→ b ∈ [y, m] and hence a y→ b ∈ [x, m]. Since a, b ∈ [x, m], a x→ b also belongs to [x, m].

The maximal element characterization of a x→ b and a y→ b on the semi-Heyting algebra [x, m] urges the two elements are equal.

Lemma 3.2.7. Let L be a skew SHADL. Then the following conditions hold:

71 (1) x ∧ (x → y) = x ∧ y ∧ m

(2) x → y = (x ∧ m) → (y ∧ m) for all x, y ∈ L.

Proof. Let L be a skew SHADL and x, y, z ∈ L. Applying Corollary 3.2.4 we obtain that for any y ∈ L, [y, m] is a semi-Heyting algebra. Then (1) Since [y ∧ m, m] is a semi-Heyting algebra we have

x ∧ (x → y) = x ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= m ∧ x ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)}

= x ∧ ((x ∨ y) ∧ m) ∧ {((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)} = x ∧ ((x ∨ y) ∧ m) ∧ (y ∧ m)

= x ∧ y ∧ m.

(2) Simply applying (2) of Theorem 3.2.3 yields that

x → y = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m ∧ m) (y∧m)→ (y ∧ m ∧ m)

= {((x ∧ m) ∨ (y ∧ m)) ∧ m} (y∧m∧m)→ (y ∧ m ∧ m) = (x ∧ m) → (y ∧ m).

The following theorem is another characterization of a skew SHADL which is an immediate consequence of Theorem 3.1.8.

Theorem 3.2.8. Let L be a skew SHADL and a, z ∈ L such that a ∈ Lz. If w, x, y ∈ [a, z], then the following conditions are equivalent:

(1) L is a skew HADL

(2) x ≤ y ⇒ x a→ y = z

72 (3) w ≤ x ⇒ x a→ y ≤ w a→ y

(4) (w ∨ x)a→ y = (w a→ y) ∧ (x a→ y).

Proof. Suppose L be a skew SHADL and z ∈ L. Then Lz is a skew semi-Heyting algebra. If L is a skew HADL it is immediate that Lz is a skew Heyting algebra. Thus from Theorem 3.1.8 it follows that conditions (2)-(4) hold. Conversely, if each of conditions (2)-(4) are given, then Lz is a skew Heyting algebra so that L is a skew HADL. This proves the theorem.

To prove the next theorem first we observe the following lemma which can be verified routinely.

Lemma 3.2.9. Let L be a skew SHADL and x, y, z ∈ L such that x ∧ m = y ∧ m. Then the following statements hold:

(1) x → y = m

(2) x → z = y → z and z → x = z → y.

Theorem 3.2.10. Let L be a skew SHADL and θ defined by

θ = {(x, y) ∈ L × L|x ∧ y = y and y ∧ x = x} is a relation on L. Then the following conditions hold:

(1) θ is a congruence relation on L

(2) the congruence classes are the maximal rectangular subalgebras of L

(3) L/θ is the maximal lattice image of L

(4) the congruence class [a]θ of a for each a ∈ L is a distributive lattice.

73 Proof. Let L be a skew SHADL and θ is a relation defined as above. The proofs of (1)-(3) are similar with that of Theorem 2.2.10. For any a ∈ L and x, y, z ∈ [a]θ we have

(x ∨ z) ∧ (y ∨ z) = ((x ∨ z) ∧ y) ∨ ((x ∨ z) ∧ z)

= ((z ∨ x) ∧ y) ∨ z

= ((z ∧ y) ∨ (x ∧ y)) ∨ z

= y ∨ z

= (x ∧ y) ∨ z.

Which shows that [a]θ is a distributive lattice. This proves (4).

The next theorem is another characterization of a skew SHADL.

Theorem 3.2.11. Let L be an ADL with a maximal element m. Then L is a skew

SHADL if for any z ∈ L and w, x, y ∈ Lz the following conditions hold:

(1) y ≤ x →z y

(2) (x →z y) ∧ w = w if and only if y ∧ w ∧ x = w ∧ x.

Proof. Assume that conditions (1) and (2) hold. Clearly Lz is co-strongly distribu- tive skew lattice. For any b ∈ Lz, take c, d, e ∈ [b, z]. By (1), c ≤ e b→ c which implies that e b→ c ∈ [b, z] and then by (2) we get

d ≤ e b→ c ⇔ d = d ∧ ( e b→ c )

⇔ d = ( e b→ c ) ∧ d ⇔ d ∧ e = d ∧ e ∧ c

⇔ d ∧ e ≤ c.

Hence [b, z] is a Heyting algebra so that it is a semi-Heyting algebra.

One can define an induced binary operation →z on Lz by x →z y = (y∨x∨y) y→ y. Hence Lz is a skew semi-Heyting algebra and therefore L is a skew SHADL.

74 Lemma 3.2.12. Let L be a skew SHADL. Then for any x, y ∈ L, x = y ∧ x if and only if x → y = m.

Proof. Let L be a skew SHADL. Assume that ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = m. Theorem 3.2.3 assures that [y ∧ m, m] is a semi-Heyting almost distributive lattice and hence the proof is the same as that of Corollary 2.2.12.

Definition 3.2.13. Let L be a skew SHADL. For any x, y ∈ L we say that x is compatible with y and write x ∼ y if x ∧ y = y ∧ x and x → y = y → x.

A subset M of L is said to be compatible if x ∼ y for all x, y ∈ M. A maximal compatible set is called a maximal set.

Definition 3.2.14. Let L be a skew SHADL and M be a maximal set. Then an element x ∈ L is said to be M-amicable if there exist y ∈ M and x 6= y such that x → y = m.

Theorem 3.2.15. Let L be a skew SHADL and M be a maximal set. If x ∈ L is M-amicable such that x → y = m for some y ∈ L, then there exist z ∈ L that satisfies the condition x → z = z → y.

Proof. Suppose x be M-amicable. Then there exist y ∈ M, x 6= y such that x → y = m. Now, write z = x ∧ y. From Lemma 3.2.11 we obtain that y ∧ x = x and hence

x → z = x → (x ∧ y)

= ((x ∨ (x ∧ y)) ∧ m) (x∧y∧m)→ (x ∧ y ∧ m)

= (x ∧ m) (x∧y∧m)→ (x ∧ y ∧ m)

= (x ∧ m) (y∧x∧m)→ (y ∧ x ∧ m)

= (x ∧ m) (x∧m)→ (x ∧ m) = m.

75 On the other hand we have:

m = (y ∧ m) (y∧m)→ (y ∧ m)

= {((x ∧ y) ∨ y) ∧ m} (y∧m)→ (y ∧ m) = (x ∧ y) → y

= z → y.

Consequently, we obtain that x → z = z → y.

Theorem 3.2.16. Let L be an ADL with a maximal element m. Then L is a skew SHADL if and only if the set PI(L) of all principal ideals of L is a skew semi-Heyting algebra.

Proof. Suppose L be a skew SHADL. Clearly PI(L) is a distributive lattice. Define → on PI(L) by (x] → (y] = (x → y]. Clearly the binary operation → is well defined on PI(L). A distributive lattice PI(L) is a co-strongly distributive skew lattice with top element (m] = L. We show that an upset (x]↑ = {(y] ∈ PI(L)|(x] ⊆ (y]} where x, y ∈ L and (x], (y] ∈ PI(L) is a semi-Heyting algebra. Let a, b, c, d ∈ L, such that (b], (c], (d] ∈ (a]↑ and define (a]→ on (a]↑ by (b] (a]→ (c] = (b a→ c]. Clearly

(a] ⊆ (c]. Since [a, m] and [c, m] are SHADL for all x, y ∈ [c, m], x c→ y = x a→ y and thus (b] (a]→ (c] = (b] → (c]. Using this fact we have the following:

(1)( b] (a]→ (b] = (b] → (b] = (b → b] = (m], since [b, m] is a semi-Heyting algebra.

(2) Since (x ∧ y] = (y ∧ x] for all x, y ∈ L and from the fact that [b ∧ m, m] is a

semi-Heyting algebra for any b ∈ L we have (d] ∧ ((d] (a]→ (b]) = (d] ∧ (b] (see (3) in the proof of Theorem 2.2.14).

76 (3)( b] ∧ (((b] ∧ (d]) (a]→ ((b] ∧ (c]))

= (b] ∧ ((b ∧ d] (a]→ (b ∧ c]) = (b] ∧ ((b ∧ d] → (b ∧ c])

= (b] ∧ ((b ∧ d) → (b ∧ c)]

= (b] ∧ ((((b ∧ c) ∨ (b ∧ d) ∨ (b ∧ c)) ∧ m) (b∧c∧m)→ (b ∧ c ∧ m)]

= (b] ∧ ((b ∧ (c ∨ d ∨ c) ∧ m) (b∧c∧m)→ (b ∧ c ∧ m)]

= (b] ∧ ((b ∧ (d ∨ c) ∧ m) (b∧c∧m)→ (b ∧ c ∧ m)]

= (b] ∧ (((b ∧ m) ∧ ((d ∨ c) ∧ m)) (b∧c∧m)→ ((b ∧ m) ∧ (c ∧ m))]

= (m ∧ b] ∧ (((b ∧ m) ∧ ((d ∨ c) ∧ m)) (c∧m)→ ((b ∧ m) ∧ (c ∧ m))]

= (b ∧ m] ∧ (((b ∧ m) ∧ ((d ∨ c) ∧ m)) (c∧m)→ ((b ∧ m) ∧ (c ∧ m))]

= (b ∧ m] ∧ (((d ∨ c) ∧ m) (c∧m)→ (c ∧ m)]

= ((m ∧ b] ∧ (((d ∨ c) ∧ m) (c∧m)→ (c ∧ m)] = (b] ∧ ((d] → (c])

= (b] ∧ ((d] (a]→ (c]).

Therefore (a]↑ is a semi-Heyting algebra. Now, for any (a], (b] ∈ PI(L) we have

((b] ∨ (a] ∨ (b]) (b]→ (b] = (b ∨ a ∨ b] (b]→ (b]

= (m ∧ (b ∨ a ∨ b)] (m∧b]→ (m ∧ b]

= ((b ∨ a ∨ b) ∧ m] (b∧m]→ (b ∧ m]

= ((a ∨ b) ∧ m] (b∧m]→ (b ∧ m]

= ((a ∨ b) ∧ m) (b∧m)→ (b ∧ m)] = (a → b]

= (a] → (b].

Hence an induced binary operation → is defined on PI(L) by (a] → (b] =

((b] ∨ (a] ∨ (b]) (b]→ (b]. Therefore PI(L) is a skew semi-Heyting algebra.

77 Conversely, suppose PI(L) be a skew semi-Heyting algebra. For each x ∈ L we show that Lx is a skew semi-Heyting algebra. For a, b ∈ Lx define a →x b = c ∧ x, for some c ∈ L such that (a] → (b] = (c]. Clearly the binary operation →x is well defined on Lx (as in the proof of Theorem 2.2.14). Since Lx is a distributive lattice it is co-strongly distributive skew lattice with top x. For any y ∈ Lx, we claim that

[y, x] is a semi-Heyting algebra with respect to the binary operation y→ on [y, x].

Let a, b, c ∈ [y, x] and define a y→ b = a →x b. Clearly (a], (b], (c] ∈ (y]↑. Then

(i) Since (y]↑ is a semi-Heyting algebra, (a] → (a] = (m]. Then we have a y→

a = a →x a = m ∧ x = x.

(ii) Let (a] → (b] = (t] for some t ∈ L. Then a y→ b = a →x b = t ∧ x. Therefore

a ∧ (a y→ b) = a ∧ t ∧ x = a ∧ b ∧ x = a ∧ b(see (iii) of Theorem 2.2.14).

(iii) Let (b] → (c] = (t] for some t ∈ L and assume that (a ∧ b] → (a ∧ c] =

(q] for some q ∈ L. Then b →x c = t ∧ x. Now a ∧ (b y→ c) = a ∧ (b →x c) = a ∧ t ∧ x = t ∧ a ∧ x = t ∧ a. Thus

(a ∧ ((a ∧ b) y→ (a ∧ c))] = (a] ∧ ((a ∧ b) y→ (a ∧ c)]

= (a] ∧ ((a ∧ b) →x (a ∧ c)]

= (a] ∧ (q ∧ x], where (a ∧ b) →x (a ∧ c) = q ∧ x for some q ∈ L

= (a ∧ q ∧ x]

= (q ∧ a]

= (a ∧ q]

= (a] ∧ (q]

= (a] ∧ ((a ∧ b] → (a ∧ c])

78 = (a] ∧ (((a] ∧ (b]) → ((a] ∧ (c]))

= (a] ∧ ((b] → (c])

= (a] ∧ (t]

= (a ∧ t]

= (t ∧ a]

= (a ∧ (b y→ c)].

