Fuzzy Heyting Algebras

Derebew, Nigussie

dc.contributor.author	Derebew, Nigussie
dc.date.accessioned	2019-03-27T08:34:24Z
dc.date.available	2019-03-27T08:34:24Z
dc.date.issued	2019-03-27
dc.identifier.uri	http://hdl.handle.net/123456789/9309
dc.description.abstract	Abstract In this dissertation several results of the new notion fuzzy Heyting algebras are introduced based on the crisp theory.The new result of the concept of congruence relations on Heyting algebra using implicatively as well as multiplicatively closed subsets of H is introduced.Using the de nition of homomorphism of Heyting alge- bras, we characterized and studied some important properties of quotient Heyting algebra by the congruence classes of it. As a result of the new notion fuzzy Heyting Algebra (FHA), we further stud- ied some important properties of fuzzy Heyting algebra using fuzzy relation and fuzzy poset de ned by Chon.We also characterized fuzzy Heyting algebra using the directed above fuzzy poset and proved that any distributive fuzzy lattice is fuzzy Heyting algebra i there exists a largest element c of H(Heyting Algebra) such that A(a ^ c; b) > 0, for all a,b 2 H. This dissertation aims to introduce fuzzy congruence relations over Heyting al- gebras (HA) and give constructions of quotient Heyting algebras induced by fuzzy congruence relations on HA. The fuzzy rst,fuzzy second and fuzzy third isomor- phism theorems of HA are established.Moreover, we investigate the relationships between fuzzy ideals and fuzzy congruence relations on HA. The e ect of a homomorphism on the join,product,and intersection of two fuzzy ideals of HA are discussed.The results obtained here will be useful in studying the al- gebraic nature of fuzzy prime ( fuzzy maximal,fuzzy semiprime,fuzzy primary,fuzzy semiprimary) ideals under homomorphism. The fuzzy prime ideals,fuzzy maximal ideals,fuzzy semi primary ideals of a Heyt- ing algebra are also characterized with their level sets.We give a brief discussion on fuzzy prime ideals and fuzzy maximal ideals,fuzzy semiprime ideals and fuzzy pri- mary ideals of of a Heyting algebra, cross product of fuzzy prime ideals and some characterizations. We concentrate on fuzzy prime ideals of Heyting algebra in such away that if is a fuzzy ideal of H and is a maximal ideal of H, then is a fuzzy viii maximal ideal of H.We also proved fuzzy ideal of H H is said to be fuzzy semiprime i the level ideals ( )t; t 2 im( ) is semiprime ideal of H H: We propose the notions of 􀀀ideals and 􀀀 lters of a fuzzy Heyting algebra and characterize them by using its support and its level set.We characterize a fuzzy ideal on product between fuzzy Heyting algebras L and M and de ne fuzzy 􀀀ideals of fuzzy Heyting algebra.Here, we characterize a fuzzy 􀀀ideals of product between fuzzy Heyting algebras L and M. This dissertation also has played great role on the study of Heyting almost dis- tributive fuzzy lattices(HADFLs) based on FHA.After we de ne a Heyting Almost Distributive Fuzzy Lattices(HADFLs) as an extension of a fuzzy Heyting algebra, we give many equivalent conditions for FHAs to become an HADFL.From the def- initions and results of the above concepts, many basic properties of HADFLs has been proved. We also introduce the concept of an implicative fuzzy lters in an HADL as a fuzzy lter of the same HADL and study the properties of implicative fuzzy lters of an HADL.Lastely,we de ne some particular implicative fuzzy lters of an HADL and prove that some of their properties are preserved under homomor- phisms of HADLs.	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Fuzzy Heyting Algebras	en_US
dc.type	Thesis	en_US