This theory and lattice theory. %section{Background } The

This research is divided into three parts: Compatible Sets of Almost Distributive Fuzzy Lattice, Ideals and Filters of an Almost Distributive Fuzzy Lattice, and Fuzzy Ideals and Fuzzy Filters of an Almost Distributive Fuzzy Lattice. This research lies in the areas of fuzzy theory and lattice theory.
%section{Background }

The word fuzzy logic was first introduced with the 1965 proposal of fuzzy set theory by Zadeh cite{zad}. Fuzzy logic has been applied to many fields such as control theory, artificial intelligence, medicine, economy, etc.
In mathematics and statistics, a fuzzy variable (such as temperature, Velocity or age) is a value that could lie in a probable range defined by measurable parameters, and can be usefully defined with vague classes (such as high, medium or low).

In mathematics and computer science, the various degrees of applicable connotation of a fuzzy notion are abstracted and defined in terms of measurable relationships defined by logical operators. Such method is rarely termed $”degree-theoretic$ $semantics “$ by logicians and philosophers, but the more common term is fuzzy logic or many-valued logic. The basic idea is that a real number is allocated to every statement written in a language, in a range from 0 to 1, where 1 means that the statement is totally true and 0 means that the statement is totally false whereas in-between values characterize that the statements are partly true. This makes it possible to examine a spreading of statements for their truth-content, classify data patterns, make inferences and estimates and model how progressions operate.
Fuzzy notions frequently play a role in the inventive process of creating new ideas to comprehend something. In the most primeval sense, this can be observed in children who, through practical experience, learn to classify, differentiate and simplify the correct application of an idea, and narrate it to other ideas.
Fuzzy notions are often used to denote complex phenomena, or to express something which is emerging and varying, which might involve shedding some old meanings and acquiring new ones.
The fuzzy logic is broadly used in computer science and mathematics.

In mathematics it is broadly used in the fuzzification of mathematics concepts as groups, rings, lattices, ideals, filters and several others. According to Pedrycz and Gomide cite{ped}, fuzzy set ideas constitute one of the most essential tools of computational intelligence. The notion of fuzzy sets is rather new and intellectually inspiring and their applications are varied and innovative.

Order theory is a branch of mathematics that explores our instinctive concept of order using binary relations. It provides a proper background for describing statements such as $” thisquad isquad lessquad thanquad that”$ or $” thisquad precedesquad that”$. The possessions common to orders we see in our daily lives have been extracted and are used to describe the concepts of order. Cars waiting for the signal to change at an intersection, natural numbers ordered in the increasing order of their magnitude are just a few examples of order we encounter in our daily lives. This intuitive idea can be extended to orders on other sets of numbers, such as the integers and the reals. The order relations we are going to study here are an abstraction of those relations.

According to Zadeh cite{zad1}, a fuzzy relation, which is a generalization of a function, has a normal extension to a fuzzy set and plays a significant role in the concept of such sets and their applications. Similar to an ordinary relation, a fuzzy relation in a set X is a fuzzy set in the product space $X imes X $ . The characteristic function of a crisp relation can be generalized to allow tuples to have degrees of membership within the relation.
Thus, a fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets, may have changeable degrees of membership within the relation. The membership grade is ordinarily represented by a real number in the closed interval 0, 1, and designates the strength of the relation present between the elements of the tuple.

Orders are everywhere in mathematics and associated fields like computer science.
However, each element cannot always be compared to any other. For example, consider the inclusion ordering of sets. If set A contains all the elements of set B, then, B is said to be smaller than or equal to A. However, there are some sets that cannot be related in this way. Whenever both contain some elements that are not in the other one, the two sets are not related by inclusion. Hence, inclusion is only a partial order, as opposed to the total orders given before.

