Generalizations of Skew L-Almost Distributive Lattices and their Duals

Yeshiwas, Mebrat

dc.contributor.author	Yeshiwas, Mebrat
dc.date.accessioned	2019-03-27T08:40:46Z
dc.date.available	2019-03-27T08:40:46Z
dc.date.issued	2019-03-27
dc.identifier.uri	http://hdl.handle.net/123456789/9313
dc.description.abstract	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 de ne a congruence relation on a skew HADL L and show that each congruence class is a maximal rectangular subalgebra of L and L= 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 de ned 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 de ne a relation on a skew SHADL L so that each congruence class is the maximal rectangular subalgebra and L= 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 di erent 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 de ned 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 de ned 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.	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Generalizations of Skew L-Almost Distributive Lattices and their Duals	en_US
dc.type	Thesis	en_US