Hence (a ∧ ((a ∧ b) y→ (a ∧ c))) ∧ x = (a ∧ (b y→ c)) ∧ x so that a ∧ ((a ∧ b) y→

(a ∧ c)) = a ∧ (b y→ c). Therefore [y, x] is a semi-Heyting algebra.

On PI(L) we have (a] (b]→ (b] = (a b→ b]. Then for a, b ∈ Lx we have the following

(a] → (b] = ((b] ∨ (a] ∨ (b]) (b]→ (b]

= (b ∨ a ∨ b] (b]→ (b]

= ((b ∨ a ∨ b) b→ b].

Which implies that a →x b = ((b∨a∨b) b→ b)∧x. Consequently a →x b = (b∨a∨b) b→ b. Hence Lx is a skew semi-Heyting algebra. Therefore H is a skew SHADL.

79 Chapter 4

Skew L-Almost Distributive Lattices

An L-algebra is a Heyting algebra which satisfies the identity (a → b) ∨ (b → a) = 1 for all a, b ∈ L. Relating this concept with the concept of skew Heyting algebra, in the first section of this chapter we introduce the concept of skew L- algebra and we characterize it in terms of skew Heyting algebras. We also give a condition on which an upset in a skew L-algebra is a Stone lattice. Relating the concepts of L-ADL with the concepts of skew L-algebra, in the second section of this chapter we introduce the concept of skew L-ADL. Considering a skew HADL L and for each a ∈ L such that b ∈ La, let x, y ∈ [b, a]. If (x →a y)∨(y →a x) = a, then [b, a] becomes an L-algebra and hence La is a skew L-algebra so that L is a skew L-ADL. We give several characterizations of skew L-ADL. Further we define a congruence relation θ on a skew L-ADL L and we show that each congruence class is a maximal rectangular subalgebra of L and the quotient algebra L/θ is the maximal lattice image of L. In this section, characterization of a skew L-ADL in terms of the set of all its principal ideals is made.

80 4.1 Skew L-Algebras

In this section, we introduce the concept of skew L-algebra and give several characterization of it. Also we present some conditions on which for any a in a skew L-algebra L, each upset a↑ is a Stone lattice.

Definition 4.1.1. An algebra (L, ∨, ∧, →, 1) of type (2, 2, 2, 0) is said to be a skew L-algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 1) is a co-strongly distributive skew lattice with top 1

(2) For any a ∈ L, an operation a→ can be defined on a↑= {x ∈ L|a ≤ x} such

that (a↑, ∨, ∧, a→, 1, a) is an L-algebra with top 1 and bottom a

(3) An induced binary operation → is defined on L by x → y = (y ∨ x ∨ y) y→ y.

Example 4.1.2. Let L = {0, x, 1} be a chain with 0 < x < 1 which is the lat- tice reduct of the Heyting algebra (L, ∨, ∧, →, 0, 1) where the binary operation → is defined on L by table 4.1 below. Now for any a ∈ L define a→ on a↑ by x a→ y = x → y. Since either of x → y = 1 or y → x = 1 holds we obtain that (x a→ y) ∨ (y a→ x) = (x → y) ∨ (y → x) = 1. Hence (a↑, ∨, ∧, a→, a, 1) is an L-algebra.

Therefore one can easily check that for all a, b ∈ L, a → b = (b ∨ a ∨ b) b→ b and conclude that (L, ∨, ∧, →, 0, 1) is a skew L-algebra.

→ 0 x 1 0 1 1 1 x 0 1 1 1 0 x 1

Table 4.1: (skew L-algebra)

81 Next, we present some useful arithmetical properties of skew L-algebras which can be verified routinely.

Theorem 4.1.3. Let L be a skew L-algebra and x, y, z ∈ L. Then the following conditions hold:

(1) 1 → x = x

(2) x → 1 = 1

(3) x ≤ y ⇒ x → y = 1

(4) y ≤ x → y

(5) (x ∨ y) → x = y → x

(6) (x ∧ y) → x = 1

(7) x → (y → x) = y → (x → y).

Theorem 4.1.4. Let L be a skew Heyting algebra and for all a ∈ L, a↑ be a chain. Then L is a skew L-algebra.

Proof. Suppose L be a skew Heyting algebra such that for all a ∈ L and x, y ∈ a↑, x ≤ y. Since a↑ is a Heyting algebra we have x a→ y = 1. Thus (x a→ y) ∨ (y a→ x) = 1 ∨ (y a→ x) = 1. This shows that a↑ is an L-algebra and therefore L is a skew L-algebra.

Lemma 4.1.5. Let L be a skew L-algebra and x, y ∈ L such that x ∈ y↑. Then the following conditions hold:

(1) (x → y) ∨ (y → x) = 1

(2) (x → y) ∧ (y → x) = x y→ y.

82 Proof. Suppose L be a skew L-algebra and for x, y ∈ L, let x ∈ y↑. The proof of (1) follows from Theorem 4.1.3 (3). Clearly we have x ∨ y = x = y ∨ x. Thus

(x → y) ∧ (y → x) = ((y ∨ x ∨ y) y→ y) ∧ ((x ∨ y ∨ x) x→ x)

= (x y→ y) ∧ (x x→ x)

= (x y→ y) ∧ 1

= x y→ y.

This proves (2).

Theorem 4.1.6. Let (L, ∨, ∧, →, 0, 1) be an algebra of type (2, 2, 2, 0, 0) such that (L, ∨, ∧, 0, 1) is a co-strongly distributive skew lattice and let b ∈ L. Then (L, ∨, ∧, → , 1) is a skew L-algebra if and only if the following conditions hold:

(1) (x b→ y) ∨ (y b→ x) = 1, ∀x, y ∈ b↑

(2) (x ∧ y) b→ x = 1, ∀x, y ∈ b↑

(3) x ∧ (x b→ y) = x ∧ y, ∀x, y ∈ b↑

(4) x ∧ (y b→ z) = x ∧ ((x ∧ y) b→ (x ∧ z)), ∀x, y, z ∈ b↑

(5) y ≤ x → y, ∀x, y ∈ L

(6) x → y = (y ∨ x ∨ y) y→ y, ∀x, y ∈ L.

Proof. Assume that L be a skew L-algebra. Then for any b ∈ L, b↑ is an L-algebra and hence by Definition 1.2.2 and Definition 1.2.9 conditions (1)-(4) hold directly.

Clearly for any x, y ∈ L, y ∧ (x → y) = y ∧ ((y ∨ x ∨ y) y→ y) = ((y ∨ x ∨ y) y→ y) ∧ y = y. Since y and (y ∨ x ∨ y) y→ y both belongs to y↑, and y↑ is a lattice we have (x → y) ∧ y = y. Thus (5) holds. Also as L is skew L-algebra (6) holds from the definition. Conversely, conditions (1)-(4) shows that b↑ is an L-algebra. Now for b ∈

83 L and x, y, z ∈ b↑, set x b→ y = x → y. Clearly (5) implies that x → y ∈ y↑ ⊆ b↑.

Thus the restriction b→ of → to b↑ is well defined. Since b↑ is a lattice, con- ditions (1)-(4) assures that for each b ∈ L,(b↑, ∨, ∧, b→, b, 1) is an L-algebra. Therefore using (6) it is possible to define an induced binary operation → on L by x → y = (y ∨ x ∨ y) y→ y that makes L is a skew L-algebra.

Lemma 4.1.7. A skew L-algebra L with bottom 0 is an L-algebra.

Proof. Suppose L is a skew L-algebra with 0. Then for any a ∈ L, (a↑, ∨, ∧, a→, a, 1) is an L-algebra. Now take a = 0. Since 0↑ = L, we get (L, ∨, ∧, →, 0, 1) is an L- algebra.

Remark 4.1.1. Suppose L is a skew Heyting algebra. Then L is a skew L- algebra if and only if for all a ∈ L such that x, y ∈ a↑, (x a→ y) ∨ (y a→ x) = 1. Alternatively we have the following theorem.

Theorem 4.1.8. Let (L, ∨, ∧, →, 1) be an algebra of type (2, 2, 2, 0) such that (L, ∨, ∧, 1) is a co-strongly distributive skew lattice with top 1. Then (L, ∨, ∧, →, 1) is a skew L-algebra if and only if it satisfies the following axioms:

(1) (x a→ y) ∨ (y a→ x) = 1, ∀a ∈ L such that x, y ∈ a↑

(2) x → x = 1, ∀x ∈ L

(3) x ∧ (x → y) ∧ x = x ∧ y ∧ x, ∀x, y ∈ L

(4) y ∧ (x → y) = y = (x → y) ∧ y, ∀x, y ∈ L

(5) x → y = (y ∨ x ∨ y) → y, ∀x, y ∈ L

(6) x → (a ∨ (y ∧ z) ∨ a) = (x → (a ∨ y ∨ a)) ∧ (x → (a ∨ z ∨ a)), ∀a, x, y, z ∈ L.

Proof. Suppose L is a skew L- algebra. Then for each a ∈ L, a↑ is an L-algebra and thus (1) holds. Since a skew L-algebra is a skew Heyting algebra, conditions (2)-(6) follows directly from Theorem 3.2. on [6]. Conversely if conditions (2)-(6)

84 are satisfied by L, then by the same reason L becomes a skew Heyting algebra and condition (1) makes every a↑ is an L-algebra. Therefore L becomes a skew L-algebra.

Lemma 4.1.9. Let L be a skew L-algebra and θ be a relation on L defined by

xθy if and only if x ∨ y ∨ x = x and y ∨ x ∨ y = y.

Then θ is a congruence relation on L.

Proof. Since L is a skew Heyting algebra, θ is a congruence relation on L.

Theorem 4.1.10. Let L be a skew Heyting-algebra and a ∈ L. If y ∨ x ∨ y = y for any x, y ∈ a↑, then L is a skew L-algebra.

Proof. Suppose L be a skew Heyting algebra and a ∈ L such that y ∨ x ∨ y = y for any x, y ∈ a↑. Define a→ on a↑ by x a→ y = x → y. Thus we get x a→ y = (y ∨ x ∨ y)y→y = y y→ y = 1 and by symmetry y a→ x = 1. Consequently, we obtain that (x a→ y) ∨ (y a→ x) = 1 ∨ 1 = 1. Hence a↑ is an L-algebra and therefore L is a skew L-algebra.

Lemma 4.1.11. Let L be a skew L-algebra. Then for any x ∈ L, x↑ is a skew L-algebra.

Proof. Suppose L is a skew L-algebra. Then by definition for any x ∈ L, the algebra

(x↑, ∨, ∧, x→, x, 1) is an L-algebra so that it is a Heyting algebra. We want to show that for any b ∈ x↑, b↑ is an L-algebra. Clearly b↑ is a Heyting algebra. Define b→ on b↑ by m b→ n = m x→ n for all m, n ∈ b↑. Then (m b→ n) ∨ (n b→ m) = (m x→ n) ∨ (n x→ m) = 1. Thus b↑ is an L-algebra. Now defining the induced binary operation x→ on x↑ by y x→ z = (z ∨ y ∨ z) z→ z makes that x↑ is a skew L-algebra.

Theorem 4.1.12. Let L be a skew L-algebra and [x, y] be a chain for some x, y ∈ L. Then [x, y] is a skew L-algebra.

85 Proof. Let L be a skew L-algebra. For any x ∈ L, x↑ is a Heyting algebra and hence [x, y] is a Heyting algebra if y ∈ x↑. Since [x, y] is a chain for any z ∈

[x, y] and a, b ∈ [z, y], either a z→ b = y or b z→ a = y. Thus in either cases we have that (a z→ b) ∨ (b z→ a) = y. This implies that [z, y] is an L-algebra.

Now defining x→ on [x, y] by a x→ b = (b ∨ a ∨ b) b→ b makes [x, y] is a skew L-algebra.

A skew L-algebra may not contain a zero element. In the next lemma we use the symbol La for the set {x ∈ L|x ≤ a} where L is a skew L-algebra and a is an arbitrary element of L.

Lemma 4.1.13. Let (L, ∨, ∧, →, 1) be a skew L-algebra. Then to each a ∈ L, (La, ∨, ∧, →a, a) is a skew L-algebra where →a is a binary operation on La.