The lattice theory has several applications in the areas of information engineering, physical sciences, communication systems, and information analysis, especially in coding theory and cryptography. Cryptography algorithms based on lattice theory have gained notoriety due to the advent of quantum computers because they become unsafe methods of public key based on number theory as well as the constructions of codes and lattices through rings of semi group and decoding algorithms for codes obtained. Additional application of the notion of lattices is on image processing techniques such as magnetic resonance imaging, nuclear magnetic resonance or computed tomography, which is expected to get good results using codes constructed from lattices.

These notions can be extended to fuzzy theory. In 1971, Zadeh cite{zad1} defined a fuzzy ordering as a generalization of the notion of ordering, that is, a fuzzy ordering is a fuzzy relation that is transitive. In particular, a fuzzy partial ordering is a fuzzy ordering that is reflexive and antisymmetric. The concepts of fuzzy sublattices and fuzzy ideals of a lattice was introduced by Yuan and Wu cite{yua}. Fuzzy lattice was defined as a fuzzy algebra by Ajmal and Thomas cite{ajm} and they characterized fuzzy sublattices at the first time. In 2000, fuzzy ideal and fuzzy filters of a lattice was defined and, characterized in terms of join and meet operations by Attallah cite{att}.
More recently, In 2009, fuzzy partial order relation was characterized in terms of its level set by Chon cite{ch}. Chon, in the same paper, defined a fuzzy lattice as a fuzzy relation, and developed basic properties and characterized a fuzzy lattice by its level set. He also familiarized the concepts of distributive and modular fuzzy lattices and considered some basic properties of fuzzy lattices.

The applications of fuzzy lattices theory are very analogous to lattices theory as in coding theory, cryptography, image processing techniques, decoding algorithms for codes and neuro-computing of fuzzy lattice that occurs as a connectionist standard in the background of fuzzy lattices whose benefits include the ability to deal rigorously with different types of data such as verbal and numeric data.
While numerous diverse notions of fuzzy order relations have been presented in literature, like Belohlavek cite{bel}; Bodenhofer and Kung cite{bod}; Fodor and Roubens cite{fod}; Gerlacite{ger}; Yao and Lu cite{yao}. The notion introduced by Zadeh cite{zad1} has been broadly considered in recent years as we can find in Amroune and Davvaz cite{amr}; Beg cite{beg}; Chon cite{ch}; Mezzomo et al.cite{mez}; and Seselja and Tepavceviccite{ses}.

On the other hand, the axiomatization of Boole’s two valued propositional calculus led to the concept of Boolean algebra and the class of Boolean Algebras (Ring). This includes the ring theoretic generalizations and the lattice theoretic generalizations like Heyting Algebras and distributive lattice. U.M. Swamy and G.C.Rao cite{ym} introduced the concepts of an ADL as a common abstraction of Heyting algebra and distributive lattice. The theory of ADL was further studied by U.M. Swamy, G.C.Ra , G.N.Rao, S. Ramish, M.P.K. Kishor and otherscite{swa, swa1, swa2, swa3}, in the last few decades. The benefit of this study is that it will allow us to see the structure of an almost distributive lattice in terms of a fuzzy partial order relation. Our purpose is to use the concepts of fuzzy relation to study the properties of an almost distributive lattice.

The first notions of almost distributive lattice was studied in terms of a partial ordered relation by U.M. Swamy and G.C Raocite{ym}. On the other hand, the results of various researchers in the area strongly suggest that the fuzzyfications of those classical concepts will prove a fruitful area of research. This work will advance the state of the art in the theoretical developments of fuzzy order theory, and leads to new tools and approaches to support the study of Almost Distributive Lattice in general, and the theoretical developments of the notions of ideals and filters of an Almost Distributive Lattice in particular.

One of the main aims of my work is to extend the notions of an Almost Distributive Lattice into a new Mathematical notion, Almost Distributive Fuzzy Lattice. Often the most powerful theories in mathematics are those which links the gaps between diverse areas. Fuzzy partial order relation was one of these in that it provided a passage between Lattice theory on the one hand and fuzzy lattice theory on the other. Lattice can be defined as a poset or equivalently as an algebra. In the classical concepts of a lattice, an ordered pair may have a degree of relation only either 1 or 0, which means, the characteristics image (their degree of relation) of any arbitrary ordered pair in the cross product of R has only two possibilities, it is equal to 1 if it is a member of a partially ordered relation and 0 if it is not.

