Filters in Residuated Skew Lattices: Part Ii

Honam Mathematical J. 40 (2018), No. 3, pp. 401–431 https://doi.org/10.5831/HMJ.2018.40.3.401

(SKEW) FILTERS IN RESIDUATED SKEW LATTICES: PART II

R. Koohnavard and A. Borumand Saeid∗

Abstract. In this paper, some kinds of (skew) filters are defined and are studied in residuated skew lattices. Some relations are got between these filters and quotient algebras constructed via these filters. The Green filter is defined which establishes a connection between residuated lattices and residuated skew lattices. It is investigated that relationships between Green filter and other types of filters in residuated skew lattices and the relationship between residuated skew lattice and other skew structures are studied. It is proved that for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent too.

1. Introduction

Skew lattices were introduced for the first time by Jordan [10]. The study of skew lattices began with the 1989 paper of Leech [16]. Skew lattices are a generalization of lattices that operations ∨, ∧ are not commutative. In skew lattices, two different order concepts can be defined: the natural preorder, denoted by and the natural partial order denoted by ≤. For given a skew lattice (A, ∨, ∧) both algebraic reducts (A, ∨) and (A, ∧) are bands, that is semigroups whose elements are idempotent (x2 = x). J. Leech explained skew Boolean algebras are non-commutative one- pointed generalisations of (possibly generalised) Boolean algebras [14], each right-handed skew Boolean algebra can be embedded into a generic skew Boolean algebra of partial functions from a given set to the codomain

Received October 28, 2017. Revised May 12, 2018. Accepted May 21, 2018. 2010 Mathematics Subject Classiﬁcation. 03G10, 06A75, 03G25. Key words and phrases. residuated skew lattice, (Stonean, (positive) implicative, fantastic, easy) (skew) ﬁlter. *Corresponding author 402 R. Koohnavard and A. Borumand Saeid

{0, 1} [7], also defined normal and conormal skew lattices [12]. K. Cvetko- Vah defined skew Heyting algebra as dual skew Boolean algebra [7]. J. Pita Costa, expressed the notion of (skew) ideals and (skew) filters in skew lattices [18]. A. Borumand Saeid and R. Koohnavard defined residuated skew lattices as non-commutative generalization of residuated lattices. They defined deductive system and skew deductive system in residuated skew lattices and showed that the class of all conormal residuated skew lattices that x → y = (y ∨ x ∨ y) →y y forms a variety too. They showed that Green’s relation D is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. The authors investigated that for any filter F of a residuated skew lattice A can associate a congruence and conversely [2]. We showed that in residuated skew lattice, deductive system and filter are equivalent, but skew deductive system and skew filter are not equivalent and any skew deductive system is a skew filter under some conditions [11]. For better underestanding of residuated skew lattices, we investigate filters and use them for classification and obtain residuated skew lattices structural properties. The filter theory in residuated skew lattices plays an important role. From a logic point of view, various filters have natural interpretation as various sets of provable formulas. In this direction, we define some types of (skew) filters in residuated skew lattice and investigate relationships among them. Also we define Green filter and show that quotient algebra via this filter is a residuated lattice and we state and prove the relationship between Green filter and other types of filters. We get some relations between other types of filters and their quotient algebras, for example if F is an easy filter, then A/F is a skew semi-G-algebra. We prove that A is a skew G-algebra iff any filter of A is an implicative filter and for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent. We show that skew Boolean algebra is not a residuated skew lattice.

2. Preliminaries

In this section, we review some properties of skew lattices and residuated skew lattices which we need in the sequel.

Definition 2.1 ([1]). A skew lattice is an algebra (A, ∨, ∧) of type (2, 2) such that satisﬁes in the following identities: (i)( x ∨ y) ∨ z = x ∨ (y ∨ z) and (x ∧ y) ∧ z = x ∧ (y ∧ z), (ii) x ∧ x = x and x ∨ x = x, (Skew) ﬁlters in residuated skew lattices: Part II 403

(iii) x ∧ (x ∨ y) = x = x ∨ (x ∧ y) and (x ∧ y) ∨ y = y = (x ∨ y) ∧ y. The identities found in (i − iii) are known as the associative law, the idempotent laws and absorption laws respectively. In view of the associativity (i), we can omit parentheses when no ambiguity arises. On a given skew lattice A the natural partial order ≤ and natural preorder respectively are defined by x ≤ y iff x ∧ y = x = y ∧ x or dually x ∨ y = y = y ∨ x and x y if and only if y ∨ x ∨ y = y or equivalently x ∧ y ∧ x = x. Relation D is defined by x D y iff x ∨ y ∨ x = x and y ∨ x ∨ y = y or dually, x ∧ y ∧ x = x and y ∧ x ∧ y = y. D is called the natural equivalence and it coincides with Green’s relation D on both semigroups (A, ∧) and (A, ∨) [5]. For any elements x, y of a skew lattice A, x D y iff x ∨ y = y ∧ x [5]. A pair of natural congruences, L and R, refine D . We say that x is L- related to y (denoted xLy) if x∧y = x and y∧x = y, or dually, x∨y = y and y ∨ x = x. Likewise, x and y are R-related (x R y) if x ∧ y = y and y∧x = x, or dually, x∨y = x and y∨x = y. A skew lattice is left-handed if D = L so that x ∧ y = x = y ∨ x on each rectangular subalgebra and is right-handed if D = R so that x ∧ y = y = y ∨ x on each rectangular subalgebra [5]. Left-handed skew lattices can be defined by similar identities: x ∨ y ∨ x = y ∨x or equivalently x∧y ∧x = x∧y and right-handed skew lattices are the ones satisfying the identity x ∧ y ∧ x = y ∧ x or equivalently x ∨ y ∨ x = x ∨ y [18].

Definition 2.2 ([12]). Skew lattice A is normal if x ∧ y ∧ z ∧ w = x ∧ z ∧ y ∧ w and is conormal if x ∨ y ∨ z ∨ w = x ∨ z ∨ y ∨ w for all x, y, z, w ∈ A.

Proposition 2.3 ([12]). A skew lattice A is normal iff each sub skew lattice (↓ x) is a sub lattice of A. Dually, A is conormal iff each sub skew lattice (↑ x) is a sub lattice of A ((↑ x) = {y ∈ A|y ≥ x}, (↓ x) = {y ∈ A|y ≤ x}). A skew lattice is distributive if it satisfies x ∧ (y ∨ z) ∧ x = (x ∧ y ∧ x) ∨ (x ∧ z ∧ x) and x ∨ (y ∧ z) ∨ x = (x ∨ y ∨ x) ∧ (x ∨ z ∨ x) [13]. A skew lattice 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). A skew lattice is called strongly distributive if it satisfies the identities: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z). Any co-strongly distributive skew lattice or strongly distributive skew lattice is a distributive skew lattice [7]. 404 R. Koohnavard and A. Borumand Saeid

Proposition 2.4 ([7]). A skew lattice A is co-strongly distributive if and only if A is jointly : (i) quasi-distributive: the maximal lattice image A/D is a distributive lattice, (ii) symmetric: x ∧ y = y ∧ x if and only if x ∨ y = y ∨ x, (iii) conormal: x ∨ y ∨ z ∨ w = x ∨ z ∨ y ∨ w. All distributive skew lattices are quasi-distributive [13]. A skew chain is a skew lattice that x y or y x, for all x, y ∈ A. If A is a skew chain, then A/D is a chain [7]. Theorem 2.5 ([15]). A skew lattice A is strongly distributive iff it is distributive, normal and symmetric. Definition 2.6 ([15]). A skew Boolean algebra is an algebra (A, ∨, ∧, \, 0) such that (i)( A, ∨, ∧, 0) is a strongly distributive skew lattice with 0, (ii) \ is a binary operation satisfying (x ∧ y ∧ x) ∨ (x\y) = x and (x ∧ y ∧ x) ∧ (x\y) = 0. Definition 2.7 ([7]). A skew Heyting algebra is an algebra A = (A, ∨, ∧, →, 1) of type (2, 2, 2, 0) satisfying the following: (i)( A, ∨, ∧, 1) is a co-strongly distributive skew lattice with top 1. Thus any (↑ u) is a bounded distributive lattice, (ii) For any u ∈ A, an operation →u can be defined on (↑ u) such that (↑ u, ∨, ∧, →u, u, 1) is a Heyting algebra with top 1 and bottem u, (iii) x → y = y ∨ x ∨ y →y y. A primitive skew lattice is a skew lattice that has exactly two com- parable D-class A B. A band is a semigroup that its elements are idempotent. A rectangular band is a band that xyx = x or equivalently xyz = xz. A skew lattice is rectangular, if (A, ∨) or (A, ∧) is rectangular band [15]. Definition 2.8 ([6]). A skew lattice is rectangular if it satisfies the identity x ∨ y = y ∧ x. Definition 2.9 ([19]). A function f : A ∗ A → P (A), of the set A∗A into the set of all nonempty subsets of A, is called a hyperoperation. In the following, we recall some of the definitions and results of paper [2] that we need in this paper: Definition 2.10 ([2]). A residuated skew lattice is a nonempty set A with operations ∨, ∧, and hyperoperation → and constant element 1 that satisfying the following: (Skew) filters in residuated skew lattices: Part II 405