Proof. Suppose L is a skew L-algebra. Clearly La is co-strongly distributive skew

a lattice and for all x ∈ L , x↑ is an L-algebra. Consequently, the algebra ([x, a], ∨, ∧, →a

, x, a) is a Heyting algebra. Now, it is sufficient to prove that (y →a z) ∨ (z →a y) = a for all y, z ∈ [x, a]. For this define →a on [x, a] by y →a z = (y x→ z) ∧ a. Since x↑ is a distributive lattice with top 1 and a ∈ x↑ we obtain that

(y →a z) ∨ (z →a y) = ((y x→ z) ∧ a) ∨ ((z x→ y) ∧ a) = ((y x→ z) ∨ (z x→ y)) ∧ a = 1 ∧ a = a, for all y, z ∈ [x, a]. This implies that [x, a] is an L-algebra. Following this we define →a on La by y →a z = (y → z) ∧ a, for all y, z ∈ La.

Since L is a skew L-algebra (y → z) ∧ a = ((z ∨ y ∨ z) z→ z) ∧ a = (z ∨ y ∨ z) z→ a a a z. Thus y → z = (z ∨ y ∨ z) z→ z so that the algebra (L , ∨, ∧, → , a) is a skew L-algebra.

Theorem 4.1.14. Let L be a skew Heyting algebra. Then L is a skew L-algebra if for each a ∈ L such that x, y, z ∈ a↑, either x a→ (y ∨ z) = (x a→ y) ∨ (x a→ z) or (x ∧ y) a→ z = (x a→ z) ∨ (y a→ z).

Proof. Suppose L be a skew Heyting algebra. Thus for any a ∈ L, a↑ is a Heyting algebra. We need to prove that (x a→ y)∨(y a→ x) = 1 for each a ∈ L and x, y ∈ a↑.

86 Let x a→ (y ∨ z) = (x a→ y) ∨ (x a→ z), for all x, y, z ∈ a↑. Then

( x a→ y ) ∨ ( y a→ x ) = (( x a→ y ) ∧ ( y a→ y )) ∨ (( y a→ x ) ∧ ( x a→ x ))

= ((x ∨ y) a→ y) ∨ ((y ∨ x) a→ x)

= (x ∨ y) a→ (x ∨ y) = 1.

Let (x ∧ y) a→ z = (x a→ z) ∨ (y a→ z). Then using a similar procedure we obtain that

( x a→ y ) ∨ ( y a→ x ) = (( x a→ y ) ∧ ( x a→ x )) ∨ (( y a→ x ) ∧ ( y a→ y ))

= (x a→ (x ∧ y)) ∨ (y a→ (x ∧ y))

= (x ∧ y) a→ (x ∧ y) = 1.

In both cases we obtained that a↑ is an L-algebra. Consequentley L is a skew L-algebra.

Corollary 4.1.15. Let L be a skew L-algebra such that for all a ∈ L and x, y, z ∈ a↑, (x ∧ y) a→ z = (x a→ z) ∨ (y a→ z). Then

(1) a↑ is a Stone lattice

(2) (x → a) → x = x.

Proof. Suppose L be a skew L-algebra. Which shows that for any a ∈ L, a↑ is an L-algebra so that a↑ is a pseudo-complemented lattice. (1) Take x ∈ a↑ and since a↑ is a Heyting algebra with bottom a we use the notation

87 x∗ = x → a. Thus

∗ ∗∗ ∗ (1) x ∨ x = (x a→ a) ∨ ((x a→ a) )

= (x a→ a) ∨ ((x a→ a) a→ a)

= (x ∧ (x a→ a)) a→ a

= (x ∧ a) a→ a

= a a→ a = 1.

Hence a↑ is a Stone lattice. (2) Since a↑ is a Stone semi-Heyting algebra from the given assumption and (1) we obtained that

(x → a) → x = ((a ∨ x ∨ a) a→ a) → x

= (x a→ a) → x

= {x ∨ (x a→ a) ∨ x} x→ x

= {x ∨ (x a→ a)} x→ x

= {x a→ (x ∨ (a a→ 1))} x→ x

= (x a→ (x ∨ 1)) x→ x

= 1 x→ x = x.

Lemma 4.1.16. If L1 and L2 are two skew L-algebras, then the direct product

L1 × L2 is also a skew L-algebra under point wise operations defined by (x1, y1) ∧

(x2, y2) = (x1 ∧ x2, y1 ∧ y2), (x1, y1) ∨ (x2, y2) = (x1 ∨ x2, y1 ∨ y2) and (x1, y1) →

(x2, y2) = (x1 → x2, y1 → y2) for every (x1, y1), (x2, y2) ∈ L1 × L2.

88 4.2 Skew L-Almost Distributive Lattices

In this section we introduce the concept of skew L-almost distributive lattices and characterize it as a skew L-algebra in terms of a congruence relation defined on it. More over we investigate some of its algebraic properties.

Definition 4.2.1. An ADL L with a maximal element m is said to be a Skew

L-Almost Distributive Lattice (skew L-ADL) whenever (Lz, ∨, ∧, →z, z) is a skew L-algebra for each z ∈ L.

Example 4.2.2. Consider a relatively complemented ADL L with a maximal ele- ment m. For any a, b ∈ L such that a ∈ Lb, define a binary operation a→ on [a, b] b b by x a→ y = (x ∨ y) ∧ b, where x is the relative complement of x in [a, b]. It is simple to show that [a, b] is an L-algebra. Now, defining an induced binary operation

→b on Lb by x →b y = (y ∨ x ∨ y) y→ y makes Lb is a skew L-algebra. Hence L becomes a skew L-ADL.

Theorem 4.2.3. Let L be an ADL with a maximal element m. Then the following conditions are equivalent.

(1) L is a skew L-ADL

(2) Lm is a skew L-algebra

(3) (i) for any b ∈ L, ([b, m], ∨, ∧, b→, b, m) is a L-ADL

(ii) there exists a binary operation → on L defined by x → y = ((x ∨ y) ∧ m)

(y∧m)→ (y ∧ m).

Proof. Let L be a skew L-ADL. Then (2) holds directly from the definition. Assume that (2) holds. Since L has maximal element(s), for each b ∈ L there exist a maximal element m such that b ≤ m and hence b ∈ Lm. As Lm is a skew L-algebra b↑ = [b, m] is a L-algebra and therefore an L-ADL. Since Lm is a skew L-algebra, the induced operation →m on Lm from b→ on [b, m] is given by x →m y = (y ∨x∨y) y→ y. Thus

89 it is possible to define a binary operation → on L by x → y = (x ∧ m) →m (y ∧ m). But

(x ∧ m) →m (y ∧ m) = ((y ∧ m) ∨ (x ∧ m) ∨ (y ∧ m)) (y∧m)→ (y ∧ m)

= ((y ∨ x ∨ y) ∧ m) (y∧m)→ (y ∧ m)

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m),

and hence x → y = ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m).

Conversely, suppose condition (3). Let z ∈ L. Then Lz is a co-strongly distributive skew lattice. By (i), b ∈ Lz implies that [b, m] is an L-ADL. Since L has maximal element, z ≤ m for some maximal element m in L and hence [b, z] ⊆ [b, m]. Thus by Theorem 1.2.19, ([b, z], ∨, ∧, b→, b, z) is an L-algebra. Now by (ii) it is possible to define →z on Lz by x →z y = ((x ∨ y) ∧ z) (y∧z)→ (y ∧ z). But ((x ∨ y) ∧ z)

(y∧z)→ (y ∧ z) = ((y ∨ x ∨ y) ∧ z) (y∧z)→ (y ∧ z) = (y ∨ x ∨ y) y→ y. Therefore

(Lz, ∨, ∧, →z, z) is a skew L-algebra and hence (L, ∨, ∧, →, m) is a skew L-ADL.

Now on wards, by a skew L-ADL we mean an algebra (L, ∨, ∧, →, m) of type (2, 2, 2, 0) satisfying condition (3) of Theorem 4.2.3.

Corollary 4.2.4. Let L be a skew L-ADL. Then for any a ∈ L, [a, m] is an L- algebra.

Proof. From (1) of Theorem 4.2.3 [a, m] is an L-ADL and since [a, m] is a lattice it is an L-algebra.

The following lemma is analogous with the statement, any interval on an L- algebra is again an L-algebra.

Lemma 4.2.5. Let L is a skew L-ADL. Then for any b ∈ L, [b, m] is a skew L-ADL.

Proof. Suppose L be a skew L-ADL. Then for any y ∈ L, Ly is a skew L-algebra.

Particularly Lm is a skew L-algebra and hence [b, m] is an L-algebra for any b ∈ Lm.

Following this for any c ∈ [b, m] defining b→ on [b, c] by x b→ y = (x →m y)∧c, where

90 →m is a binary operation on [b, m], makes [b, c] is an L-algebra and consequently a skew L-algebra with top element c. Therefore [b, m] is a skew L-ADL.

Corollary 4.2.6. Let L be a skew L-ADL. If x, y ∈ L such that x ≤ y and a, b ∈

[y, m], then a x→ b = a y→ b.

Lemma 4.2.7. Let L be a skew L-ADL. Then for any x, y ∈ L the following conditions hold:

(1) x → y = (x (y∧m)→ y) ∧ m

(2) ((x → y) ∨ (y → x)) ∧ m = m.

Proof. Let L be a skew L-ADL and x, y ∈ L. (1) Since [y ∧ m, m] is a Heyting algebra we have

(x (y∧m)→ y) ∧ m = (m (y∧m)→ (x (y∧m)→ y)) ∧ m

= ((x ∧ m) (y∧m)→ y) ∧ m ∧ m

= ((x ∧ m) (y∧m)→ y) ∧ ((x ∧ m) (y∧m)→ m) ∧ m

= ((x ∧ m) (y∧m)→ (y ∧ m)) ∧ ((y ∧ m) (y∧m)→ (y ∧ m))

= ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = x → y.

(2) Clearly x, x ∧ m, y and y ∧ m belongs to an L-ADL [x ∧ y ∧ m, m]. Thus, using condition (1) we obtain that

((x → y) ∨ (y → x)) ∧ m = ((x y∧m→ y) ∧ m) ∨ ((y x∧m→ x) ∧ m)

= ((x x∧y∧m→ y) ∧ m) ∨ ((y x∧y∧m→ x) ∧ m)

= ((x x∧y∧m→ y) ∨ (y x∧y∧m→ x)) ∧ m = m.

91 Theorem 4.2.8. Let L be a skew HADL and z ∈ L. Then L is a skew L-ADL if either x b→ (y ∨ z) = (x b→ y) ∨ (x b→ z) or (x ∧ y) b→ z = (x b→ z) ∨ (y b→ z), for any b ∈ Lz such that x, y ∈ [b, z].

Proof. Suppose L be a skew HADL. Then for any z ∈ L, Lz is a skew Heyting algebra. Thus for any b ∈ Lz,[b, z] is a Heyting algebra. Hence it is sufficient to prove that [b, z] is an L-algebra. Let x, y ∈ [b, z]. Since (x b→ y) and (y b→ x) belongs to [b, z] we have

(x b→ y) ∨ (y b→ x) = ((x b→ y) ∧ z) ∨ ((y b→ x) ∧ z)

= ((x b→ y) ∧ (y b→ y)) ∨ ((y b→ x) ∧ (x b→ x))

= ((x ∨ y) b→ y) ∨ ((y ∨ x) b→ x)

= (x ∨ y) b→ (x ∨ y) = z or

(x b→ y) ∨ (y b→ x) = ((x b→ y) ∧ z) ∨ ((y b→ x) ∧ z)

= ((x b→ y) ∧ (x b→ x)) ∨ ((y b→ x) ∧ (y b→ y))

= (x b→ (y ∧ x)) ∨ (y b→ (x ∧ y))

= (x ∧ y) b→ (x ∧ y) = z.

This shows that [b, z] is an L-algebra and hence Lz is a skew L-algebra. Therefore L is a skew L-ADL.

The next theorem characterizes a skew L-ADL with regards to a congruence relation θ defined on it.

Theorem 4.2.9. Let L be a skew L-ADL and θ defined by

92 θ = {(x, y) ∈ L × L|x ∧ y = y and y ∧ x = x} is a relation on L. Then

(1) θ is a congruence relation on L

(2) The congruence classes are the maximal rectangular subalgebras of L

(3) L/θ is the maximal lattice image of L

Skew L-algebra has a top element and need not to contain bottom element.

Theorem 4.2.10. Let L be an ADL with a maximal element m. Then L is a skew

L-ADL if and only if for any a, z ∈ L such that a ∈ Lz and w, x, y ∈ [a, z], the following conditions hold:

(1) y ≤ x a→ y

(2) (y a→ w) ∧ x = x if and only if w ∧ x ∧ y = x ∧ y

(3) (x a→ y) ∨ (y a→ x) = z.