So, the need of fuzzyfing the notions of lattice theory is to fill such a gap of giving the same degree of memberships of two different ordered pairs in the cross product of a given set R. In the case of fuzzy lattice theory, instead of a partial order relation we use fuzzy partial order relation. The degree of relations between two arbitrary elements will be assigned by any arbitrary real number in 0, 1. On the other hand, Almost Distributive Lattice is also a branch of order theory, and it can be defined as an algebra. Since the two binary operations $wedge $ “meet” and $vee $ “join” are characterized by a partial order relation $ “leq^” $ , the degree of relationship between two arbitrary elements of set R will be either 0 or 1 only. And it indicates that, the relation is vague. This motivates us to extend the concepts of an almost distributive lattice to almost distributive fuzzy lattice. As far as my knowledge is concerned, still now the concepts of an ADL has not been fuzzified. This work has four Chapters. The first Chapter talks about Preliminary, the second about fuzzy compatible sets of an Almost Distributive Fuzzy Lattices. Where as the third Chapter talks about Ideals and Filters of an Almost Distributive Fuzzy Lattice. Finally, in the fourth Chapter, we discuss about Fuzzy Ideals and Fuzzy Filters of an Almost Distributive Fuzzy Lattice.

Thus, the first aim of my study is to introduce a new Mathematical notion of an Almost Distributive Fuzzy Lattice (ADFL) and to study its property analogous to the classical concepts of an Almost Distributive Lattice.
U.Suamy and G.C. Rao, in 1981 cite{ym}, defined the notions of an ADL an algebra $(R, vee, wedge, 0)$ of type (2, 2, 0) which satisfies all the
axioms of a distributive lattice with smallest element 0 except possibly the
commutativity of $ vee $ and $ wedge $ and the right distributivity of $ vee $ over $ wedge $. In this work we fuzzify this notion (ADL), and introduce a new Mathematical notion of an ADFL in terms of a fuzzy partial order relation.
By considering a fuzzy partial order relation A forms a non-empty set R to 0, 1, we restate all the six axioms of an Almost Distributive Lattice in terms of a given fuzzy relation, we call a non empty set with a fuzzy partial order relation (R, A) an Almost Distributive Fuzzy Lattice if it satisfies all the axioms of a distributive fuzzy lattice except possibly the commutativity of $ vee $ and $ wedge $ and the right distributivity of $ vee $ over $ wedge $.
We also characterize an almost distributive fuzzy lattice (R, A) in terms of an almost distributive lattice$ ( R, vee, wedge, 0)$. In addition, we characterize an almost distributive fuzzy lattice (R, A) in terms of an almost distributive lattice $(R, A_alpha)$ where $A_alpha = { (x, y) in R^2 : A(x, y) > 0}$ is the support set of A.
In addition, we investigate some basic properties of an ADFL (R, A) analogues to the corresponding properties of an ADL $( R, vee, wedge, 0) $. Moreover, we fuzzify the concepts of amicable sets, and compatible sets, of an ADL analogues to the corresponding classical concepts. Finally, we define homomorphisms of an ADFL, and investigate some basic properties analogues to the classical concepts of homomorphisms of an ADL.

The other aim of this research is ideals and filters of an ADFL: to define the notions of ideals and filters of an almost distributive fuzzy lattice and investigate their properties analogous to the corresponding classical ideals and filters of an ADL. This part of my research will help advance the study of properties of ideals and filters of an ADFL.
In mathematical order theory, an ideal is a characteristic subset of a partially ordered set (poset). Although this word was historically derived from the idea of a ring ideal of abstract algebra, it has successively been generalized to a different notion. Ideals have great importance for many foundations in order and lattice theory. The dual notion of an ideal is known as filter. The main aims of this part of these research is to extend the notions of ideals and filters of an ADFL, and to study their properties in terms of a fuzzy partial ordered relation analogous to the corresponding classical ideals and filters of an almost distributive lattice properties.