(i)( A, ∨, ∧, 1) is a skew lattice with top 1 (for all x ∈ A, x ≤ 1), (ii)( A, , 1) is a commutative monoid, (iii) and → form an adjoint pair, i.e. z x → y iff x z y, for all x, y, z ∈ A. The relations between the pair of operations and → expressed by (iii), is a special case of the law of residuation and for every x, y ∈ A, x → y = sup{z ∈ A|x z y}. Supremum of a set in a pre-ordered set is not a unique element, x → y may be a D-class. Two D-classes have D-relationship when all of their members have D-relationship with each other. Relation between two D-classes is defined member to member (i.e. B C iff ∀c ∈ C, ∀b ∈ B, b c). Also each of the ∨, ∧, , →, between two D-classes are defined member to member [2]. Lemma 2.11 ([2]). If A is a residuated skew lattice and x, y, z ∈ A, then (1)1 → x = Dx and x → x = 1, (2) x y x, y hence x y x ∧ y, y ∧ x, y x → y, (3) x y x → y, (4) x y iff x → y = 1 and x D y iff x → y = y → x = 1, (5) x → 1 = 1, (6) x (x → y) y, x (x → y) → y, ((x → y) → y) → y)D(x → y), (7) x → y x z → y z, (8) x y implies x z y z, (9) x → y (z → x) → (z → y), (10) x → y (y → z) → (x → z), (11) x y implies z → x z → y and y → z x → z, (12) x (y → z) y → (x z) x y → x z, (13)( x → (y → z)) D (x y → z) D (y → (x → z)), (14) x1 → y1 (y2 → x2) → [(y1 → y2) → (x1 → x2)], (15) x ∨ y (x → y) → y ∧ (y → x) → x ∧ (x → y) → y, (16) x (x → y) (x ∧ y), x (x → y) (y ∧ x). Corollary 2.12 ([2]). Let A be a residuated skew lattice with 0. For every x ∈ A and n ≥ 1 we have (x∗∗)n (xn)∗∗, ((x∗∗)n = x∗∗ · · · x∗∗), (x∗ = x → 0). Definition 2.13 ([2]). A branchwise residuated skew lattice A is an algebra A = (A, ∨, ∧, , →, 1) of type (2, 2, 2, 2, 0) satisfying the following: (i)( A, ∨, ∧, 1) is a distributive skew lattice with top 1 (for all x ∈ A, x ≤ 1), 406 R. Koohnavard and A. Borumand Saeid

(ii)( A, , 1) is a commutative monoid, (iii) For any u ∈ A, two operations →u, u can be defined on (↑ u) such that (↑ u, ∨, ∧, u, →u, u, 1) is a distributive residuated lattice by top 1 and bottom u (↑ u = {x ∈ A|x ≥ u}), (iv) x → y = (y ∨ x ∨ y) →y y, (v) x y D x u y, for all u ∈ A, x, y ∈ (↑ u). Theorem 2.14 ([2]). If A is a conormal distributive residuated skew lattice which x y D x u y, for every u ∈ A, x, y ∈ (↑ u), then branchwise residuated skew lattice and residuated skew lattice are equivalent. Definition 2.15 ([2]). Let A be a residuated skew lattice. A nonempty subset D ⊆ A is called a deductive system (for short ds) of A, if the following conditions are satisfied: (i)1 ∈ D, (ii) If x ∈ D, x → y ⊆ D, then y ∈ D. Definition 2.16 ([2]). Let A be a residuated skew lattice. A nonempty subset D ⊆ A is called a skew deductive system of A, if the following conditions are satisfied: (i)1 ∈ D, (ii) If x ∈ D, x → y ⊆ D, then y ∧ (x → y) ∧ y ⊆ D. Definition 2.17 ([2]). A deductive system of a residuated skew lattice A is maximal if it is proper and it is not contained in any other proper deductive system. Corollary 2.18 ([2]). If M is a proper deductive system of residuated skew lattice A, then the following are equivalent: (i) M is a maximal deductive system, (ii) For any x ∈ A, x∈ / M iff (xn)∗ ⊆ M, for some n ≥ 1. Definition 2.19 ([11]). A (skew) filter F of residuated skew lattice A is called prime (skew) filter of (i) type (I) (PF 1), if x ∨ y ∨ x ∈ F , then x ∈ F or y ∈ F , for all x, y ∈ A, (ii) type (II) (PF 2), if x → y ⊆ F or y → x ⊆ F , for all x, y ∈ A, (iii) type (III) (PF 3), if (x → y) ∨ (y → x) ∨ (x → y) ⊆ F , for all x, y ∈ A. If F is a filter of A, then we define A/F = {[x]|x ∈ A} and [x] = x/F = {y ∈ A|x → y, y → x ⊆ F }. ∨, ∧, , → are defined on A/F as (Skew) filters in residuated skew lattices: Part II 407 follows: [x] ∨ [y] = [x ∨ y], [x] ∧ [y] = [x ∧ y], [x] [y] = [x y] and [x] → [y] = [x → y]. A/F is a residuated skew lattice [11]. Residuated skew lattice A is called prelinear, if (x → y) ∨ (y → x) ∨ (x → y) = 1, for all x, y ∈ A [11].

3. Green, strong and ultra (skew) ﬁlters

From here to the end of the paper, let A be a residuated skew lattice, unless otherwise stated. The Green’s equivalence relation communi- cates between lattice and skew lattice. A skew lattice can have different Green’s equivalence relations. We want to communicate between the residuated lattice and the residuated skew lattice by using a specific filter that is derived from the Green’s relation: Definition 3.1. The smallest filter of A that contains all Green’s equivalence relations , is called Green filter of A i.e. F = T F , Di Dj ⊆Fi∀j i (Fi is a filter of A, ∀i ∈ I). Example 3.2. Let A = {0, a, b, n, c, d, m, 1} be a skew lattice such that 0 < a, b < n < c, d < m < 1, c D d and Dc = {c, d}. A = (A, ∨, ∧, , →, 0, 1) is a residuated skew lattice with 0, with the following operations:

→ 0 a b n c d m 1 0 a b n c d m 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 a b 1 b 1 1 1 1 1 a 0 a 0 a a a a a b a a 1 1 1 1 1 1 b 0 0 b b b b b b n 0 a b 1 1 1 1 1 n 0 a b n n n n n c 0 a b Dc 1 1 1 1 c 0 a b n n n c c d 0 a b Dc 1 1 1 1 d 0 a b n n n d d m 0 a b n Dc Dc 1 1 m 0 a b n c d m m 1 0 a b n Dc Dc m 1 1 0 a b n c d m 1 ∨ 0 a b n c d m 1 ∧ 0 a b n c d m 1 0 0 a b n c d m 1 0 0 0 0 0 0 0 0 0 a a a n n c d m 1 a 0 a 0 a a a a a b b n b n c d m 1 b 0 0 b b b b b b n n n n n c d m 1 n 0 a b n n n n n c c c c c c d m 1 c 0 a b n c c c c d d d d d c d m 1 d 0 a b n d d d d m m m m m m m m 1 m 0 a b n c d m m 1 1 1 1 1 1 1 1 1 1 0 a b n c d m 1 408 R. Koohnavard and A. Borumand Saeid

• 1

m •

c • • d

n •

a • • b

0 • F = {n, c, d, m, 1} is a Green ﬁlter of A.

Theorem 3.3. If F is a Green ﬁlter of A, then A/F is a residuated lattice.

Proof. The proof is similar to Proposition 3.7 of [11] and since F contains all Green’s equivalence relations Di and ∨, ∧ in A/Di are commutative, then ∨, ∧ in A/F are commutative too. Converse of above theorem is not true, because in Example 3.2, F = {b, n, c, d, m, 1} is a filter of A that includes all Green’s equivalence relations and A/F is a residuated lattice, but F is not the smallest filter of A. Thus F is not a Green filter of A.

Corollary 3.4. {1} is a Green ﬁlter of A iﬀ A is a residuated lattice.

Definition 3.5. A (skew) ﬁlter F of a residuated skew lattice A with 0 is called strong (skew) ﬁlter, if (x∗∗ → x)∗∗ ⊆ F , for all x ∈ A.

Proposition 3.6. Let A be a residuated skew lattice with 0. Then (i) any strong filter of A is a strong skew filter, (ii) if F is a strong (skew) filter and G is a (skew) filter of A and F ⊆ G, then G is a strong (skew) filter.

Proof. (i) Since any filter of A is a skew filter, it is clear. (ii) Since for all x ∈ A, (x∗∗ → x)∗∗ ⊆ F ⊆ G, it is clear. (Skew) filters in residuated skew lattices: Part II 409

Example 3.7. Let A = {0, a, b, 1} be a skew lattice such that a D b, Da = {a, b}. A = (A, ∨, ∧, , →, 1) is a residuated skew lattice with the following operations:

→ 0 a b 1 0 a b 1 0 1 1 1 1 0 0 0 0 0 a 0 1 1 1 a 0 a a a b 0 1 1 1 b 0 a b b 1 0 Da Da 1 1 0 a b 1 ∨ 0 a b 1 ∧ 0 a b 1 0 0 a b 1 0 0 0 0 0 a a a b 1 a 0 a a a b b a b 1 b 0 b b b 1 1 1 1 1 1 0 a b 1

• 1

a • • b

•0

F2 = {b, 1} is a strong skew filter of A but is not a strong filter (because F2 is a skew filter but is not a filter).