Proof. Suppose L be a skew L-ADL. It is clear that for any z ∈ L, Lz is a skew L- algebra. For any a ∈ Lz,[a, z] is an L-algebra. Then (1) is direct. Let w, x, y ∈ [a, z].

To show that (2) holds first we assume that (y a→ w) ∧ x = x. Then x ∧ y = ((y a→ w) ∧ x) ∧ y = y ∧ ((y a→ w) ∧ x) = (y ∧ w) ∧ x = w ∧ x ∧ y.

On the other hand given that w ∧ x ∧ y = x ∧ y, we obtain y a→ (x ∧ y) = y a→

(w ∧ x ∧ y). Hence (y a→ x) ∧ z = (y a→ w) ∧ (y a→ x) ∧ z. Therefore

(y a→ w) ∧ x = (y a→ w) ∧ (y a→ x) ∧ x

= (y a→ w) ∧ (y a→ x) ∧ z ∧ x

= (y a→ x) ∧ z ∧ x

= z ∧ (y a→ x) ∧ x = z ∧ x

= x.

93 Since Lz is a skew L-algebra we obtain that [a, z] is an L-algebra for all a ∈ Lz and therefore (3) holds.

Conversely assume that (1)-(3) hold. Now for any a ∈ Lz, take c, d, e ∈ [a, z].

By (1), c ≤ e a→ c such that e a→ c ∈ [a, z] and then by (2) we get

d ≤ e a→ c ⇔ d = d ∧ (e a→ c)

⇔ d = (e a→ c) ∧ d ⇔ d ∧ e = c ∧ d ∧ e = d ∧ e ∧ c

⇔ d ∧ e ≤ c.

Hence [a, z] is a Heyting algebra and by (3) it is an L-algebra. We now define an induced binary operation →z on Lz by x →z y = (y ∨ x ∨ y) y→ y so that Lz is a skew L-algebra. Consequently L is a skew L-ADL.

Corollary 4.2.11. On a skew L-ADL L, ((x ∨ y) ∧ m) (y∧m)→ (y ∧ m) = m if and only if y ∧ x = x.

Proof. See Corollary 2.2.12.

Theorem 4.2.12. Let L be an ADL with a maximal element m. Then L is a skew L-ADL if and only if the set PI(L) of all principal ideals of L is a skew L-algebra.

Proof. Suppose L be a skew L-ADL. It is obvious that PI(L) is a distributive lattice. Define a binary operation → on PI(L) by (x] → (y] = (x → y] where x, y ∈ L. In the proof of Theorem 2.2.14 we have shown that the binary operation → is well defined on PI(L). Clearly PI(L) is a co-strongly distributive skew lattice with top element (m] = L. We show that an upset (a]↑ = {(b] ∈ PI(L)|(a] ⊆ (b]} where a, b ∈ L and (a], (b] ∈ PI(L), is an L-algebra. Further from Theorem 2.2.14 we have shown that the upset (a]↑ is a Heyting algebra. So it suffices to show that (a]↑ satisfies ((b]

94 (a]→ (c]) ∨ ((c] (a]→ (b]) = L for all (b], (c] ∈ (a]↑ where a, b ∈ L. Now

(m] = (((b ∧ m) (b∧c∧m)→ (c ∧ m)) ∨ ((c ∧ m) (b∧c∧m)→ (b ∧ m))]

= (((b ∧ m) (b∧c∧m)→(c ∧ m)]) ∨

(((c ∧ m) (b∧c∧m)→ (b ∧ m)])

= (((b ∧ m) (c∧m)→ (c ∧ m)) ∧ m] ∨

(((c ∧ m) (b∧m)→ (b ∧ m)) ∧ m]

= (((b ∧ m) (c∧m)→ (c ∧ m)) ∧ ((c ∧ m) (c∧m)→ (c ∧ m))] ∨

(((c ∧ m) (b∧m)→ (b ∧ m)) ∧ ((b ∧ m) (b∧m)→ (b ∧ m))]

= (((b ∨ c) ∧ m) (c∧m)→ (c ∧ m)] ∨

(((b ∨ c) ∧ m) (b∧m)→ (b ∧ m)] = ((b] → (c]) ∨ ((c] → (b])

= ((b] (a]→ (c]) ∨ ((c] (a]→ (b]).

Hence (a]↑ is an L-algebra. On PI(L) one can simply obtain that (a] → (b] =

((b] ∨ (a] ∨ (b]) (b]→ (b] where a, b ∈ L, and hence PI(L) is a skew L-algebra.

Conversely, suppose PI(L) be a skew L-algebra. For each x ∈ L we show that Lx is a skew L-algebra. For a, b ∈ Lx define a →x b = c ∧ x, for some c ∈ H such that (a] → (b] = (c]. As justified in the proof of Theorem 2.2.14, the binary operation

→x is well defined on Lx.

Since Lx is a distributive lattice it is co-strongly distributive skew lattice with top x. For any y ∈ Lx, we claim that y↑ is an L-algebra. Let a, b, c ∈ [y, x]. Clearly (a], (b], (c] ∈ (y]↑. From Theorem 2.2.14 it is proved that [y, x] is a Heyting algebra. Since (y]↑ is an L-algebra we have ((b] → (c]) ∨ ((c] → (b]) = (m] so that (u] ∨ (v] = (m] for some u, v ∈ H. Then (u ∨ v) ∧ x = m ∧ x. From the definition

95 of y→ on y↑, b y→ c = u ∧ x whenever (b] → (c] = (u]. Then we have

(b y→ c) ∨ (c y→ b) = (u ∧ x) ∨ (v ∧ x) = (u ∨ v) ∧ x

= m ∧ x

= x

Therefore [y, x] is an L-algebra.

Consequently the induced binary operation →x on Lx can be expressed as a →x b = (b ∨ a ∨ b) b→ b. Hence Lx is a skew L-algebra and therefore L is a skew L-ADL.

96 Chapter 5

Dual Skew Heyting Almost Distributive Lattices

Let (P, ≤) be a poset. Thus (P, ≥) is also a poset called the dual of (P, ≤). If φ is a statement about posets and in φ we replace all occurrences of ≤ by ≥, we get the dual of φ. The principle of duality states that ”If a statement φ is true in all posets, then its dual is also true in all posets ”. Likewise if we let φ be a statement about lattices expressed in terms of the binary operations ∧ and ∨, then the dual of φ is the statement we get from φ by interchanging ∧ and ∨. In this case, the principle of duality is given by ”if φ is true for all lattices, then the dual of φ is also true for all lattices ”. Unlike in lattices, the dual of an ADL is not an ADL in general. Clearly for

a each a ∈ L,(La, ∨, ∧) is a distributive lattice with top a but the ADL (L , ∨, ∧) is not necessarily a distributive lattice. Thus studying about the concepts of the duals of skew HADLs, skew SHADLs and skew L-ADLs is not a simple upside down process. For this reason, in this chapter we introduce the concepts of dual skew HADLs, dual skew SHADLs and dual skew L-ADLs. Through out this chapter for a non empty set L and for any a, b ∈ L we use the following notations:

97 (1) La = {a ∨ b|b ∈ L}, for any a ∈ L

(2) ←a is the binary operation defined on [0, a]

a (3) a← is the binary operation defined on L .

5.1 Dual Skew Heyting Almost Distributive Lat- tices

In this section, we introduce the concept of dual skew Heyting almost distributive lattices and characterize it in terms of dual skew Heyting algebras and congruence relations defined on it. We define an equivalence relation θ on a dual skew HADL L and we prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence classes are maximal rectangular subalgebras and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. We also investigate some of its algebraic properties. First we define the concept of dual skew Heyting algebra.

Definition 5.1.1. An algebra (L, ∨, ∧, ←, 0) of type (2, 2, 2, 0) is said to be a Dual Skew Heyting Algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 0) is a strongly distributive skew lattice with bottom 0

(2) For any z ∈ L a binary operation ←z can be defined on z↓ = {x ∈ L|x ≤ z}

so that (z↓, ∨, ∧, ←z, 0, z) is a dual Heyting algebra with bottom 0 and top z

(3) An induced binary operation ← is defined on L by x ← y = (y ∧ x ∧ y) ←y y.

Definition 5.1.2. An ADL L with 0 is said to be a Dual Skew Heyting Almost Distributive Lattice (Dual Skew HADL) if to each z ∈ L the algebra (Lz, ∨, ∧, z←, z) is a dual skew Heyting algebra.

98 Example 5.1.3. Let L be a skew ADL with 0. For any z ∈ L, Lz is a strongly distributive skew lattice with bottom z. Take b ∈ Lz and define a binary operation

←b on [z, b] by  z if x ≥ y x ←b y = y otherwise.

It is easy to show that ([z, b], ∨, ∧, ←b, z, b) is a dual Heyting algebra. z On L one can define z← by x z← y = x ←y y. Hence

x z← y = x ←y y

= (x ←y y) ∨ 0

= (x ←y y) ∨ (y ←y y)

= (x ∧ y) ←y y

= (y ∧ x ∧ y) ←y y,

z and we have (L , ∨, ∧, z←, z) is a dual skew Heyting algebra. Therefore (L, ∨, ∧, ← , 0) is a dual skew HADL.

Remark 5.1.1. A dual skew HADL is a dual skew Heyting algebra. The following theorem characterize dual skew HADL.

Theorem 5.1.4. Let L be an ADL with 0. Then the following conditions are equivalent.

(1) L is a dual skew HADL

(2) (i) for any z ∈ L, Lz is a skew lattice

(ii) for any b ∈ L, [z, b] is a dual HADL

(iii) there exists a binary operation ← on L defined by x ← y = (x ∧ y) ←y y.

Proof. Assume L be a dual skew HADL. This implies that for any z ∈ L, Lz is a dual skew Heyting algebra and hence it is a skew lattice. In particular L0 = L is a

99 dual skew Heyting algebra. Consequently, from the definition of dual skew Heyting

z algebra we have seen that for any b ∈ L , ([z, b], ∨, ∧, ←b, z, b) is a dual Heyting algebra. Hence [z, b] is a dual HADL. Since Lz is a dual skew Heyting algebra, the

z induced operation z← on L from ←a on [z, a], is given by xz←y = (y ∧x∧y) ←y y.

Thus it is possible to define a binary operation ← on L by x ← y = (y ∧ x ∧ y) ←y y = (x ∧ y) ←y y. Conversely, suppose condition (2) hold and let z ∈ L. Since meet in an ADL is distributive over the join and by (i), Lz is a strongly distributive skew lattice with bottom z. By (ii) for any b ∈ Lz, [z, b] is a dual HADL. Since [z, b] is a lattice ([z, b], ∨, ∧, ←b, z, b) is a dual Heyting algebra. Now using (ii) it is possible z to define z← on L by x z← y = (x ∧ y) ←y y. But x ∧ y = y ∧ x ∧ y. Hence x z← z y = (y ∧ x ∧ y) ←y y. Therefore (L , ∨, ∧, z←, z) is a dual skew Heyting algebra and hence L is a dual skew HADL.

Corollary 5.1.5. Let L be a dual skew HADL. Then for any a ∈ L, La is a dual Heyting algebra.

Proof. Clear by Theorem 5.1.4.

Lemma 5.1.6. Let L be a dual skew HADL. Then for any z ∈ L, Lz is a dual skew HADL.

Proof. Suppose L be a dual skew HADL. Then for any z ∈ L, Lz is a dual skew Heyting algebra. Take any y ∈ Lz as y ∈ L, Ly is also a dual skew Heyting algebra. Therefore Lz is a dual skew HADL.

Here is an important result that we use in the sequel.

Corollary 5.1.7. Let L be a dual skew HADL. If x, y ∈ L such that x ≤ y and a, b ∈ Lx, then a ←x b = a ←y b.

Proof. Let L be a dual skew HADL and x, y ∈ L such that x ≤ y. Then Lx ⊆ Ly and by Corollary 5.1.5 Lx and Ly are dual Heyting algebras. If a, b ∈ Lx, then

100 a ←x b ∈ Lx and hence a ←x b ∈ Ly. Since a, b ∈ Ly, a ←y b also belongs to Ly.

The minimal element characterization of a ←x b and a ←y b on the dual Heyting algebra Ly forces the two elements are equal.

Lemma 5.1.8. Let L be a dual skew HADL. Then the following conditions are satisfied:

(1) (x ← y) ∨ y = y

(2) (x ∧ y) ← z = (x ← z) ∨ (y ← z) for all x, y, z ∈ L.

Proof. Let L be a dual skew HADL and x, y, z ∈ L. Then (1) from Theorem 5.1.4(3) and Corollary 5.1.5 we obtain that (x ← y)∨y = ((x∧y) ←y y) ∨ y = y.