We introduce the notion of ideal and filters in an ADFL (R, A) in terms of a fuzzy partial order relation which agrees with the usual notion of ideal and filters in a lattice in the case when R is an almost distributive lattice and we investigate some basic properties that coincides with the properties of the usual properties of ideals and filters of an ADL, R. We also define the smallest ideals $(S_A$ and the smallest filters $S)_A$ of an ADFL (R, A) generated by a non empty subset S of R as:
(S_{A}={x in R:A(x, (igvee s_{i})_{i=1}^{n} wedge x) > 0, quad where quad s_{i} in S quad and quad n in Z^{+} } quad and,
F)_{A}={x in R: A({igwedge }_{i=1}^{n}f_{i}, xwedge ({igwedge }_{i=1}^{n}f_{i})) > 0, quad for quad some quad f_{i} in F quad and quad n in Z^{+} }.
We also define prime ideals, prime filters, maximal ideals, maximal filters, minimal ideal, and minimal filters of an ADFL. In addition, we investigate and prove some basic properties of prime ideals and filters of an ADFL analogous to the properties of prime ideals (filters) and maximal (minimal) ideals (filters) of an ADL.
In addition, We define the notions of principal ideals and principal filters of an Almost Distributive Fuzzy Lattice generated by any arbitrary element $ain R$ in terms of a fuzzy relation A as : (a_{A}= {xin R/ A(x, awedge x) > 0}, quad and
a)_{A}= {xin R/ A(a, xwedge a) > 0} respectively.
We investigate some basic relationships of principal ideals and principal filters of an Almost Distributive Fuzzy Lattice. Moreover, we studied the basic relationships between principal ideals and amicable elements of an ADFL L.
Moreover, we introduce the concepts of a fuzzy lattice by defining a fuzzy relation B on the set containing all ideals $I_{A}(L)$, and a fuzzy relation C on the set containing all filters $F_{A}(L)$ of a given Almost Distributive Fuzzy Lattice L. In addition, we proved that a fuzzy poset $(PI_{A}(L), B)$ and $(PF_{A}(L), C)$ forms a fuzzy distributive lattice, where $PI_{A}(L)$ is the set containing all principal ideals, and $PF_{A}(L)$ denotes the set containing all principal filters of a given ADFL L respectively. Moreover, we introduce ideals and filters of the two fuzzy distributive lattice $(PI_{A}(L), B)$ and $(PF_{A}(L), B)$ induced by arbitrary ideal I and filter F of an ADFL L, where $I_{A}(L)$ and $F_{A}(L)$ represent the set containing all ideals and the set containing all filters of an ADFL L respectively.