Example 3.8. Let A = {0, a, b, n, c, d, m, 1} be a skew lattice such that 0 < a, b < n < c, d < m < 1, a D b and Da = {a, b}. A = (A, ∨, ∧, , →, 0, 1) is a residuated skew lattice with 0, with the following operations:

→ 0 a b n c d m 1 0 a b n c d m 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 a Da 1 1 1 1 1 1 1 a 0 0 0 a a a a a b Da 1 1 1 1 1 1 1 b 0 0 0 b b b b b n 0 Da Da 1 1 1 1 1 n 0 a b n n n n n c 0 Da Da d 1 d 1 1 c 0 a b n c n c c d 0 Da Da c c 1 1 1 d 0 a b n n d d d m 0 Da Da n c d 1 1 m 0 a b n c d m m 1 0 Da Da n c d m 1 1 0 a b n c d m 1 410 R. Koohnavard and A. Borumand Saeid

∨ 0 a b n c d m 1 ∧ 0 a b n c d m 1 0 0 a b n c d m 1 0 0 0 0 0 0 0 0 0 a a a b n c d m 1 a 0 a a a a a a a b b a b n c d m 1 b 0 b b b b b b b n n n n n c d m 1 n 0 a b n n n n n c c c c c c m m 1 c 0 a b n c n c c d d d d d m d m 1 d 0 a b n n d d d m m m m m m m m 1 m 0 a b n c d m m 1 1 1 1 1 1 1 1 1 1 0 a b n c d m 1

• 1

m •

c • • d

n •

a • • b

0 •

F1 = {n, c, d, m, 1} is a strong filter and a strong skew filter of A. Proposition 3.9. Let A be a residuated skew lattice with 0. If F is a filter of A and for all 1 6= x ∈ A, x∗ ⊆ F , then (i) F is a strong filter, (ii) F is a prime filter of type (III).

Proof. (i) Since for all 1 6= x ∈ A, x∗ ⊆ F , then x∗ ∨ x ⊆ F and since x ∨ x∗, x∗ ∨ x (x∗ → x) → x (x∗∗ → x), then (x∗∗ → x) ⊆ F . Therefore (x∗∗ → x)∗∗ ⊆ F . (ii) Since x∗ ⊆ F for all 1 6= x ∈ A, then x ∨ x∗ ⊆ F for all x ∈ A. x y → x and x∗ x → y imply (x ∨ x∗) ⊆ F (y → x) ∨ (x → y). Therefore (y → x) ∨ (x → y) ⊆ F . Thus (x → y) ∨ (y → x) ∨ (x → y) ⊆ F .

Definition 3.10. A residuated skew lattice A with 0 is called strong residuated skew lattice, if (x∗∗ → x)∗ = 0, for all x ∈ A. (Skew) ﬁlters in residuated skew lattices: Part II 411

Example 3.11. In Example 3.8, A is a strong residuated skew lattice.

Theorem 3.12. If A is a residuated skew lattice with 0, then the following are equivalent: (i) {1} is a strong filter, (ii) Every filter of A is a strong filter, (iii) A is a strong residuated skew lattice.

Proof. (i → ii) The proof holds by Proposition 3.6. (ii → iii) {1} is a strong ﬁlter, so for all x ∈ A, (x∗∗ → x)∗∗ ⊆ {1}, that ∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗∗ ∗ is, (x → x) = 1. Hence (x → x) D (x → x) = 1 = 0, so A is a strong residuated skew lattice. (iii → i) Since A is a strong residuated skew lattice, then for all x ∈ A, (x∗∗ → x)∗ = 0. It follows that (x∗∗ → x)∗∗ = 0∗ = 1 ∈ {1}. Hence {1} is a strong ﬁlter.

Definition 3.13. A (skew) filter F of a residuated skew lattice A with 0 is called ultra (skew) filter, if F is a proper (skew) filter of A and x ∈ F or x∗ ⊆ F , for all x ∈ A.

Example 3.14. F1 = {b, n, c, d, m, 1} of Example 3.2, is an ultra ﬁlter of A.

Proposition 3.15. Let A be a residuated skew lattice with 0. Then (i) any ultra filter of A is an ultra skew filter, (ii) if F is an ultra (skew) filter and G is a (skew) filter of A and F ⊆ G, then G is an ultra (skew) filter.

Proof. (ii) Since x ∈ F or x∗ ⊆ F ⊆ G for all x ∈ A, it is clear.

Proposition 3.16. Let A be a residuated skew lattice with 0. If F is a proper ﬁlter of A, then x → y ⊆ F, y → x ⊆ F , for any x, y∈ / F iﬀ for all x ∈ A, if x∈ / F then there exists n ≥ 1 such that (x∗)n ⊆ F .

Proof. Suppose that x∈ / F . Since F is a proper ﬁlter, thus 0 ∈/ F and hence 1 = 0 → x ∈ F and x∗ = x → 0 ⊆ F . So (x∗)n ⊆ F , for n = 1. Conversely, let x, y∈ / F . We will show that x → y ⊆ F and y → x ⊆ F . By the hypothesis (x∗)n ⊆ F and (y∗)m ⊆ F , for some n, m ≥ 1. We know that (x∗)n x∗ and (y∗)m y∗. Therefore x∗ ⊆ F and y∗ ⊆ F . By Lemma 2.11, we have x∗ = x → 0 x → y and y∗ = y → 0 y → x, for all x, y ∈ A. Hence x → y ⊆ F and y → x ⊆ F . 412 R. Koohnavard and A. Borumand Saeid

Theorem 3.17. Let A be a residuated skew lattice with 0. F is an ultra filter iff F is a proper and x, y∈ / F implies x → y ⊆ F and y → x ⊆ F . Proof. Assume that x∈ / F . By assumption we get that (x∗)n ⊆ F , for some n ≥ 1. We know that (x∗)n x∗ thus x∗ ⊆ F i.e. F is an ultra filter of A. Conversely, let F be an ultra filter of A and x∈ / F . It is enough to show that (x∗)n ⊆ F , for some n ≥ 1. By the hypothesis, we get that x∗ ⊆ F . Theorem 3.18. Let A be a residuated skew lattice with 0. Then (i) any ultra filter of A is a maximal filter, (ii) any ultra filter of A is a strong filter. Proof. (i) Suppose that F is an ultra filter which is not maximal. So there exists a proper filter G such that F $ G. Let x ∈ G\F . Then (x∗)n ⊆ F for some n ≥ 1. We know that (x∗)n x∗. Therefore x∗ ⊆ F , thus also x∗ ⊆ G. So x ∈ G, x∗ ⊆ G, hence x x∗ = 0 ∈ G, which is a contradiction to the fact that G is proper. (ii) Since x ∈ F or x∗ ⊆ F , therefore x ∨ x∗ ⊆ F (x∗ → x) → x x∗∗ → x. Therefore (x∗∗ → x)∗∗ ⊆ F . In Example 3.8, F = {n, c, d, m, 1} is a maximal filter but is not a ultra ∗ filter, because a∈ / F, a = Da = {a, b} * F . In Example 3.8, F = {1} is a strong filter but is not a ultra filter, because c∈ / F, c∗ = 0 ∈/ F . Proposition 3.19. Let A be a prelinear residuated skew lattice with 0. Then (i) if F is an ultra filter of A, then A/F is a residuated skew chain, (ii) if F is a prime filter of type (I) and x ∨ x∗ ∨ x ⊆ F , then F is an ultra filter of A. Proof. (i) Since any ultra filter is a maximal filter and any maximal filter is a prime filter of type (I) and by Theorem 3.2 [11] and Remark 3.1 [11], A/F is a residuated skew chain.

4. Implicative, positive implicative and Boolean (skew) ﬁl- ters

In this section, we want to investigate some of the (skew) ﬁlters, some relations among them and study their quotient algebras too. We show (Skew) ﬁlters in residuated skew lattices: Part II 413 that skew Boolean algebras and residuated skew lattices do not have any relationship to each other.

Definition 4.1. A (skew) ﬁlter F of A is called implicative (skew) ﬁlter, if x → (y → z) ⊆ F, x → y ⊆ F , then x → z ⊆ F .