(2) Since Lz is a dual Heyting algebra we have

(x ∧ y) ← z = (x ∧ y ∧ z) ←z z

= ((x ∧ z) ∧ (y ∧ z)) ←z z

= ((x ∧ z) ←z z) ∨ ((y ∧ z) ←z z) = (x ← z) ∨ (y ← z).

The next theorem characterizes a dual skew HADL in terms of a congruence relation θ defined on it. First observe the following lemma.

Lemma 5.1.9. Let L be a dual skew HADL and x ∧ y = y for any x, y ∈ L. Then the following statements hold:

(1) x ← y = 0

(2) x ← z = y ← z, for any z ∈ L.

101 Proof. Let L be a dual skew HADL and x, y ∈ L such that x ∧ y = y. Clearly, x ∨ y = x ∨ (x ∧ y) = x and hence y ∧ x = x. Then

(1) x ← y = x ← (x ∧ y) = (x ∧ (x ∧ y)) ←(x∧y) (x ∧ y) = (x ∧ y) ←(x∧y) (x ∧ y) = 0. (2) Take z ∈ L, we obtain that

x ← z = (x ∧ z) ←z z

= ((y ∧ x) ∧ z) ←z z

= ((x ∧ y) ∧ z) ←z z

= (y ∧ z) ←z z = y ← z.

Lemma 5.1.10. Let L be a relatively complemented ADL with 0. If L is a dual skew HADL and θ is defined by

θ = {(x, y) ∈ L × L|x ∨ y = x and y ∨ x = y}, then θ is an equivalence relation on L.

Proof. Clearly L is a skew ADL. Suppose L be a dual skew HADL. Let x, y, z ∈ L. It is easily seen that θ is reflexive and symmetric. Assume that (x, y) ∈ θ and (y, z) ∈ θ. Then x ∨ y = x, y ∨ x = y, y ∨ z = y and z ∨ y = z. Hence z ∨ x = (z∨y)∨x = z∨(y∨x) = z∨y = z and x∨z = (x∨y)∨z = x∨(y∨z) = (x∨y) = x, it follows that (x, z) ∈ θ. Consequently θ is transitive and therefore it is an equivalence relation.

Next we show that θ is a congruence relation on each of the equivalence classes.

Theorem 5.1.11. Let a relatively complemented ADL L with 0 be a dual skew HADL. For each b ∈ L, consider the equivalence class [b]θ where θ is a relation given by Lemma 5.1.10. Then the following conditions hold:

102 (1) θ is a congruence relation on [b]θ

(2) For each x ∈ [b]θ, the congruence class [x]θ is the maximal rectangular subal- gebra of [b]θ

(3) [x]θ/θ is the maximal lattice image of [x]θ.

Proof. To show that θ is a congruence relation on [b]θ, for any x, y, a ∈ [b]θ we need to check that ((x ← a), (y ← a)) ∈ θ, ((a ← x), (a ← y)) ∈ θ and the substitution properties are satisfied. Clearly for any a, x, y ∈ [b]θ,(x∧a)∨(y ∧a) = (x∨y)∧a = x ∧ a and (y ∧ a) ∨ (x ∧ a) = (y ∨ x) ∧ a = y ∧ a. Thus ((x ∧ a), (y ∧ a)) ∈ θ. Also

(x ∨ a) ∧ (y ∨ a) = ((x ∨ a) ∧ y) ∨ ((x ∨ a) ∧ a)

= ((x ∨ a) ∧ y) ∨ a

= ((a ∨ x) ∧ y) ∨ a

= ((a ∧ y) ∨ (x ∧ y)) ∨ a

= ((a ∧ y) ∨ y) ∨ a

= y ∨ a.

Using the same procedure we have (y ∨ a) ∧ (x ∨ a) = x ∨ a. From these results we obtain that (x ∨ a) ∨ (y ∨ a) = (x ∨ a) ∨ ((x ∨ a) ∧ (y ∨ a)) = x ∨ a and (y ∨ a) ∨ (x ∨ a) = (y ∨ a) ∨ ((y ∨ a) ∧ (x ∨ a)) = y ∨ a. Thus ((x ∨ a), (y ∨ a)) ∈ θ and ((y ∨ a), (x ∨ a)) ∈ θ. This shows that θ satisfies substitution property. One can simply observe that x ∧ y = (x ∨ y) ∧ y = y and y ∧ x = (y ∨ x) ∧ x = x. Indeed (2) of Lemma 5.1.9 assures that ((x ← a), (y ← a)) ∈ θ. Also using Lemma 5.1.9(1) we obtain that (a ← x) ∨ (a ← y) = (a ← x) ∨ 0 = a ← x and (a ← y) ∨ (a ← x) = (a ← y) ∨ 0 = a ← y. Thus ((a ← x), (a ← y)) ∈ θ. Hence θ is a congruence relation on each equivalence class. Suppose y, z ∈ [b]θ. It is obvious that (y, z) ∈ θ. Following this we obtain that z ∧ y = (z ∨ y) ∧ (y ∨ z) = (y ∨ z) ∧ (y ∨ z) = y ∨ z so that each congruence class is

103 rectangular. Now take an arbitrary element x ∈ [b]θ and consider the congruence class [x]θ. Let T be a rectangular subalgebra of [b]θ such that [x]θ ⊆ T . Let r ∈ T . Since x ∈ T and T is a rectangular subalgebra of [b]θ we have r ∨ x = x ∧ r and x ∨ r = r ∧ x. Thus x ∧ r = (r ∧ x) ∧ r = (x ∨ r) ∧ r = r which implies that r ∨ x = x ∧ r = r and x ∨ r = x ∨ (x ∧ r) = x. Hence r ∈ [x]θ. Therefore T ⊆ [x]θ and we conclude that [x]θ = T . Hence [x]θ is maximal, i.e., each congruence class [x]θ is a maximal rectangular subalgebra of [b]θ. Next, to show that [x]θ/θ is a lattice we need to have the following results. Let y, z ∈ [x]θ. Then

z ∨ y = ((y ∨ z) ∧ z) ∨ y

= ((y ∨ z) ∧ z) ∨ ((z ∨ y) ∧ y)

= ((y ∨ z) ∧ z) ∨ ((y ∨ z) ∧ y)

= (y ∨ z) ∧ (z ∨ y).

Similarly,

y ∨ z = ((z ∨ y) ∧ y) ∨ z

= ((z ∨ y) ∧ y) ∨ ((y ∨ z) ∧ z)

= ((z ∨ y) ∧ y) ∨ ((z ∨ y) ∧ z)

= (z ∨ y) ∧ (y ∨ z).

Hence (z ∨ y) ∨ (y ∨ z) = (z ∨ y) ∨ ((z ∨ y) ∧ (y ∨ z)) = z ∨ y and (y ∨ z) ∨ (z ∨ y) = (y ∨ z) ∨ ((y ∨ z) ∧ (z ∨ y)) = y ∨ z. Therefore ((z ∨ y), (y ∨ z)) ∈ θ. Also we have

(z ∧ y) ∨ (y ∧ z) = (z ∧ y) ∨ (z ∧ y ∧ z)

= (z ∧ y) ∨ ((z ∧ y) ∧ z)

= z ∧ y

104 and

(y ∧ z) ∨ (z ∧ y) = (y ∧ z) ∨ (y ∧ z ∧ y)

= (y ∧ z) ∨ ((y ∧ z) ∧ y)

= y ∧ z.

Hence ((z ∧ y), (y ∧ z)) ∈ θ. Now assume that [y]θ, [z]θ ∈ [x]θ/θ. Define [y]θ ∨ [z]θ = [y ∨ z]θ and [y]θ ∧ [z]θ = [y ∧ z]θ. Then t ∈ [y ∨ z]θ ⇔ (t, (y ∨ z)) ∈ θ ⇔ (t, (z ∨ y)) ∈ θ ⇔ t ∈ [z ∨ y]θ. Thus [y]θ ∨ [z]θ = [z]θ ∨ [y]θ. Similarly, [y]θ ∧ [z]θ = [z]θ ∧ [y]θ. Therefore [x]θ/θ is a lattice. Let β be a congruence relation on [x]θ such that [x]θ/β is a lattice. Suppose y, z ∈ [x]θ. Clearly (y, z) ∈ θ. Also,

y ∨ z = y and z ∨ y = z ⇒ [y ∨ z]β = [y]β and [z ∨ y]β = [z]β

⇒ [y]β ∨ [z]β = [y]β and [z]β ∨ [y]β = [z]β

⇒ [y]β = [z]β, since [y]β ∨ [z]β = [z]β ∨ [y]β

⇒ (y, z) ∈ β.

Therefore θ ⊆ β. Suppose H be a lattice image of [x]θ. Then there exist an epimorphism f :[x]θ −→ H. Define the kernel K of f by

K = {(y, z) ∈ [x]θ × [x]θ|f(y) = f(z)}.

K is reflexive, symmetric and transitive. Let (x1, y1), (x2, y2) ∈ K. Since f is homomorphism we have f(x1 ∨ x2) = f(x1) ∨ f(x2) = f(y1) ∨ f(y2) = f(y1 ∨ y2) so that ((x1 ∨ x2), (y1 ∨ y2)) ∈ K. Using a similar procedure for ∧ and ← we obtain that ((x1 ∧ x2), (y1 ∧ y2)) ∈ K and ((x1 ← x2), (y1 ← y2)) ∈ K. This shows that K is a congruence relation on [x]θ. Hence we obtain that [x]θ/Kerf ∼= f([x]θ) = H. Therefore any lattice image of [x]θ is of the form [x]θ/φ for some congruence relation φ on [x]θ. Consider the set L = {[x]θ/φ|φ is a congruence on [x]θ and θ ⊆ φ } of lattices.

105 Now take congruence relations θ and β discussed as above and assume that [x]θ/θ ⊆ [x]θ/β. Let a, c ∈ L such that (a, c) ∈ β. Since θ ⊆ β and [x]θ/θ ⊆ [x]θ/β there exist a0, c0 ∈ [x]θ such that a ∈ [a]θ = [a0]β and c ∈ [c]θ = [c0]β. Thus (a, a0), (c, c0) ∈ θ and (a, a0), (c, c0) ∈ β. Hence (a, c), (c, c0) ∈ β ⇒ (a, c0) ∈ β ⇒ a ∈ [c0]β = [c]θ ⇒ (a, c) ∈ θ. Therefore β ⊆ θ and hence θ = β. Consequently, [x]θ/θ = [x]θ/β so that [x]θ/θ is the maximal lattice image of [x]θ.

Lemma 5.1.12. Let L be an ADL with 0 and z ∈ L. If L is a dual skew HADL, then for any w, x, y ∈ Lz the following conditions hold:

(1) x ≥ w ←z x

(2) (x ←z y) ∨ w = w if and only if y ∨ w ∨ x = w ∨ x.

Proof. Suppose L be a dual skew HADL and z ∈ L such that w, x, y ∈ Lz. Then by definition Lz is a dual skew Heyting algebra. This implies that L0 = L is a dual skew Heyting algebra so that Lz is a dual Heyting algebra. Thus (1) follows. Now, assume that (x ←z y) ∨ w = w. Then

w ∨ x = ((x ←z y) ∨ w) ∨ x

= ((x ←z y) ∨ x) ∨ w = x ∨ y ∨ w

= y ∨ w ∨ x.

On the other hand given that y ∨ w ∨ x = w ∨ x, we obtain x ←z (w ∨ x) = x ←z

(y ∨ w ∨ x). Hence (x ←z w) = (x ←z y) ∨ (x ←z w). Thus

(x ←z y) ∨ w = (x ←z y) ∨ ((x ←z w) ∨ w)

= ((x ←z y) ∨ (x ←z w)) ∨ w

= (x ←z w) ∨ w = w.

Therefore (2) is satisfied.

106 The following theorem gives a condition for an ADL to be a skew dual HADL.

Theorem 5.1.13. Let L be an ADL with 0 and for any z ∈ L, Lz is a skew lattice. Then L is a dual skew HADL if for any b ∈ Lz and w, x, y ∈ [z, b] the following conditions are satisfied:

(1) x ≥ w ←b x

(2) (x ←b y) ∨ w = w if and only if y ∨ w ∨ x = w ∨ x.

Proof. Assume that conditions (1) and (2) hold. For any b ∈ Lz and w, x, y ∈ [z, b] we have by (1), y ≥ x ←b y such that x ←b y ∈ [z, b] and then by (2) we get

w ≥ x ←b y ⇔ w = (x ←b y) ∨ w ⇔ w ∨ x = y ∨ w ∨ x

⇔ w ∨ x ≥ y.

z Hence [z, b] is a dual Heyting algebra. Define a binary operation z← on L by x z z← y = (y ∧ x ∧ y) ←y y makes L is a dual skew Heyting algebra and therefore L is a dual skew HADL.