The main objective of these research in the notions of fuzzy ideals and fuzzy filters are: to define fuzzy ideals and fuzzy filters of an ADFL as a fuzzy set $mu$ from a set R to 0, 1, and to study their properties as a fuzzy set.
In cite{ym} U.M.Swamy and G.C.Rao, define the notions of ideals and filters of an ADL as a non empty subsets of a set R, and similarly, we also define ideals and filters of an ADFL as a non empty subset of R too with respect to a fuzzy relation A. In both cases, the notions of ideals and filters are crisp subsets of R, and the degree of memberships of their elements are either 0 or 1 only. Here, as a main result of this research work, we define the notions of fuzzy ideals and filters of an ADFL as a fuzzy set. The result of this research may contribute a significant development of the study of ideals and filters of an ADFL.
We introduce the notion of fuzzy ideal and fuzzy filter in an ADFL (R, A) as a fuzzy set $mu$ in terms of a fuzzy partial order relation which agrees with the notion of ideal and filter of an almost distributive lattice, and we investigate some basic properties that coincide with the properties of ideals and filters of an ADFL. In addition, we characterize a fuzzy ideal(filter) $mu$ of an ADFL by the it’s support set $S(mu)$.
We also define the smallest fuzzy ideal and filter of an ADFL induced by any arbitrary non-zero fuzzy set. We deal with the smallest ideal $(S_A$ and the smallest filter $S)_A$ of an ADFL (R, A) generated by any non-empty subset S of R , where, $(S_{A}={x in R:A(x, (igvee s_{i})_{i=1}^{n} wedge x) > 0$, where $s_{i} in S$ and $n in Z^{+} }$, and $F)_{A}={x in R: A({igwedge }_{i=1}^{n}f_{i}, xwedge ({igwedge }_{i=1}^{n}f_{i})) > 0$, for some $f_{i} in F$ and $n in Z^{+} }$. Since we intend to define a fuzzy ideal(filter) of an ADFL as a fuzzy set, in this section we also deal with the smallest fuzzy ideal(filter) of an ADFL induced by a non zero fuzzy set $mu$ from R to 0, 1. Consider any arbitrary non-zero fuzzy set $mu$ on R and define a fuzzy set $mu^i_{S(mu)}$, and $mu^f_{S(mu)}$ on R with respect to the support set $S(mu)$ as follows:
for all $xin R$, $mu^i_{S(mu)}(x) = min{mu(s_i)} $, if there exist some $s_i’$s in $S(mu)$ such that $A(x, (igvee s_{i})_{i=1}^{n} wedge x) > 0$, and $mu^i_{S(mu)}(x) = 0$, otherwise. Similarly, for all $xin R$, $mu^f_{S(mu)}(x) = min{mu(f_i)}$ if there exist some $f_i’$s in $S(mu)$ such that $A({igwedge }_{i=1}^{n}f_{i}, xwedge ({igwedge }_{i=1}^{n}f_{i})) > 0$, and $mu^i_{S(mu)}(x)= 0$, otherwise. As the main objective of this research, we proved that $mu^i_{S(mu)}$ is a fuzzy ideal and $mu^f_{S(mu)}$ is a fuzzy filter of an ADFL (R, A) containing $mu$. In addition, since the images of $mu^i_{S(mu)}$ and $mu^f_{S(mu)}$ are not directly defined in terms of $A$, we also define additional fuzzy set $mu^{idownarrow}_{S(mu)}$ and $mu^{fuparrow}_{S(mu)}$ from a set R to $0, 1$ in terms of $mu^i_{S(mu)}$, and $mu^f_{S(mu)}$ respectively as \mu^{idownarrow}_{S(mu)}(x) = Sup_{rin R} min{mu^i_{S(mu)}(r), A(x, r)}, and \mu^{fuparrow}_{S(mu)}(x) = Sup_{rin R} min{mu^f_{S(mu)}(r), A(r, x)}.
As we have seen, both fuzzy sets $mu^{idownarrow}_{S(mu)}$ and $mu^{fuparrow}_{S(mu)}$ are defined as a function of a given fuzzy relation. Here also, we proved that $mu^{idownarrow}_{S(mu)}$ is a fuzzy ideal and $mu^{fuparrow}_{S(mu)}$ is a fuzzy filter of an ADFL (R, A) containing $mu^i_{S(mu)}$, and $mu^f_{S(mu)}$ respectively. In addition, we studied the properties of operations (intersection and union) on fuzzy ideals and fuzzy filters of a given ADFL.
We also define the notions of prime fuzzy ideals and filters of an Almost Distributive Fuzzy Lattice, and investigate some basic properties. We intent to characterize fuzzy prime ideals (filters) $mu$ of (R, A) of an ADFL in terms of its support set $S(mu)$ of $mu$.
Moreover, we studied the properties of homomrphisms of an ADFL on fuzzy ideals and fuzzy filters. \
Finally we studied properties of a homomorphism on fuzzy ideals and fuzzy filters of an
ADFL, and investigate and proof some basic results.

BACK TO TOP