Example 4.2. Let A = {0, 00, m, a, b, p, n, c, d, 1} be a skew lattice 0 0 such that 0 , 0 < m < a, b < p < n < c, d < 1, D0 = {0, 0 }, Da = {a, b} 0 and 0 D 0 , a D b. A = (A, ∨, ∧, , →, 1) is a residuated skew lattice with the following operations:

0 00 m a b p n c d 1 0 00 m a b p n c d 1 →0 1111111111 0 0 0 0 0 0 0 0 0 0 0 00 1 1 1 1 1 1 1 1 1 1 00 0 00 00 00 00 00 00 00 00 00 m D0 D0 1 1 1 1 1 1 1 1 m 0 00 m m m m m m m m a D0 D0 Da 1 1 1 1 1 1 1 a 0 00 m m m a a a a a b D0 D0 Da 1 1 1 1 1 1 1 b 0 00 m m m b b b b b p D0 D0 m Da Da 1 1 1 1 1 p 0 00 m a b p p p p p n D0 D0 m Da Da p 1 1 1 1 n 0 00 m a b p n n n n c D0 D0 m Da Da p d 1 d 1 c 0 00 m a b p n c n c d D0 D0 m Da Da p c c 1 1 d 0 00 m a b p n n d d 1 D0 D0 m Da Da p n c d 1 1 0 00 m a b p n c d 1

0 00 m a b p n c d 1 0 00 m a b p n c d 1 ∨ ∧ 0 0 00 m a b p n c d 1 0 0 0 0 0 0 0 0 0 0 0 00 0 00 m a b p n c d 1 00 00 00 00 00 00 00 00 00 00 00 m mmmabpncd1 m 0 00 m m m m m m m m a aaaabpncd1 a 0 00 m a a a a a a a b bbbabpncd1 b 0 00 m b b b b b b b p ppppppncd1 p 0 00 m a b p p p p p n nnnnnnncd1 n 0 00 m a b p n n n n c cccccccc11 c 0 00 m a b p n c n c d ddddddd1d1 d 0 00 m a b p n n d d 1 1111111111 1 0 00 m a b p n c d 1

1 •

c d • •

n •

p •

a b • •

m •

0 00 • • 414 R. Koohnavard and A. Borumand Saeid

0 F1 = {m, a, b, p, n, c, d, 1} is an implicative filter. F2 = {0 , m, a, b, p, n, c, d, 1} is an implicative skew filter of A but is not an implicative filter because is not a filter. Proposition 4.3. If F is a filter of A, then the following conditions are equivalent: (i) F is an implicative filter, (ii) For any a ∈ A, Fa = {x ∈ A|a → x ⊆ F } is a filter of A, (iii) x → (x → y) ⊆ F implies x → y ⊆ F , for all x, y ∈ A, (iv) x → (y → z) ⊆ F implies (x → y) → (x → z) ⊆ F , for all x, y, z ∈ A, (v) y → (y → z) ⊆ F implies y → z ⊆ F , for all y, z ∈ A. Proof. (i → ii) Suppose F is an implicative filter. Since a ∈ A, a → 1 = 1 ∈ F , then 1 ∈ Fa. If x ∈ Fa, x → y ⊆ Fa, then a → x, a → (x → y) ⊆ F . Since F is an implicative filter, then a → y ⊆ F hence y ∈ Fa, i.e. Fa is a filter of A. (ii → i) Since 1 ∈ F1 = F we deduce that 1 ∈ F . Let x, y, z ∈ A be such that x → (y → z), x → y ⊆ F . Then y ∈ Fx, y → z ⊆ Fx. Since Fx is a filter, then z ∈ Fx and so x → z ⊆ F , therefore F is an implicative filter. (ii → iii) If x, y ∈ A, x → (x → y) ⊆ F , then x → y ⊆ Fx. Since x → x = 1 ∈ F we deduce that x ∈ Fx, x → y ⊆ Fx. Since Fx is a filter, then y ∈ Fx, that is x → y ⊆ F . (iii → iv) Let x, y, z ∈ A be such that x → (y → z) ⊆ F . We have y → z (x → y) → (x → z) that implies x → (y → z) x → ((x → y) → (x → z)) hence x → ((x → y) → (x → z)) ⊆ F . It implies x → (x → ((x → y) → z)) ⊆ F , therefore (x → y) → (x → z) ⊆ F . (iv → v) Consider y, z ∈ A such that y → (y → z) ⊆ F . By hypothesis (y → y) → (y → z) ⊆ F . That is y → z ⊆ F . (v → i) Consider x, y, z ∈ A such that x → (y → z) ⊆ F, x → y ⊆ F . We must show that x → z ⊆ F . We have x → (y → z) D (y → (x → z)) (x → y) → (x → (x → z)) hence (x → y) → (x → (x → z)) ⊆ F . Since F is a filter, then x → (x → z) ⊆ F . So x → z ⊆ F . Proposition 4.4. The following are true. (i) Any implicative filter of A is an implicative skew filter of A, (ii) Suppose F and G are filters of A and F ⊆ G. If F is an implicative filter, then G is an implicative filter. Proof. (ii) Let u = z → (y → x) ⊆ G. By Lemma 2.11, z → (y → (u → x)) D z → (u → (y → x)) D u → (z → (y → x)) = 1. Thus (Skew) filters in residuated skew lattices: Part II 415 z → (y → (u → x)) ⊆ F . By Proposition 4.3 (iv), we get that

(z → y) → (z → (u → x)) ⊆ F ⊆ G. (F)

Lemma 2.11 implies u → ((z → y) → (z → x)) D (z → y) → (u → (z → x)) D (z → y) → (z → (u → x)). By (F) we have u → ((z → y) → (z → x)) ⊆ G. Since u ⊆ G and G is a filter, (z → y) → (z → x) ⊆ G and so by Proposition 4.3, G is an implicative filter. Corollary 4.5. If F is a filter, then the following conditions are equivalent: (i) F is an implicative filter, (ii) x → x2 ⊆ F , for all x ∈ A. 2 2 2 Proof. (i → ii) We have x → (x → x ) D x → x = 1 ∈ F hence by Proposition 4.3, x → x2 ⊆ F . (ii → i) Consider x, y ∈ A such that x → (x → y) ⊆ F hence x2 → y ⊆ F . It implies x → y ⊆ F hence F is an implicative filter of A (by Proposition 4.3 (iii)). Definition 4.6. A residuated skew lattice A is called skew G- 2 algebra, if x D x, for all x ∈ A. Example 4.7. In Example 3.7, A is a skew G-algebra. Theorem 4.8. In A, the following conditions are equivalent: (i) A is a skew G-algebra, (ii) Any filter of A is an implicative filter of A, (iii) {1} is an implicative filter of A. Proof. (i → ii) Let A be a skew G-algebra and F be an arbitrary filter of A. To show that F is an implicative filter, we use of Proposition 2 4.3. Since A is a skew G-algebra y → (y → x) D y → x D y → x, hence y → x ⊆ F i.e. F is an implicative filter. (ii → iii) is clear. (iii → i) Since x → (x → x2) = 1, for all x ∈ A and {1} is an implicative filter, we have x → x2 = 1. Hence x x2, for all x ∈ A. Proposition 4.9. In A, the following conditions are equivalent: (i) F is an implicative filter, (ii) A/F is a skew G-algebra. Proof. (i → ii) It is enough to show that F is an implicative filter of A iff the filter [1] of A/F is an implicative filter. Let F be an implicative filter of A and x, y ∈ A be such that [y] → ([y] → [x]) = [1]. Then y → 416 R. Koohnavard and A. Borumand Saeid

(y → x) ⊆ F . By Proposition 4.3, y → x ⊆ F . Hence [y] → [x] = [y → x] = [1], which proves that [1] is an implicative filter. (ii → i) suppose that every filter of A/F is an implicative filter and x, y ∈ A be such that y → (y → x) ⊆ F . Then we have [y] → ([y] → [x]) = [y → (y → x)] = [1]. Since [1] is an implicative filter, then [y → x] = [y] → [x] = [1] that is, y → x ⊆ F . Hence F is an implicative filter.

Definition 4.10. A (skew) ﬁlter F of A is called positive implicative (skew) ﬁlter, if (y → z) → y ⊆ F , then y ∈ F .

Example 4.11. In Example 3.2, F = {n, c, d, m, 1} is a positive implicative ﬁlter.

Theorem 4.12. Any positive implicative ﬁlter of A is an implicative ﬁlter.

Proof. Let F be a positive implicative ﬁlter. Consider x, y ∈ A be such that x → (x → y) ⊆ F and must prove that x → y ⊆ F . We have x → (x → y) ((x → y) → y) → (x → y) hence ((x → y) → y) → (x → y) ⊆ F . By assumption we deduce that x → y ⊆ F .

In Example 3.7, F = {1} is an implicative ﬁlter but is not a positive implicative ﬁlter, because (a → 0) → a = 0 → a = 1 ∈ F , but a∈ / F .

Theorem 4.13. Let A be a residuated skew lattice with 0 and F be a filter of A. Consider the following assertions: (1) F is a positive implicative filter, (2) If x ∈ A, x∗ → x ⊆ F , then x ∈ F , (3) If x, y ∈ A and (x → y) → y ⊆ F , then (y → x) → x ⊆ F . Then (i) (1) ↔ (2), (ii) (1) → (3), (iii) If F is an implicative filter, then (1) ↔ (2) ↔ (3).