Corollary 5.1.14. On a dual skew HADL L, (x ∧ y) ←y y = 0 if and only if x ∨ y = x.

Proof. Suppose that L be a dual skew HADL and (x ∧ y) ←y y = 0. Since Ly is a dual Heyting algebra (Corollary 5.1.5), we obtain that

x = x ∨ (x ∧ y)

= x ∨ ((x ∧ y) ∨ 0)

= x ∨ ((x ∧ y) ∨ ((x ∧ y) ←y y)) = x ∨ ((x ∧ y) ∨ y)

= x ∨ y.

Hence x ∨ y = x. The converse is straight forward.

107 Theorem 5.1.15. Let L be a skew ADL with 0. If the set PI(L) of all principal ideals of L is a dual skew Heyting algebra, then L is a dual skew HADL.

Proof. Suppose PI(L) be a dual skew Heyting algebra. For each x ∈ L we show

x x that L is a dual skew Heyting algebra. Define x← on L by a x← b = x ∨ t, for some t ∈ L such that (a] ← (b] = (t]. Let (u] = (v] for some u, v ∈ L. Since (u] ⊆ (v] and (v] ⊆ (u] implies that v ∧ u = u and u ∧ v = v respectively, we have u ∧ x = v ∧ u ∧ x = u ∧ v ∧ x = v ∧ x. Now take a, b, c, d ∈ Lx such that a = c, b = d. Then we obtain that (a] = (c] and (b] = (d]. Consequently (a] ← (b] = (c] ← (d].

Let (a] ← (b] = (e], (c] ← (d] = (f] for some e, f ∈ L, we have a x← b = x ∨ e and c x← d = x ∨ f. Hence x ∨ e = x ∨ f implies that a x← b = c x← d. Therefore x x the binary operation x← is well defined on L . It is clear that L is a strongly distributive skew lattice with bottom x. For any y ∈ Lx, we claim that [x, y] is a dual Heyting algebra. Let a, b, c ∈ [x, y] and define ←y on [x, y] by a ←y b = (a x← b) ∧ y. Clearly (a], (b], (c] ∈ (y]↓ and (y]↓ is a dual Heyting algebra. Then

(i)( a] ← (a] = (0]. Then we have a ←y a = (a x← a)∧y = (x∨0)∧y = x∧y = x.

(ii) Let (a] ← (b] = (t] for some t ∈ L. Then a x← b = x ∨ t. Therefore

(a ←y b) ∨ b = ((a x← b) ∧ y) ∨ (b ∧ y)

= ((a x← b) ∨ b) ∧ y = ((x ∨ t) ∨ b) ∧ y

= ((x ∨ b) ∨ t) ∧ y

= (b ∨ t) ∧ y

= b ∧ y

= b

This is because of (a] ← (b] ⊆ (b] ⇒ (t] ⊆ (b] ⇒ b ∧ t = t and hence b ∨ t = b ∨ (b ∧ t) = b.

108 (iii) Let (a] ← (b] = (t] for some t ∈ L. Then a x← b = x ∨ t. Now (a ∨ t] = (a] ∨ (t] = (a]∨((a] ← (b]) = (a]∨(b] = (a∨b]. Hence (a∨t)∧y = (a∨b)∧y = a∨b. Therefore

a ∨ (a ←y b) = a ∨ ((a x← b) ∧ y)

= (a ∧ y) ∨ ((a x← b) ∧ y)

= (a ∨ (a x← b)) ∧ y = (a ∨ (x ∨ t)) ∧ y

= ((a ∨ x) ∨ t) ∧ y

= (a ∨ t) ∧ y

= a ∨ b.

(iv) Let (a] ← (c] = (u] and (b] ← (c] = (v] for some u, v ∈ L. Then we obtain

that a x← c = x ∨ u and b x← c = x ∨ v. Further

(a ∧ b] ← (c] = ((a] ∧ (b]) ← (c]

= ((a] ← (c]) ∨ ((b] ← (c])

= (u] ∨ (v]

= (u ∨ v].

Hence we obtain that

(a ∧ b) ←y c = ((a ∧ b) x← c) ∧ y = (x ∨ (u ∨ v)) ∧ y

= ((x ∨ u) ∨ (x ∨ v)) ∧ y

= ((x ∨ u) ∧ y) ∨ ((x ∨ v) ∧ y)

= ((a x← c) ∧ y) ∨ ((b x← c) ∧ y)

= (a ←y c) ∨ (b ←y c).

109 (v) Let (a] ← (b] = (u] and (a] ← (c] = (v] for some u, v ∈ L. Then a x← b = x∨u

and a x← c = x ∨ v. Since

(a] ← (b ∨ c] = (a] ← ((b] ∨ (c])

= ((a] ← (b]) ∨ ((a] ← (c])

= (u] ∨ (v]

= (u ∨ v],

we get

a ←y (b ∨ c) = (a x← (b ∨ c)) ∧ y = (x ∨ (u ∨ v)) ∧ y

= ((x ∨ u) ∧ y) ∨ ((x ∨ v) ∧ y)

= ((a x← b) ∧ y) ∨ ((a x← c) ∧ y)

= (a ←y b) ∨ (a ←y c).

Hence [x, y] is a dual Heyting algebra.

x On PI(L) we have (a] ←(b] (b] = (a ←b b]. Then for a, b ∈ L we have the following

(a] ← (b] = ((b] ∧ (a] ∧ (b]) ←(b] (b]

= (b ∧ a ∧ b] ←(b] (b]

= ((b ∧ a ∧ b) ←b b].

Which implies that a x← b = x ∨ ((b ∧ a ∧ b) ←b b). Thus a x← b = (b ∧ a ∧ b) ←b b. Hence Lx is a dual skew Heyting algebra. Therefore L is a dual skew HADL.

110 5.2 Dual Skew Semi-Heyting Almost Distribu- tive Lattices

In this section, we introduce the concept of dual skew semi-Heyting almost dis- tributive lattices (dual skew SHADLs) and, characterize it in terms of dual skew semi-Heyting algebras and congruence relations defined on it. We define an equiv- alence relation θ on a dual skew SHADL L and we prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence classes are maximal rectangular subalgebras and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. We also investigate some of its algebraic properties. To define a skew HADL we use the concept of skew Heyting algebra, similarly to define dual skew SHADL we need to introduce the concept of dual skew semi- Heyting algebra. Thus we define the concept of dual skew semi-Heyting algebra as follows.

Definition 5.2.1. A lattice (L, ∨, ∧, 0, 1) is called a dual semi-Heyting algebra if its dual is a semi-Heyting algebra.

The following theorem is an immediate consequence of Definition 5.2.1.

Theorem 5.2.2. A lattice (L, ∨, ∧, 0, 1) is a dual semi-Heyting algebra if and only if there exists a binary operation ← satisfying the following conditions:

(1) x ∨ (x ← y) = x ∨ y

(2) x ∨ (y ← z) = x ∨ (x ∨ y ← x ∨ z)

(3) x ← x = 0 for all x, y, z ∈ L.

Definition 5.2.3. Let (L, ∨, ∧, 0, m) be an ADL with a maximal element m. An algebra (L, ∨, ∧, ←, 0, m) of type (2, 2, 2, 0, 0) is said to be a Dual Semi-Heyting

111 Almost Distributive Lattice (Dual SHADL) if for all x, y, z ∈ L the following con- ditions are satisfied:

(1)( x ∨ (x ← y)) ∧ m = (x ∨ y) ∧ m

(2)( x ∨ (y ← z)) ∧ m = [x ∨ ((x ∨ y) ← (x ∨ z))] ∧ m

(3)( x ← y) ∧ m = ((x ∧ m) ← (y ∧ m)) ∧ m

(4) x ← x = 0.

Theorem 5.2.4. Let L be an ADL with 0 and a maximal element m. Then the following are equivalent:

(1) L is dual semi-HADL

(2) [0, z] is a dual semi-Heyting algebra for any z ∈ L

(3) [0, m] is a dual semi-Heyting algebra.

Proof. Suppose L be a dual SHADL and z ∈ L. The dual of a bounded distribu- tive lattice ([0, z], ∨, ∧) is a bounded distributive lattice. Now we define a binary operation ←z on [0, z] by x ←z y = (x ← y) ∧ z. Consequently we have the following:

(1) x ∨ (x ←z y) = x ∨ ((x ← y) ∧ z) = (m ∧ x ∧ z) ∨ ((x ← y) ∧ m ∧ z)

= (x ∧ m ∧ z) ∨ {((x ∧ m) ← (y ∧ m)) ∧ m ∧ z}

= {(x ∧ m) ∨ {((x ∧ m) ← (y ∧ m)) ∧ m}} ∧ z

= {{(x ∧ m) ∨ ((x ∧ m) ← (y ∧ m))} ∧ m} ∧ z

= {((x ∧ m) ∨ (y ∧ m)) ∧ m} ∧ z

= (x ∨ y) ∧ m ∧ z

= x ∨ y.

112 (2) Using the definition of ←z on [0, z] we have

x ∨ (y ←z w) = x ∨ ((y ← w) ∧ z) = (x ∧ m ∧ z) ∨ ((y ← w) ∧ m ∧ z)

= ((x ∨ (y ← w)) ∧ m) ∧ z

= {{x ∨ ((x ∨ y) ← (x ∨ w))} ∧ m} ∧ z

= {x ∨ ((x ∨ y) ← (x ∨ w))} ∧ m ∧ z

= {x ∨ ((x ∨ y) ← (x ∨ w))} ∧ z

= (x ∧ z) ∨ {((x ∨ y) ← (x ∨ w)) ∧ z}

= x ∨ ((x ∨ y) ←z (x ∨ w)).

(3) x ←z x = (x ← x) ∧ z = 0 ∧ z = 0.

Hence [0, z] is a dual semi-Heyting algebra. This proves that (1) ⇒ (2). To prove (2) ⇒ (3) we take z = m and the result follows directly. Finally to show (3) ⇒ (1) we assume that [0, m] is a dual semi-Heyting algebra. Consider the binary operation ←m on [0, m]. Let w, x, y ∈ L and define ← on L by x ← y =

((x ∧ m) ←m (y ∧ m)) ∧ m. Thus we have the following results: (i) (x ∨ (x ← y)) ∧ m

= (x ∧ m) ∨ ((x ← y) ∧ m)

= (x ∧ m) ∨ {{((x ∧ m) ←m (y ∧ m)) ∧ m} ∧ m}

= (x ∧ m ∧ m) ∨ {((x ∧ m) ←m (y ∧ m)) ∧ m}

= {(x ∧ m) ∨ ((x ∧ m) ←m (y ∧ m))} ∧ m = ((x ∧ m) ∨ (y ∧ m)) ∧ m

= (x ∨ y) ∧ m ∧ m

= (x ∨ y) ∧ m.

113 (ii) (x ∨ (y ← w)) ∧ m

= (x ∧ m) ∨ ((y ← w) ∧ m)

= (x ∧ m) ∨ {{((y ∧ m) ←m (w ∧ m)) ∧ m} ∧ m}

= {(x ∧ m) ∨ ((y ∧ m) ←m (w ∧ m))} ∧ m

= {(x ∧ m) ∨ {((x ∧ m) ∨ (y ∧ m)) ←m ((x ∧ m) ∨ (w ∧ m))}} ∧ m

= {(x ∧ m) ∨ {((x ∨ y) ∧ m) ←m ((x ∨ w) ∧ m)}} ∧ m

= ((x ∧ m) ∧ m) ∨ {((x ∨ y) ∧ m) ←m ((x ∨ w) ∧ m)} ∧ m}

= ((m ∧ x) ∧ m) ∨ {{((x ∨ y) ∧ m) ←m ((x ∨ w) ∧ m)} ∧ m ∧ m}

= {(m ∧ x) ∨ {{((x ∨ y) ∧ m) ←m ((x ∨ w) ∧ m)} ∧ m}} ∧ m = {x ∨ ((x ∨ y) ← (x ∨ w))} ∧ m.

(iii) Using the definition of ← given above we have

((x ∧ m) ← (y ∧ m)) ∧ m = {{((x ∧ m) ∧ m) ←m ((y ∧ m) ∧ m)} ∧ m} ∧ m

= {((x ∧ m) ←m (y ∧ m)) ∧ m} ∧ m = (x ← y) ∧ m.

(iv) x ← x = ((x ∧ m) ←m (x ∧ m)) ∧ m = 0 ∧ m = 0. Hence (i)-(iv) above shows that L is a dual SHADL.

