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Generalizations of Skew L-Almost Distributive Lattices and Their Duals DSpace Institution DSpace Repository http://dspace.org 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 2 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 2 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.
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