Proof. (i) (1 → 2) is clear, since x∗ → x = (x → 0) → x. (2 → 1) Let y, z ∈ A be such that (y → z) → y ⊆ F . From 0 z we deduce that y → 0 y → z, (y → z) → y (y∗ → y), so y∗ → y ⊆ F , hence y ∈ F . Therefore F is a positive implicative ﬁlter. (ii) (1 → 3) Let x, y ∈ A and denote a = (x → y) → y, b = (y → x) → x. Then we want prove that a (b → y) → b. Indeed from x b we deduce that b → y x → y hence

(x → y) → b (b → y) → b (F) (Skew) ﬁlters in residuated skew lattices: Part II 417 on the other hand by (F), we have a = (x → y) → y (y → x) → ((x → y) → x) D (x → y) → ((y → x) → x) = (x → y) → b (b → y) → b. So if a ⊆ F , then (b → y) → b ⊆ F so b ⊆ F . (iii) Let x ∈ A be such that x∗ → x ⊆ F . We have x∗ → x x∗ → x∗∗ hence x∗ → x∗∗ ⊆ F . Since x∗ → x∗∗ = x∗ → (x∗ → 0) and F is an implicative ﬁlter, then we deduce that x∗ → 0 ⊆ F i.e. x∗∗ ⊆ F . But ∗∗ x = (x → 0) → 0 and by hypothesis we deduce that Dx = (0 → x) → x ⊆ F , therefore x ∈ F . So 3 → 2 ↔ 1 hence 1 ↔ 3.

Proposition 4.14. The following are true. (i) Any positive implicative filter of A is a positive implicative skew filter, (ii) If F is a positive implicative filter, then every filter G containing F is also a positive implicative filter. Proof. (ii) By Theorem 4.12, we know that F is an implicative filter. By Proposition 4.4, G is an implicative filter. Suppose (y → x) → x ⊆ G, it is sufficient to show that (x → y) → y ⊆ G. Denote u = (y → x) → x, since u → ((y → x) → x) = 1 ∈ F and F is an implicative filter, it follows from Theorem 4.3 and Lemma 2.11, (u → (y → x)) → (u → x) D (y → (u → x)) → (u → x). Hence (y → (u → x)) → (u → x) ⊆ F , by Theorem 4.13, we get that ((u → x) → y) → y ⊆ F , thus ((u → x) → y) → y ⊆ G. On the other hand (y → x) → x (((y → x) → x) → x) → x = (u → x) → x (x → y) → ((u → x) → y) (((u → x) → y) → y) → ((x → y) → y), since G is a filter so ((u → x) → y) → ((x → y) → y) ⊆ G. Using ((u → x) → y) → y ⊆ G we get that (x → y) → y ⊆ G.

Theorem 4.15. Let A be a residuated skew lattice with 0. If F is a ﬁlter of A, then the following conditions are equivalent: (i) x∗ ∨ x ∨ x∗ ⊆ F (or x ∨ x∗ ∨ x ⊆ F ), (ii) x∗ → x ⊆ F implies x ∈ F . 418 R. Koohnavard and A. Borumand Saeid

Proof. (i → ii) By Theorem 3.4 of [2], ∗ ∗ ∗ ∗ (x ∨ x ∨ x ) → x D (x → x) ∧ (x → x) ∧ (x → x) = (x∗ → x) ∧ 1 ∧ (x∗ → x) = (x∗ → x) and (x∗ → x) ⊆ F , then (x∗ ∨ x ∨ x∗) → x ⊆ F . Which implies x ∈ F . ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ (ii → i) Let a = x ∨ x ∨ x . Since a = (x ∨ x ∨ x ) D x ∧ x ∧ x x∗ a, then (a∗ → a) = 1 ∈ F . Therefore a ∈ F i.e. x∗ ∨x∨x∗ ⊆ F . Corollary 4.16. F is a positive implicative filter iff x∗∨x∨x∗ ⊆ F (or x ∨ x∗ ∨ x ⊆ F ). Proposition 4.17. The following conditions are equivalent: (i) {1} is a positive implicative filter, (ii) Every filter F of A is a positive implicative filter, (iii) For every a ∈ A, [a) is a positive implicative filter. Proof. The equivalence (i ↔ ii) follows from Proposition 4.14. (ii → iii) Since {1} is a positive implicative filter, then by Theorem 4.12, {1} is an implicative filter. Clearly, 1 ∈ [a). To prove that [a) is a filter, consider x, y ∈ A such that x ∈ [a), x → y ⊆ [a). Then a → (x → y) D a → x = 1 ∈ {1}. Since {1} is an implicative filter, then a → y ⊆ {1}, hence a → y = 1, that is y ∈ [a). By (ii), [a) is a positive implicative filter. (iii → i) is clear. Theorem 4.18. Let A be a residuated skew lattice with 0 and F be a filter of A. For any x, y ∈ A, the following conditions are equivalent: (i) F is a maximal and positive implicative filter, (ii) F is a maximal and implicative filter, (iii) F is an ultra filter. Proof. (i → ii) By Theorem 4.12. (ii → iii) Let y∈ / F . At first, we show that Fy = {u ∈ A|y → u ⊆ F } is a filter. 1 ∈ F , since y → 1 = 1 ∈ F . If x ∈ Fy, x → t ⊆ Fy, then y → x, y → (x → t) ⊆ F , hence, since F is an implicative filter, it follows that y → t ⊆ F , thus t ∈ Fy. If x ∈ F , since x y → x, we get y → x ⊆ F , thus x ∈ Fy hence F ⊆ Fy ⊆ A. Now, since y∈ / F , but y → y = 1 ∈ F and hence y ∈ Fy, it follows that F ⊂ Fy, since F is a maximal filter and y∈ / F , we get that Fy = A. Then every x ∈ A satisfies x ∈ Fy, and so y → x ⊆ F . Similarly, if x, y ∈ A such that x∈ / F , then x → y ⊆ F . Therefore by Theorem 3.17, F is an ultra filter. (iii → i) Assume on the contrary that F is not a positive implicative (Skew) filters in residuated skew lattices: Part II 419

filter, thus there exist x, y ∈ A such that (x → y) → x ⊆ F and x∈ / F . We consider two cases: y ∈ F or y∈ / F . If y ∈ F , then x → y ⊆ F , hence x ∈ F . If y∈ / F by the hypothesis we have x → y ⊆ F and again x ∈ F . In every cases we get a contradiction. Hence F is a positive implicative filter. According to Theorem 3.18, F is a maximal filter.

Proposition 4.19. Let A be a residuated skew lattice with 0. If F is a maximal filter of A, then F is an implicative filter iff x ∨ x∗ ∨ x ⊆ F (or x∗ ∨ x ∨ x∗ ⊆ F ).

Proof. By Theorem 4.18, we imply x ∈ F or x∗ ⊆ F that implies x ∨ x∗ ∨ x ⊆ F . Conversely, by Corollary 4.16, it is clear.

Theorem 4.20. Let A be a residuated skew lattice with 0. If F is a filter of A, then any ultra filter of A is a positive implicative filter.

Proof. Since F is a positive implicative filter iff x∗ ∨ x ∨ x∗ ⊆ F and since x, x∗ x ∨ x∗ ∨ x, x∗ ∨ x ∨ x∗ and x ∈ F or x∗ ⊆ F . In Example 3.2, F = {n, c, d, m, 1} is a positive implicative filter but is not an ultra filter, because a∈ / F, a∗ = b∈ / F .

Theorem 4.21. Let A be a residuated skew lattice with 0. Then (i) if F is a positive implicative (skew) filter and a prime (skew) filter of type (I) of A, then F is an ultra (skew) filter, (ii) any positive implicative filter of A is a prime filter of type (III). Proof. (ii) Without loss of generality, let A be a left-handed residuated skew lattice with 0. Therefore x∗ ∨ x ∨ x∗ = x ∨ x∗. Let F be a positive implicative filter. Since x y → x and x∗ x → y, then (x ∨ x∗) ⊆ F (y → x) ∨ (x → y). Therefore (y → x) ∨ (x → y) ⊆ F . Thus (x → y) ∨ (y → x) ∨ (x → y) ⊆ F . (If A be a right-handed residuated skew lattice with 0, then it is proven similarly).

In Example 4.2, F = {n, c, d, 1} is a prime ﬁlter of type (III) but is not a positive implicative ﬁlter. Because (a → 0) → a = D0 → a = 1 ∈ F , but a∈ / F .

Theorem 4.22. Let A be a residuated skew lattice with 0. If F is a Green filter of A such that for any x ∈ A, x 6= 1, x∗ ⊆ F , then F is a positive implicative filter. Proof. Let F be a Green filter and x∗ → x ⊆ F . Since any Green filter is a filter and by assumption, then we have x ∈ F . 420 R. Koohnavard and A. Borumand Saeid

In Example 4.2, F = A\{0, 00} is a positive implicative filter but is not a Green filter, because does not contain all the Green equivalence relations D. Definition 4.23. A (skew) filter F of a residuated skew lattice A with 0 is called Boolean (skew) filter, if x∗∗ → x ⊆ F , for all x ∈ A. Example 4.24. In Example 3.8, F = {n, c, d, m, 1} is a Boolean filter. Proposition 4.25. The following are true. (i) Any Boolean filter of A is a Boolean skew filter, (ii) If F is a Boolean filter, then every filter G containing F is also a Boolean filter. Theorem 4.26. Let A be a residuated skew lattice with 0. Then (i) if F is a filter of A, then any Boolean filter is a strong filter of A, ∗∗ (ii) if F is a strong filter of A and x D x, for all x ∈ A, then F is a Boolean filter. Proof. (i) Since (x∗∗ → x) (x∗∗ → x)∗∗, it is clear. (ii) It is clear. In Example 3.2, F = {m, 1} is a strong filter of A but is not Boolean filter. Because n∗∗ → n = 1 → n = n∈ / F .