Theorem 5.2.5. Let (L, ∨, ∧, ←, 0, m) be a dual SHADL and a, b, c, d, x ∈ L. Then the following conditions hold:

(1) (0 ← a) ∧ m = a ∧ m

(2) If a ≥ b, then [a ∨ (b ← c)] ∧ m = (a ∨ c) ∧ m

(3) If a ≥ b, then a ∧ m ≥ (b ← 0) ∧ m

(4) If a ≥ b, then a ∧ m ≥ (a ← b) ∧ m

(5) (a ∨ b) ∧ m ≥ (a ← b) ∧ m

114 (6) If a ≥ b, then a ∧ m ≥ (b ← a) ∧ m

(7) If a ≥ b and a ≥ c, then a ∧ m ≥ (b ← c) ∧ m

(8) If x ≥ a ← b, then (x ∨ a) ∧ m ≥ b ∧ m

(9) If a ← b = 0, then a ∧ m ≥ b ∧ m

(10) a ∧ m ≥ ((a ∨ b) ← b) ∧ m

(11) If a ≥ b ≥ c, then [b ∨ (a ← c)] ∧ m = [b ∨ (a ← b)] ∧ m

(12) a ∧ m ≥ [(a ← b) ← b] ∧ m

(13) a ∧ m ≥ [((a ∨ b) ← b) ← a] ∧ m

(14) a ∧ m ≥ [a ← (b ← (a ∨ b))] ∧ m

(15) If a ≤ b, then (b ← a) ∧ m ≥ (a ← b) ∧ m

(16) (a ← b) ∧ m ≥ (a ← (a ∨ b)) ∧ m

(17) If a ← (b ∧ m) = 0, then (a ∨ b) ∧ m = a ∧ m

(18) b ∧ m ≥ (a ←) ∧ m

(19) If b ≥ a ← m, then (a ∨ b) ∧ m = m

(20) (x ← y) ∧ m ≤ (x ∨ y) ∧ m.

Next we introduce the concept of dual skew SHADLs. First we define the concept of dual skew semi-Heyting algebra.

Definition 5.2.6. An algebra (L, ∨, ∧, ←, 0) of type (2, 2, 2, 0) is said to be a dual skew semi-Heyting algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 0) is a strongly distributive skew lattice with bottom 0

115 (2) For any z ∈ L a binary operation ←z can be defined on z↓ = {x ∈ L|x ≤ z}

such that (z↓, ∨, ∧, ←z, 0, z) is a dual semi-Heyting algebra with bottom 0 and top z

(3) An induced binary operation ← is defined on L by x ← y = (y ∧ x ∧ y) ←y y.

Definition 5.2.7. An ADL L with 0 is said to be a Dual Skew Semi-Heyting Almost Distributive Lattice (Dual Skew SHADL) if to each z ∈ L the algebra (Lz, ∨, ∧, ←, z) is a dual skew semi-Heyting algebra.

Example 5.2.8. Let L be a dual semi-Heyting algebra. Consider L = L\{1} and take an element z ∈ L. Define a binary operation ←z on Lz by x ←z y = (y ∧ x ∧ y) ←y y we obtain that (Lz, ∨, ∧, ←z, z) is a dual skew semi-Heyting algebra. This is due to the fact that L is strongly distributive skew lattice and for any a ∈ L, La is a dual semi-Heyting algebra. In turn this shows that for each w ∈ Lz, Lw is a dual semi-Hyeting algebra. Therefore we can deduce that L on which for each z ∈ L the binary operation ←z can be defined on Lz as above is a dual skew SHADL.

Theorem 5.2.9. Let L be an ADL with 0. Then the following conditions are equivalent.

(1) L is a dual skew SHADL

(2) (i) for any z ∈ L, Lz is a skew lattice

(ii) for any b ∈ L, Lb is a dual SHADL

(iii) there exists a binary operation ← on L defined by x ← y = (x ∧ y) ←y y.

Proof. Assume L be a dual skew SHADL. This implies that for any z ∈ L, Lz is a dual skew semi-Heyting algebra and hence it is a skew lattice. In particular L0 = L is a dual skew semi-Heyting algebra. Consequently, from the definition of dual skew

z semi-Heyting algebra we have seen that for any b ∈ L , ([z, b], ∨, ∧, ←b, z, b) is a dual semi-Heyting algebra. Hence [z, b] is a dual SHADL. Since Lz is a dual skew

116 z semi-Heyting algebra, the induced operation z← on L from ←a on [z, a], is given by xz←y = (y ∧ x ∧ y) ←y y. Thus it is possible to define a binary operation ← on

L by x ← y = (y ∧ x ∧ y) ←y y = (x ∧ y) ←y y. Conversely, suppose condition (2) hold and let z ∈ L. Since meet in an ADL is distributive over the join and by (i), Lz is a strongly distributive skew lattice with bottom z. By (ii) for any b ∈ Lz, [z, b] is a dual SHADL. Since [z, b] is a lattice

([z, b], ∨, ∧, ←b, z, b) is a dual semi-Heyting algebra. Now using (ii) it is possible z to define z← on L by x z← y = (x ∧ y) ←y y. But x ∧ y = y ∧ x ∧ y. Hence x z z← y = (y ∧ x ∧ y) ←y y. Therefore (L , ∨, ∧, z←, z) is a dual skew semi-Heyting algebra and hence L is a dual skew SHADL.

Corollary 5.2.10. Let L be a dual skew SHADL. Then for any z ∈ L, Lz is a dual semi-Heyting algebra.

Proof. It is clear by Theorem 5.2.4 and Theorem 5.2.9 (2).

The following lemma is analogous with the statement, any interval on a dual Heyting algebra is again a dual Heyting algebra.

Lemma 5.2.11. Let L be a dual skew SHADL. Then for any z ∈ L, Lz is a dual skew SHADL.

Proof. Suppose L be a dual skew SHADL. Then for any z ∈ L, Lz is a dual skew semi-Heyting algebra with bottom element z. Take any y ∈ Lz as y ∈ L, Ly is also a dual skew semi-Heyting algebra. Therefore Lz is a dual skew SHADL.

Corollary 5.2.12. Let L be a dual skew SHADL. If x, y ∈ L such that x ≤ y and a, b ∈ Lx, then a ←x b = a ←y b.

Proof. Let L be a dual skew SHADL and x, y ∈ L such that x ≤ y. Then Lx ⊆ Ly.

If a, b ∈ Lx, then a ←x b ∈ Lx and hence a ←x b ∈ Ly. Since a, b ∈ Ly, a ←y b also belongs to Ly. The minimal element characterization of a ←x b and a ←y b on the dual semi-Heyting algebra Ly forces the two elements are equal.

117 The next theorem characterizes a dual skew SHADL in terms of a congruence relation θ defined on the equivalence class [x]θ. To prove the Theorem we use the following lemma.

Lemma 5.2.13. Let L be a dual skew SHADL and for any x, y ∈ L, x ∧ y = y. Then the following statements hold:

(1) x ← y = 0

(2) x ← z = y ← z.

Theorem 5.2.14. Let a relatively complemented ADL L with 0 be a dual skew SHADL. Let θ be an equivalence relation on L. For each b ∈ L, consider the equivalence class [b]θ. Then the following conditions hold:

(1) θ is a congruence relation on [b]θ

(2) For each a ∈ [b]θ, the congruence class [a]θ is the maximal rectangular subal- gebra of [b]θ

(3) [a]θ/θ is the maximal lattice image of [a]θ.

Theorem 5.2.15. Let L be an ADL with 0 and for any z ∈ L, Lz is a skew lattice. Then L is a dual skew SHADL if for any b ∈ Lz and w, x, y ∈ [z, b] the following conditions are satisfied:

(1) x ≥ w ←b x

(2) (x ←b y) ∨ w = w if and only if y ∨ w ∨ x = w ∨ x.

Proof. From the assumption we have Lz is a strongly distributive skew lattice. Suppose conditions (1) and (2) hold. For any b ∈ Lz and w, x, y ∈ [z, b] by (1) we have, y ≥ x ←b y such that x ←b y ∈ [z, b] and then by (2) we get

w ≥ x ←b y ⇔ w = (x ←b y) ∨ w ⇔ w ∨ x = y ∨ w ∨ x

⇔ w ∨ x ≥ y.

118 Hence [z, b] is a dual Heyting algebra. This implies that [z, b] is a dual semi-

z Heyting algebra. Define a binary operation z← on L by x z← y = (y ∧ x ∧ y) ←y y which makes Lz a dual skew semi-Heyting algebra and therefore L is a dual skew SHADL.

Corollary 5.2.16. On a dual skew SHADL L, (x ∧ y) ←y y = 0 if and only if x ∧ y = y.

Proof. Suppose that L be a dual skew SHADL and (x ∧ y) ←y y = 0 for any x, y ∈ L. Then

x ∧ y = (x ∧ y) ∨ 0

= (x ∧ y) ∨ ((x ∧ y) ←y y) = (x ∧ y) ∨ y

= y.

Hence x ∧ y = y. The converse is straight forward.

Theorem 5.2.17. Let L be a skew ADL with 0. If the set PI(L) of all principal ideals of L is a dual skew semi-Heyting algebra, then L is a dual skew SHADL.

Proof. Suppose PI(L) be a dual skew semi-Heyting algebra. For each x ∈ L we

x x show that L is a dual skew semi-Heyting algebra. Define x← on L by a x← b = x ∨ c, for some t ∈ L such that (a] ← (b] = (t]. It can be seen easily that the

x binary operation x← is well defined on L . It is clear that Lx is a strongly distributive skew lattice with bottom x. For any y ∈ Lx, we claim that [x, y] is a dual semi-Heyting algebra. Let a, b, c ∈ [x, y] and define ←y on [x, y] by a ←y b = (a x← b) ∧ y. Clearly (a], (b], (c] ∈ (y]↓ and (y]↓ is a dual semi-Heyting algebra. Then

(i) Since (a] ← (a] = (0]. We have a ←y a = (a x← a)∧y = (x∨0)∧y = x∧y = x.

119 (ii) Let (a] ← (b] = (t] for some t ∈ L. Then a x← b = x ∨ t. Now (a ∨ t] = (a] ∨ (t] = (a]∨((a] ← (b]) = (a]∨(b] = (a∨b]. Hence (a∨t)∧y = (a∨b)∧y = a∨b. Therefore

a ∨ (a ←y b) = a ∨ ((a x← b) ∧ y)

= (a ∧ y) ∨ ((a x← b) ∧ y)

= (a ∨ (a x← b)) ∧ y = (a ∨ (x ∨ t)) ∧ y

= ((a ∨ x) ∨ t) ∧ y

= (a ∨ t) ∧ y

= a ∨ b.

(iii) Let (b] ← (c] = (t] for some t ∈ L. Then b x← c = x ∨ t which implies that

b ←y c = (b x← c) ∧ y = (x ∨ t) ∧ y. Now (a] ∨ ((b] ← (c]) = (a] ∨ (t] = (a ∨ t] ⇒ (a] ∨ ((a ∨ b] ← (a ∨ c]) = (a ∨ t].

120 Thus

(a ∨ ((a ∨ b) ←y (a ∨ c))] = (a] ∨ ((a ∨ b) ←y (a ∨ c)]

= (a ∧ y] ∨ (((a ∨ b) x← (a ∨ c)) ∧ y]

= {(a] ∨ (((a ∨ b) x← (a ∨ c))]} ∧ (y]

= (a ∨ ((a ∨ b) x← (a ∨ c))] ∧ (y]

= (a ∨ (x ∨ q)] ∧ (y], where (a ∨ b)x←(a ∨ c) = x ∨ q for some q ∈ L such that (a ∨ b]←(a ∨ c] = (q]

= ((x ∨ a) ∨ q] ∧ (y]

= (a ∨ q] ∧ (y]

= ((a] ∨ (q]) ∧ (y]

= ((a] ∨ ((a ∨ b] ← (a ∨ c])) ∧ (y]

= {(a] ∨ (((a] ∨ (b]) ← ((a] ∨ (c]))} ∧ (y]

= ((a] ∨ ((b] ← (c])) ∧ (y]

= ((a] ∨ (t]) ∧ (y]

= (((a ∨ x) ∨ t) ∧ y]

= (((a ∨ (x ∨ t)) ∧ y]

= ((a ∧ y) ∨ ((x ∨ t) ∧ y)]

= (a ∨ (b ←y c)].

