Abstract:
Abstract
In this research work our aim is to introduce implicative almost distributive lattices
as a generalization of implicative algebras in the class of almost distributive lattices.
Our motivation is based on the following ideas.
Gratzer, G. discussed some theory's in the development of lattice theory. Af-
ter two decades Xu, Y. proposed a logical algebra - lattice implication algebra by
combining algebraic lattice and implication algebra, which is an important form of
non-classical logical algebra. This lattice valued logic is becoming a research eld
which strongly in
uences the development of algebraic logic, computer science and
arti cial intelligence.
Further, Venkateswarlu, K. and Berhanu, B. introduced the concept of implica-
tive algebras (IAs). Moreover, they proved that implicative algebra is equipped
with a structure of a bounded lattice and it is also a lattice implication algebra.
The notion of an almost distributive lattice (ADL) was introduced by Swamy,
U.M. and Rao, G.C. as a common abstraction to most of the existing ring theoretic
and lattice theoretic generalizations of Boolean algebra without considering the
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right distributivity of _ over ^, commutativity of _ and ^ and (x ^ y) _ x = x for
x; y 2 L of an ADL L.
As a result we have identi ed gaps between IAs and ADLs. And there was no
study conducted regarding this issue. Therefore, we are motivated to extend IAs
to implicative ADLs in the class of ADLs. Here we apply methodology of Swamy,
U.M. and Rao, G.C. in order to produce our results in this research work.
In this dissertation, the theory of implicative almost distributive lattices (IADLs)
is introduced and we have developed di erent results related with IADLs.
In the rst chapter of this study we list some preliminary results that will be
useful for reference in the coming chapters.
In the second chapter of this study, we discuss the concept of IADLs as a gener-
alization of implicative algebra in the class of ADLs and some characterizations of
IADL are investigated. In addition, we de ne the concept of autometrized implica-
tive almost distributive lattices (AIADLs) as extension of autometrized algebra in
the class of IADLs. Regular autometrized IADLs are also discussed. Moreover, we
introduce the concept of H-implicative almost distributive lattices (H-IADLs) and
homomorphisms in IADLs and study their properties.
In the third chapter of this dissertation, we study the concept of positive im-
plicative lters, associative lters, transitive and absorbent lters of IADLs. We
prove that every positive implicative lter is an implicative lter and every associa-
tive lter is a lter. In addition, we give equivalent conditions for both a positive
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implicative lter and associative lter in IADLs. Further more, necessary and suf-
cient condition is derived for every lter to become a transitive lter. Also, a set
of equivalent conditions is given for a lter to become an absorbent lter.
In the fourth chapter of this dissertation, we study LI-ideals and congruence
relations in IADLs. We prove that every LI-ideal of L is an ADL ideal of L. How-
ever we show that the converse doesn't hold in IADL unless the IADL is H-IADL.
In addition, we discuss the relationship between lters and LI-ideals , generate an
LI-ideal by a set, construct quotient structure by using LI-ideal and study prop-
erties of LI-ideals related to implicative homomorphisms on IADLs. Besides, the
basic properties and the structures of general congruence relations on IADLs are
discussed. Also, we investigate that an IADL is congruence -per-mutable.
Finally, in the fth chapter of this dissertation, the notion of ILI-ideals, prime LI-
ideals and maximal LI-ideals of IADLs are introduced. The properties of ILI-ideals,
prime LI-ideals and maximal ideals are investigated. Several characterizations of
ILI-ideals and prime LI-ideals are given. The extension theorem of ILI-ideals is
obtained. Some relations among ILI-ideals, LI-ideals, implicative lter, prime LI-
ideals and maximal LI-ideals are observed. Some classes of IADLs are characterized
by their ILI-ideals (respectively prime LI-ideals, maximal LI-ideals).