Proposition 4.27. Let A be a residuated skew lattice with 0. Then (i) if {1} is a Boolean filter of A, then A is a strong residuated skew lattice, (ii) if any filter of A is a Boolean filter, then A is a strong residuated skew lattice, (iii) if F is a positive implicative filter of A, then F is a Boolean filter. Proof. (i, ii) According to Theorems 4.26 and 3.12, it is clear. (iii) Without loss of generality let A be a left-handed residuated skew lattice with 0. Since (x∗ ∨ x ∨ x∗) = x ∨ x∗, (x ∨ x∗ ∨ x) = x∗ ∨ x ⊆ F and x ∨ x∗, x∗ ∨ x (x∗ → x) → x (x∗∗ → x), then (x∗∗ → x) ⊆ F . (If A be a right-handed residuated skew lattice with 0, then it is proven similarly). In Example 3.8, F = {n, c, d, m, 1} is a Boolean filter, but is not a ∗ positive implicative filter. Because a → a = Da → a = 1 ∈ F , but a∈ / F . (Skew) filters in residuated skew lattices: Part II 421

Definition 4.28. A skew Hilbert algebra is an algebra (A, →, 1) of type (2, 0) that satisfying the following identities: (i) x → (y → x) = 1, (ii)( x → (y → z)) → ((x → y) → (x → z)) = 1, (iii) If x → y = y → x = 1, then x D y. Example 4.29. In Example 3.7, A is a skew Hilbert algebra. Proposition 4.30. If A is residuated skew lattice, then the following assertions are equivalent: 2 (i) x D x, for all x ∈ A, (ii) x (x → y) D x y D x ∧ y, for all x, y ∈ A. Proof. (i → ii) Let x, y ∈ A. Then x (x → y) (x x) → (x y) 2 iﬀ x (x → y) x → (x y) iﬀ x → y x → (x → (x y)) D x → (x y) D x → (x y), which implies x (x → y) x y. Since y x → y, then x y x (x → y), so x (x → y) D x y. Clearly x y x, y. To prove x y D x ∧ y, let t ∈ A be such that t x and 2 t y. Then t D t x y that is x y D x ∧ y. (ii → i) In particular 2 for x = y we obtain x D x. Proposition 4.31. If A is a residuated skew lattice, then the following are equivalent: (i)( A, →, 1) is a skew Hilbert algebra, (ii)( A, ∨, ∧, , →, 1) is a skew G-algebra. Proof. (i → ii) Suppose that (A, →, 1) is a skew Hilbert algebra, then for every x, y, z ∈ A we have x → (y → z) D (x → y) → (x → z). From Lemma 2.11, we have x → (y → z) D (x y) → z and (x → y) → (x → z) D (x (x → y)) → z, so we obtain (x y) → z D (x (x → y)) → z 2 hence x y D x (x → y), for x = y we obtain x D x, therefore A is a skew G- algebra. (ii → i) Follows from Proposition 4.30 and Lemma 2.11.

According to Proposition 4.30, in a skew G-algebra, we have x y D x∧y. Any Boolean algebra is a residuated lattice, but skew Boolean algebras and residuated skew lattices do not have any relationship to each other. In Example 4.2, A is a residuated skew lattice but is not a skew Boolean algebra, because A is not normal (a = 1 ∧ a ∧ b ∧ 1 6= 1 ∧ b ∧ a ∧ 1 = b). On the other hand, by Example 4.1.1 [15], for given a rectangular skew lattice B, a primitive skew Boolean algebra is formed by B = {B, ∨, ∧, \, 0} upon setting x\y = x if y = 0 and 0 otherwise. 422 R. Koohnavard and A. Borumand Saeid

B = {B, ∨, ∧, \, 0} is not a residuated skew lattice, because B does not have top element 1 (skew Boolean algebra with top element 1 is a lattice).

5. Stonean, fantastic and easy (skew) ﬁlters

In this section, we want to investigate some of the (skew) filters, some relations among them and study their quotient algebras too. Definition 5.1. Let A be a residuated skew lattice with 0. Skew filter F of A is called Stonean skew filter of (i) type (I), if x∗∗ ∨ x∗ ∨ x∗∗ ⊆ F , for all x ∈ A, (ii) type (II), if x∗ ∨ x∗∗ ∨ x∗ ⊆ F , for all x ∈ A. Remark 5.2. In above definition, if F is a filter, then types (I) and (II) become coincident and F is called a Stonean filter. Proposition 5.3. The following are true. (i) Any Stonean filter of A is a Stonean skew filter (of types (I) and (II)),

(ii) If F is a Stonean skew filter of type (I) (or (II)) and G is a skew filter of A such that F ⊆ G, then G is a Stonean skew filter of type (I) (or (II)) of A.

Example 5.4. In Example 3.2, F1 = {n, c, d, m, 1} is a Stonean filter. F2 = {b, 1} of Example 3.7 is a Stonean skew filter of types (I) and (II) but is not a Stonean filter. Theorem 5.5. Let A be a residuated skew lattice with 0. Then (i) if F is a Stonean filter (of type (I) or (II)), then x∗∗ → x∗ ⊆ F implies x∗ ⊆ F , (ii) any positive implicative filter of A is a Stonean filter, (iii) if F is a maximal filter and an implicative filter of A, then F is a Stonean filter. Proof. (i) Without loss of generality, let A be a left-handed residuated skew lattice with 0 and F be a Stonean filter. Therefore x∗∗ ∨ x∗ ∨ x∗∗ = x∗ ∨x∗∗. Now since x∗ ∨x∗∗ ⊆ F (x∗ → x∗∗) → x∗∗ ∧(x∗∗ → x∗) → x∗, then (x∗∗ → x∗) → x∗ ⊆ F , therefore by assumption x∗ ⊆ F . (If A be a right-handed residuated skew lattice with 0, then it is proven similarly). (ii) Since x x∗∗, then x∗ ∨ x ∨ x∗ ⊆ F x∗ ∨ x∗∗ ∨ x∗, it is clear. (iii) By Theorem 4.18, x ∈ F or x∗ ⊆ F for all x ∈ A, therefore x∗∗ ∨ x∗ ∨ x∗∗ ⊆ F . (Skew) filters in residuated skew lattices: Part II 423

In Example 3.7, F = {1} is a Stonean filter but is not a positive implicative filter, because b∗ → b = 1 ∈ F but b∈ / F . Definition 5.6. Let A be a residuated skew lattice with 0. A is called Stonean residuated skew lattice, if x∗∗ ∨x∗ ∨x∗∗ = 1 (or x∗ ∨x∗∗ ∨ x∗ = 1), for all x ∈ A. Since residuated skew lattice A has a top element 1 and x∗∗ ∨ x∗ ∨ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ x D x ∨ x ∨ x , then x ∨ x ∨ x = 1 iff x ∨ x ∨ x = 1. Example 5.7. In Example 3.7, A is a Stonean residuated skew lattice. Theorem 5.8. Let A be a residuated skew lattice with 0. Then A is Stonean iff any skew filter of A is a Stonean skew filter of type (I) (or (II)). Proof. Let A be Stonean. Therefore for any x ∈ A, x∗ ∨ x∗∗ ∨ x∗ = 1 (or x∗∗ ∨ x∗ ∨ x∗∗ = 1), then x∗ ∨ x∗∗ ∨ x∗ ⊆ F (or x∗∗ ∨ x∗ ∨ x∗∗ ⊆ F ), for any skew filter F of A. Therefore any skew filter is a Stonean skew filter of type (II) (or (I)). Conversely, if any skew filter is a Stonean skew filter of A of type (II) (or (I)), then F = {1} also is a Stonean skew filter of type (II) (or (I)) i.e. for any x ∈ A, x∗ ∨ x∗∗ ∨ x∗ = 1 (or x∗∗ ∨ x∗ ∨ x∗∗ = 1), therefore A is a Stonean residuated skew lattice. Corollary 5.9. Let A be a residuated skew lattice with 0. Then A is Stonean iff {1} is a Stonean skew filter of type (I) (or (II)). Theorem 5.10. Let A be a residuated skew lattice with 0. Then (i) if F is a (skew) filter of A, then F is a Stonean (skew) filter of A iff A/F is a Stonean residuated skew lattice, ∗ ∗ (ii) if (x → y) D (y → x) , for any 0 6= x, y ∈ A, then A is Stonean. Proof. (i) Let F is a Stonean skew filter of type (II). Therefore x∗ ∨ x∗∗ ∨ x∗ ⊆ F iff [x]∗ ∨ [x]∗∗ ∨ [x]∗ = [1] iff A/F is a Stonean residuated skew lattice. (ii) According to assumption, for y = 1 we have (x → ∗ ∗ ∗ ∗ ∗∗ ∗ 1) D (1 → x) therefore 0 = x . So x ∨ x ∨ x = 1. Proposition 5.11. Let A be a residuated skew lattice with 0. (i) If F is a Stonean filter and a prime filter of type (I) and a positive implicative filter, then F is an ultra filter, (ii) If F is a positive implicative filter, then A/F is a Stonean residuated skew lattice. 424 R. Koohnavard and A. Borumand Saeid

Proof. (i) Since x∗ ∨ x∗∗ ∨ x∗ ⊆ F , then by assumption x∗ ⊆ F or x∗∗ ⊆ F . If x∗∗ ⊆ F and since x∗∗ x∗ → x, then we imply x∗ → x ⊆ F that implies x ∈ F . (ii) By Theorems 5.5 and 5.10, it is clear.