Hence (a ∨ ((a ∨ b) ←y (a ∨ c))) ∧ y = (a ∨ (b ←y c)) ∧ y so that a ∨ (b ←y c) = a ∨ ((a ∨ b) ←y (a ∨ c)). Therefore [x, y] is a dual semi-Heyting algebra. x On PI(L) we have (a] ←(b] (b] = (a ←b b]. Then for a, b ∈ L we have the following

(a] ← (b] = ((b] ∧ (a] ∧ (b]) ←(b] (b]

= (b ∧ a ∧ b] ←(b] (b]

= ((b ∧ a ∧ b) ←b b].

121 Consequently, a x← b = x∨((b∧a∧b) ←b b). Thus a x← b = (b∧a∧b) ←b b. Hence Lx is a dual skew semi-Heyting algebra. Therefore, L is a dual skew SHADL.

5.3 Dual Skew L-Almost Distributive Lattices

In this section, we introduce the concept of dual skew L-ADLs. We define an equivalence relation θ on a dual skew L-ADL L and we prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence classes are maximal rectangular subalgebras and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. We also investigate some of its algebraic properties.

Definition 5.3.1. An algebra (L, ∨, ∧, ←, 0) of type (2, 2, 2, 0) is said to be a dual skew L-algebra whenever the following conditions are satisfied:

(1)( L, ∨, ∧, 0) is a strongly distributive skew lattice with bottom 0

(2) For any z ∈ L, an operation ←z can be defined on z↓ = {x ∈ L|x ≤ z} such

that (z↓, ∨, ∧, ←z, 0, z) is a dual L-algebra with top z and bottom 0

(3) An induced binary operation ← is defined on L by x ← y = (y ∧ x ∧ y) ←y y.

Theorem 5.3.2. Let (L, ∨, ∧, 0) be a strongly distributive skew lattice with bottom 0. Then (L, ∨, ∧, ←, 0) is a dual skew L-algebra if and only if it satisfies the following axioms:

(a) (x ←z y) ∧ (y ←z x) = 1, ∀z ∈ L such that x, y ∈ z↑

(b) x ← x = 1, ∀x ∈ L

(c) x ∨ (x ← y) ∨ x = x ∨ y ∨ x, ∀x, y ∈ L

(d) y ∨ (x ← y) = y = (x ← y) ∨ y, ∀x, y ∈ L

122 (e) x ← y = (y ∧ x ∨ y) ← y, ∀x, y ∈ L

(f) x ← (w ∧ (y ∨ z) ∧ w) = (x ← (w ∧ y ∧ w)) ∨ (x ← (w ∧ z ∧ w))

for all w, x, y, z ∈ L.

Theorem 5.3.3. Let L be a dual skew Heyting algebra. Then L is a dual skew

L-algebra if for each z ∈ L such that w, x, y ∈ z↓, either x ←z (y ∧ w) = (x ←z y) ∧ (x ←z w) or (x ∨ y) ←z w = (x ←z w) ∧ (y ←z w).

Definition 5.3.4. Let L be an ADL with 0 and z ∈ L. Then L is said to be a dual Skew L-Almost Distributive Lattice( dual skew L-ADL), whenever the algebra

z (L , ∨, ∧, z←, z) is a dual skew L-algebra.

Example 5.3.5. Let L be an ADL with 0. For any z ∈ L we define a binary

z operation z← on L by  z if x ≥ y x z← y = y otherwise.

z For any b ∈ L if we define a binary operation ←b on [z, b] by x ←b y = (x z← y) ∧ b. Either of (x ←b y) or (y ←b x) is z and the other is y. Thus we obtain that

(x ←b y) ∧ (y ←b x) = ((x z← y) ∧ b) ∧ ((y z← x) ∧ b)

= ((xz← y) ∧ (y z← x)) ∧ b = z ∧ y ∧ b

= z ∧ b

= z

so that [z, b] is a dual L-algebra. Now, if we take b = y the binary operation ←y on

123 [z, y] is given by x ←y y = (x z← y) ∧ y = x z← y. Hence

x z← y = x ←y y

= (x ←y y) ∨ 0

= (x ←y y) ∨ (y ←y y)

= (x ∧ y) ←y y

= (y ∧ x ∧ y) ←y y,

z and we have (L , ∨, ∧, z←, z) is a dual skew L-algebra. Hence (L, ∨, ∧, ←, 0) is a dual skew L-ADL.

Theorem 5.3.6. Let L be an ADL with 0. Then the following conditions are equivalent.

(1) L is a dual skew L-ADL

(2) (i) for any z ∈ L, Lz is a skew lattice

(ii) for any b ∈ L, Lb is a dual L-ADL

(iii) there exists a binary operation ← on L defined by x ← y = (x ∧ y) ←y y.

Proof. Assume L be a dual skew L-ADL. This implies that for any z ∈ L, Lz is a dual skew L-algebra and hence it is a skew lattice. In particular L0 = L is a dual skew L-algebra. Consequently, from the definition of dual skew L-algebra we have

z seen that for any b ∈ L , ([z, b], ∨, ∧, ←b, z, b) is a dual L-algebra. Hence [z, b] is a z z dual L-ADL. Since L is a dual skew L-algebra, the induced operation z← on L from ←a on [z, a], is given by xz←y = (y ∧x∧y) ←y y. Thus it is possible to define a binary operation ← on L by x ← y = (y ∧ x ∧ y) ←y y = (x ∧ y) ←y y. Conversely, suppose condition (2) hold and let z ∈ L. Since meet in an ADL is distributive over the join and by (i), Lz is a strongly distributive skew lattice with bottom z. By (ii) for any b ∈ Lz, [z, b] is a dual L-ADL. Since [z, b] is a lattice ([z, b], ∨, ∧, ←b, z, b) is a dual L-algebra. Now using (ii) it is possible to

124 z define z← on L by x z← y = (x ∧ y) ←y y. But x ∧ y = y ∧ x ∧ y. Hence x z← z y = (y ∧x∧y) ←y y. Therefore (L , ∨, ∧, z←, z) is a dual skew L-algebra and hence L is a dual skew L-ADL.

Corollary 5.3.7. Let L be a dual skew L-ADL. Then for any z ∈ L, Lz is a dual L-algebra.

Proof. It is a direct consequence of Theorem 5.3.6 (2).

The following lemma is analogous with the statement, any interval on a dual L-algebra is again a dual L-algebra.

Lemma 5.3.8. Let L be a dual skew L-ADL. Then for any z ∈ L, Lz is a dual skew L-ADL.

Proof. Suppose L is a dual skew L-ADL. Then for any z ∈ L, Lz is a dual skew L- algebra. Take any y ∈ Lz. Since y ∈ L, Ly is also a dual skew L-algebra. Therefore Lz is a dual skew L-ADL.

Corollary 5.3.9. Let L be a dual skew L-ADL. If x, y ∈ L such that x ≤ y and a, b ∈ Lx, then a ←x b = a ←y b.

Proof. Let L be a dual skew L-ADL and x, y ∈ L such that x ≤ y. Then Lx ⊆ Ly.

If a, b ∈ Lx, then a ←x b ∈ Lx and hence a ←x b ∈ Ly. Since a, b ∈ Ly, a ←y b also belongs to Ly. The minimal element characterization of a ←x b and a ←y b on the dual L- algebra Ly forces the two elements are equal.

In the next theorem we give an equivalence relation defined on a dual skew L-ADL. The proof is similar to Lemma 5.1.10.

Theorem 5.3.10. Let L be a relatively complemented ADL with 0. If L is a dual skew L-ADL, then a relation θ defined by

θ = {(x, y) ∈ L × L|x ∨ y = x and y ∨ x = y}

125 is an equivalence relation on L.

The following theorem shows that the relation given by Theorem 5.3.10 is a congruence relation on each equivalence class.

Theorem 5.3.11. Let L be a relatively complemented ADL with 0 such that it is a dual skew L-ADL. For each b ∈ L, consider the equivalence class [b]θ, where θ is given by the above theorem. Then the following conditions hold:

(1) θ is a congruence relation on [b]θ

(2) For each a ∈ [b]θ, the congruence classes [a]θ are the maximal rectangular subalgebras of [b]θ

(3) [a]θ/θ is the maximal lattice image of [a]θ.

Remark 5.3.1. Dual skew L-algebra is a generalization of dual L-algebra. It has a bottom element and need not to contain top element. If a dual skew L-algebra contains a top element, then it is a dual L-ADL.

Lemma 5.3.12. Let L be an ADL with 0 and z ∈ L. If L is a dual skew L-ADL, then for any w, x, y ∈ Lz the following conditions hold:

(1) x ≥ w ←z x

(2) (x ←z y) ∧ (y ←z x) = 0

(3) (x ←z y) ∨ w = w if and only if y ∨ w ∨ x = w ∨ x.

Proof. Suppose L be a dual skew L-ADL and z ∈ L such that w, x, y ∈ Lz. Then by definition Lz is a dual skew L-algebra. This implies that L0 = L is a dual skew

L-algebra so that Lz is a dual L-algebra. Thus (1) and (2) follows. Now, assume

126 that (x ←z y) ∨ w = w. Then

w ∨ x = ((x ←z y) ∨ w) ∨ x

= ((x ←z y) ∨ x) ∨ w = x ∨ y ∨ w

= y ∨ w ∨ x.

On the other hand given that y ∨ w ∨ x = w ∨ x, we obtain x ←z (w ∨ x) = x ←z

(y ∨ w ∨ x). Hence (x ←z w) = (x ←z y) ∨ (x ←z w). Thus

(x ←z y) ∨ w = (x ←z y) ∨ ((x ←z w) ∨ w)

= ((x ←z y) ∨ (x ←z w)) ∨ w

= (x ←z w) ∨ w = w.

Therefore (3) is satisfied.

Theorem 5.3.13. Let L be an ADL with 0 and for any z ∈ L, Lz is a strongly distributive skew lattice. Then L is a dual skew L-ADL if for any b ∈ Lz and w, x, y ∈ [z, b] the following conditions are satisfied:

(1) x ≥ w ←b x

(2) (x ←b y) ∨ w = w if and only if y ∨ w ∨ x = w ∨ x.

(3) (x ←b y) ∧ (y ←b x) = 0

Proof. Assume that conditions (1) and (2) hold. For any b ∈ Lz and w, x, y ∈ [z, b] we have by (1), y ≥ x ←b y such that x ←b y ∈ [z, b] and then by (2) we get

w ≥ x ←b y ⇔ w = (x ←b y) ∨ w ⇔ w ∨ x = y ∨ w ∨ x

⇔ w ∨ x ≥ y.

127 Hence [z, b] is a dual Heyting algebra. Thus (3) assures that [z, b] is a dual L-

z algebra. Now define a binary operation z← on L by x z← y = (y ∧ x ∧ y) ←y y makes Lz is a dual skew L-algebra and therefore L is a dual skew L-ADL.

Theorem 5.3.14. Let L be a skew ADL with 0. If the set PI(L) of all principal ideals of L is a dual skew L-algebra, then L is a dual skew L-ADL.

Proof. Suppose PI(L) be a dual skew L-algebra. For each x ∈ L we show that Lx

x is a dual skew L-algebra. Define x← on L by a x← b = x ∨ t, for some t ∈ L such that (a] ← (b] = (t]. In Theorem 5.1.15 we have proved that the binary operation

x x x← is well defined on L . It is clear that L is a strongly distributive skew lattice with bottom element x. For any y ∈ Lx, we claim that [x, y] is a dual L-algebra.

Let c, d ∈ [x, y] and define ←y on [x, y] by c ←y d = (c x← d)∧y. Let (c], (d] ∈ (y]↓.

Let (c] ← (d] = (u] and (d] ← (c] = (v] for some u, v ∈ L. Then c x← d = x ∨ u and d x← c = x ∨ v. Since (y]↓ is a dual L-algebra we have ((c] ← (d]) ∧ ((d] ← (c]) = (u] ∧ (v] implies that (0] = (u ∧ v]. Hence u ∧ v = 0 so that

(c ←y d) ∧ (d ←y c) = ((c x← d) ∧ y) ∧ ((d x← c) ∧ y) = ((x ∨ u) ∧ y) ∧ ((x ∨ v) ∧ y)

= (x ∨ (u ∧ v)) ∧ y

= (x ∨ 0) ∧ y

= x ∧ y

= x.

This shows that [x, y] is a dual L-algebra. On PI(L) we have (a] ←(b] (b] = (a ←b b]. x Then for a, b ∈ L ,(a] ← (b] = ((b] ∧ (a] ∧ (b]) ←(b] (b] = (b ∧ a ∧ b] ←(b] (b] =

((b ∧ a ∧ b) ←b b]. Consequently, a x← b = x ∨ ((b ∧ a ∧ b) ←b b). Thus a x← x b = (b ∧ a ∧ b) ←b b. Hence L is a dual skew Heyting algebra. Therefore L is a dual skew L-ADL.

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