Implicative Almost Distributive Lattices

Tilahun, Mekonnen

dc.contributor.author	Tilahun, Mekonnen
dc.date.accessioned	2019-03-27T08:38:57Z
dc.date.available	2019-03-27T08:38:57Z
dc.date.issued	2019-03-27
dc.identifier.uri	http://hdl.handle.net/123456789/9312
dc.description.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 viii 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 ix 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).	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Implicative Almost Distributive Lattices	en_US
dc.type	Thesis	en_US