Definition 5.12. A (skew) ﬁlter F of A is called fantastic (skew) ﬁlter if x → y ⊆ F , then ((y → x) → x) → y ⊆ F .

Example 5.13. In Example 3.8, F1 = {n, c, d, m, 1} is a fantastic filter. In Example 4.2, F2 = {0, m, a, b, p, n, c, d, 1} is a fantastic skew filter but is not a fantastic filter.

Proposition 5.14. The following are true. (i) Any fantastic filter of A is a fantastic skew filter, (ii) Suppose F ⊆ G where F is a fantastic filter and G is a filter of A. Then G is a fantastic filter. Proof. (ii) Let x, y ∈ A and y → x ⊆ G, we show that ((x → y) → y) → x ⊆ G. Consider y → ((y → x) → x) D (y → x) → (y → x) = 1 ∈ F . We get (y → x) → ((((y → x) → x) → y) → y) → x) D ((((y → x) → x) → y) → y) → ((y → x) → x) then ((((y → x) → x) → y) → y) → ((y → x) → x) ⊆ F ⊆ G, by Lemma 2.11, hence (((((y → x) → x) → y) → y) → x) ⊆ G. Now consider (((((y → x) → x) → y) → y) → x) → (((x → y) → y) → x) ((x → y) → y) → ((((y → x) → x) → y) → y) (((y → x) → x) → y) → (x → y) x → ((y → x) → x) D (y → x) → (x → x) = (y → x) → 1 = 1. Since G is a filter and 1 ∈ G we get (((((y → x) → x) → y) → y) → x) → (((x → y) → y) → x) ⊆ G. On the other hand (((((y → x) → x) → y) → y) → x) ⊆ G and since G is a filter we get that ((x → y) → y) → x ⊆ G. Therefore G is a fantastic filter.

Theorem 5.15. Let A be a residuated skew lattice with 0. Then (i) any positive implicative filter of A is a fantastic filter, (ii) any fantastic (skew) filter of A is a Boolean (skew) filter. Proof. (i) Let x → y ⊆ F . We must show that ((y → x) → x) → y ⊆ F . Since y ((y → x) → x) → y we have (((y → x) → x) → y) → x (Skew) filters in residuated skew lattices: Part II 425 y → x. Also (y → x) → x (((y → x) → x) → y) → x) → x and x → y ((y → x) → x) → ((y → x) → y) D (y → x) → ((y → x) → x) → y) ((((y → x) → x) → y) → x) → (((y → x) → x) → y). According to x → y ⊆ F we deduce ((((y → x) → x) → y) → x) → (((y → x) → x) → y) ⊆ F . Therefore (((y → x) → x) → y) ⊆ F . (ii) Let x → y ⊆ F . We must show that y∗∗ → y ⊆ F . By assumption since x → y ⊆ F , then ((y → x) → x) → y ⊆ F with put x = 0 we have y∗∗ → y ⊆ F . In Example 3.8, F = {n, c, d, m, 1} is a fantastic filter of A but is not a positive implicative filter. Because a∗ → a = 1 ∈ F but a∈ / F . Proposition 5.16. If F is an ultra filter, then F is a fantastic filter but the converse is not true. Proof. If F is an ultra filter of A, then F is an implicative filter of A (by Theorem 4.18) and F is a positive implicative filter so F is a fantastic filter of A (by Theorem 5.15). In Example 3.8, F = {n, c, d, m, 1} is a ∗ fantastic filter but is not an ultra filter, because a∈ / F and a = Da = {a, b} * F . Definition 5.17. Let A be a residuated skew lattice with 0. Then a (skew) filter F of A is called easy (skew) filter if x∗∗ → (y → z) ⊆ F, (x∗∗ → y) ⊆ F , then x∗∗ → z ⊆ F for every x, y, z ∈ A.

Example 5.18. In Example 3.7, F1 = {b, 1} is an easy skew filter that is not a easy filter. F2 = {a, b, 1} is an easy filter. Proposition 5.19. Let A be a residuated skew lattice with 0. Then for filter F of A, the following conditions are equivalent: (i) F is an easy filter, (ii) x∗∗ → (y → z) ⊆ F implies (x∗∗ → y) → (x∗∗ → z) ⊆ F , for every x, y, z ∈ A, (iii) x∗∗ → (x∗∗ → y) ⊆ F implies x∗∗ → y ⊆ F , for every x, y ∈ A. Proof. (i → ii) Let F be an easy filter of A and x, y, z ∈ A be such that x∗∗ → (y → z) ⊆ F . By Lemma 2.11, since x∗∗ → (x∗∗ → ((x∗∗ → y) → ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ z)) D x → ((x → y) → (x → z)) x → (x → (y → z)) x∗∗ → (y → z) we deduce that x∗∗ → (x∗∗ → ((x∗∗ → y) → z)) ⊆ F . 426 R. Koohnavard and A. Borumand Saeid

Since x∗∗ → x∗∗ = 1 ∈ F and F is an easy filter, then x∗∗ → ((x∗∗ → ∗∗ ∗∗ ∗∗ ∗∗ y) → z) ⊆ F . Since x → ((x → y) → z) D (x → y) → (x → z) we deduce that (x∗∗ → y) → (x∗∗ → z) ⊆ F . (ii → iii) is clear. (iii → i) Consider x, y, z ∈ A such that x∗∗ → (y → ∗∗ ∗∗ ∗∗ ∗∗ z), x → y ⊆ F . Since (x → (y → z)) D (y → (x → z)) (x → y) → (x∗∗ → (x∗∗ → z)) and x∗∗ → y ⊆ F , then x∗∗ → (x∗∗ → z) ⊆ F . By hypothesis we deduce that x∗∗ → z ⊆ F , hence F is an easy filter. Proposition 5.20. Let A be a residuated skew lattice with 0. Then (i) any easy filter of A is an easy skew filter, (ii) if F, G are filters of A, F ⊆ G and F is an easy filter, then G is an easy filter. Proof. (ii) By Proposition 5.19 (iii), let x, y ∈ A be such that a = ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ x → (x → y) ⊆ G. Since x → (x → (a → y)) D (x → ∗∗ ∗∗ ∗∗ (a → (x → y))) D (a → (x → (x → y))) D (a → a) = 1 ∈ F , then x∗∗ → (x∗∗ → (a → y)) ⊆ F . By Proposition 5.19 (iii), we deduce that x∗∗ → (a → y) ⊆ F ⊆ G so x∗∗ → (a → y) ⊆ G. Then a → (x∗∗ → y) ⊆ G. From a ∈ G, a → (x∗∗ → y) ⊆ G we conclude that x∗∗ → y ⊆ G. Theorem 5.21. Let A be a residuated skew lattice with 0. Then for filter F of A, the following conditions are equivalent: (i) F is an easy filter, (ii) x∗∗ → (x∗∗)2 ⊆ F , for all x ∈ A. ∗∗ ∗∗ ∗∗ 2 ∗∗ 2 Proof. (i → ii) For x ∈ A we have x → (x → (x ) ) D (x ) → (x∗∗)2 = 1 ∈ F . By Proposition 5.19 (iii), we deduce that x∗∗ → (x∗∗)2 ⊆ F . (ii → i) Consider x, y ∈ A such that x∗∗ → (x∗∗ → y) ⊆ F . Then (x∗∗)2 → y ⊆ F . Since x∗∗ → (x∗∗)2 ⊆ F , then x∗∗ → y ⊆ F therefore F is an easy filter (by Proposition 5.19 (iii)). Theorem 5.22. Let A be a residuated skew lattice with 0. Then any implicative filter of A is an easy filter. Proof. Let x∗∗ → (y → z), x∗∗ → y ⊆ F . By Proposition 4.3 (iv), we deduce that (x∗∗ → y) → (x∗∗ → z) ⊆ F . Since x∗∗ → y ⊆ F we deduce that x∗∗ → z ⊆ F . Therefore F is an easy filter. Proposition 5.23. Let A be a residuated skew lattice with 0. Then the following conditions are equivalent: (i) {1} is an easy filter, (ii) Any filter of A is an easy filter, (Skew) filters in residuated skew lattices: Part II 427

