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.