∗∗ 2 ∗∗ (iii)( x ) D x , for all x ∈ A. Proof. (i ↔ ii) is clear, from Proposition 5.20. (i ↔ iii). As in the case of Theorem 4.8 by replacing x with x∗∗. Proposition 5.24. Let A be a residuated skew lattice with 0. If ∗∗ ∗∗ 2 2 ∗ ∗ x D (x ) , for all x ∈ A, then (x ) D x , for all x ∈ A. ∗∗ 2 2 ∗∗ ∗∗ ∗∗ 2 Proof. For x ∈ A by Corollary 2.12, (x ) (x ) . So if x D (x ) , then we deduce successively x∗∗ (x2)∗∗ and (x2)∗ x∗. Since x∗ 2 ∗ 2 ∗ ∗ (x ) we deduce that (x ) D x . Definition 5.25. A residuated skew lattice A with 0 is called skew 2 ∗ ∗ semi-G-algebra, if (x ) D x , for all x ∈ A. Example 5.26. In Example 3.7, A is a skew semi-G-algebra. Proposition 5.27. Let A be a residuated skew lattice with 0. Then (i) if F is an easy filter, then A/F is a skew semi-G-algebra, (ii) if F is an easy filter and x ∈ A such that ((x∗∗)n)∗ ⊆ F for some n ≥ 1, then x∗ ⊆ F . Proof. (i) If x ∈ A, then by Theorem 5.21, we deduce that (x/F )∗∗ ∗∗ 2 ∗∗ ∗∗ ∗∗ 2 ((x/F ) ) (x/F ) , that is (x/F ) D ((x/F ) ) so by Proposition 2 ∗ ∗ 5.24, we deduce that ((x/F ) ) D (x/F ) that is A/F is a skew semi-G- algebra. ∗∗ n ∗ ∗∗ ∗∗ ∗∗ (ii) We have ((x ) ) D x → (x → · · · → (x → 0)) and if we apply inductively Proposition 5.19 (iii), finally we obtain that ∗∗ ∗∗∗ ∗ x → 0 = x D x ⊆ F . Proposition 5.28. Let A be a residuated skew lattice with 0. Then M is an easy and maximal filter iff M is a prime filter of type (I) and a positive implicative filter. Proof. If M is an easy and maximal filter, then M 6= A and by Propo- sition 3.11 of [11] M is a prime filter of type (I). Consider x ∈ A such that x∈ / M. Then (xn)∗ ⊆ M, for some n ≥ 1. On the other hand (x∗∗)n (xn)∗∗, hence (xn)∗ ((x∗∗)n)∗. By above proposition we deduce that x∗ ⊆ M, hence x∨x∗ ∨x ⊆ M, that M is a positive implicative filter. Conversely, consider now M 6= A such that M is a positive implicative filter, hence x ∨ x∗ ∨ x ⊆ M, for all x ∈ A. Since M is a prime filter of type (I), therefore if x∈ / M, then x∗ ⊆ M, so M is a maximal filter. To prove that M is an easy filter, consider x, y ∈ A such that x∗∗ → (x∗∗ → y) ⊆ M and we must show that x∗∗ → y ⊆ M. If by 428 R. Koohnavard and A. Borumand Saeid

∗∗ ∗∗ ∗ ∗∗ contrary, x → y * M, then (x → y) ⊆ M. Since y x → y, then (x∗∗ → y)∗ y∗, hence y∗ ⊆ M. Since x∗ x∗∗ → y, then (x∗∗ → y)∗ x∗∗, hence x∗∗ ⊆ M. From x∗∗, x∗∗ → (x∗∗ → y) ⊆ M we deduce successively x∗∗ → y ⊆ M, y ∈ M, a contradiction (since from y ∈ M, y∗ ⊆ M we deduce that 0 ∈ M, hence M = A).

Theorem 5.29. Any skew G-algebra is a skew semi-G-algebra.

prime filter (I) prime filter (III) o / prime filter (II) o / prime filter (I) 9 prime filter (III) \

Boolean ﬁlter o / strong ﬁlter O e x⇤⇤ D x

positive implicative prime ﬁlter (I)

positive implicative ﬁlter / fantastic ﬁlter o 7 = O e O

prime ﬁlter (I) maximal ﬁlter

% ✏ maximal filter / maximal filter implicative filter o ultraO filterX h

implicative ﬁlter

✏ ✏ ⌃ x =1,x⇤ F easy ﬁlter maximal maximal ﬁlter 8 6 ✓

) y Stonean_ filter Green filter positive implicative filter, prime filter (I)

x =1,x⇤ F 8 6 ✓

implicative ﬁlter

Figure 1. The relationship among various kinds of ﬁl- ters in residuated skew lattices (Skew) ﬁlters in residuated skew lattices: Part II 429

prime skew filter (III) positive implicative skew filter fantastic skew filter

prime skew ﬁlter (I) prime skew ﬁlter (I)

prime skew filter (II) ultra skew filter Boolean skew filter

Figure 2. The relationship among various kinds of skew ﬁlters in residuated skew lattices

skew Heyting algebra O

conormal, distributive, symmetric and x y D x u y x y x y x y x y ∧ D ∧ D

% skew Hilbert algebra symmetric branchwise residuated skew lattice O

residuated skew lattice

skew G-algebra o / skew semi-G-algebra O x∗ D x

2 x D x

(x∗∗ x)∗ = 0 → / residuated skewd lattice strong residuated skew lattice

x∗ x∗∗ x∗ = 1 or x∗∗ x∗ x∗∗ = 1 ∨ ∨ ∨ ∨

Stonean residuated skew lattice

Figure 3. The relationship among some algebraic structures 430 R. Koohnavard and A. Borumand Saeid

6. Conclusion

In this paper, some types of (skew) filters were defined in residuated skew lattices and some relationships were investigated among them. Some relations were considered between these filters and quotient algebras constructed via these filters too. The Green filter was defined and was shown that its quotient algebra is a residuated lattice. Also it was shown that if F is an ultra filter, then its quotient algebra is a residuated skew chain. The relationship between residuated skew lattice and other skew structures was studied. It was shown that a residated skew lattice A is Stonean iff {1} is a Stonean skew filter of A and was shown that if F is a positive implicative filter, then A/F is a Stonean residuated skew lattice and a skew G-algebra too. It was shown that if any filter of a residuated skew lattice A is a Boolean filter, then A is a strong residuated skew lattice.

7. Acknowledgment

We would like to warmly thank the referees for their helpful comments and suggestions for the improvement of this paper.

References

[1] R. J. Bignall, J. Leech, Skew Boolean algebras and discriminator varieties, Alge- bra Univ, 33 (1995), 387-398. [2] A. Borumand Saeid, R. Koohnavard, On residuated skew lattices, An. St. Univ. Ovidius Constanta (to appear). [3] A. Borumand Saeid, M. Pourkhatoun, Obstinate filters in residuated lattices, Bull Math. Soc Sci. Math Roum Tome, 55 103(4) (2012), 413-422. [4] D. Busneag, D. Piciu, Some types of filters in residuated lattices, Soft Comput, 18(5) (2014), 825-837. [5] K. Cvetko-Vah, M. Kinyon, J. Leech, and M. Spinks, Cancellation in skew lattices, Order, 28 (2011), 9-32. [6] K. Cvetko-Vah, A. Salibra, The connection of skew Boolean algebras and discriminator varieties to Church algebras, Algebra Univ, 73 (2015), 369-390. [7] K. Cvetko-Vah, On skew Heyting algebra, Ars Math, Contemp, 12 (2017), 37-50. [8] P. Hajek, Metamathematics of fuzzy logic, Trends Log. Stud. Log. Libr 4, Dor: Kluwer Acad Publ (1998). [9] M. Haveshki, A. Borumand Saeid, E. Eslami, Some types of filters in BL-algebras, Soft Comput, 10(8) (2006), 657-664. [10] P. Jordan, Under nichtkommutative verbande, Arch. Math, 2 (1949), 56-59. [11] R. Koohnavard, A. Borumand Saeid, (Skew) filters in residuated skew lattices, Sci. An. Comput. Sci, (to appear). (Skew) filters in residuated skew lattices: Part II 431

[12] J. Leech, Normal skew lattices, Semigroup Forum, 44 (1992), 1-8. [13] J. Leech, Recent developments in the theory of skew lattices, Semigroup Forum, 52(1) (1996), 7-24. [14] J. Leech, Skew Boolean algebras, Algebra Univ, 27 (1990), 497-506. [15] J. Leech, Skew lattices, Unpublished Book. [16] J. Leech, Skew lattices in rings, Algebra Univ, 26 (1989), 48-72. [17] D. Piciu, Algebras of fuzzy logic, Univ Craiova Publ House, Craiova, Romania, (2007). [18] J. Pita Costa, On ideals of a skew lattice, Gen Algebra. Appl, 32 (2012), 5-21. [19] O. Zahiri, R. Borzooei, M. Bakhshi, Quotient hyper residuated lattices, Quasi- groups and Related Systems, 20(1) (2012), 125-138.

Roghayeh Koohnavard Department of Pure Mathematics, Faculty of Mathematics and Com- puter, Shahid Bahonar University of Kerman, Kerman, Iran. E-mail: [email protected]

Arsham Borumand Saeid Department of Pure Mathematics, Faculty of Mathematics and Com- puter, Shahid Bahonar University of Kerman, Kerman, Iran. E-mail: [email protected]