Fuzzy Amicable sets of an Almost Distributive Fuzzy Lattice

In this paper, we introduce the concept of Fuzzy Amicable sets, we prove some properties of Fuzzy Amicable set, too. We also prove that two Fuzzy compatible elements of an Almost distributive Fuzzy Lattice (ADFL) are equal if and only if their corresponding unique Fuzzy amicable elements are equal. We define the homomorphism of two Almost Distributive Fuzzy lattices(ADFL) and finally we observe that any two Fuzzy amicable set in an Almost Distributive Fuzzy Lattice (ADFL) are isomorphic.


I. INTRODUCTION
T HE 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. Samy and G.C. Rao [1] introduced the concepts of an ADL as a common abstraction of distributive lattice.
On the other hand, Zadeh [2] was the first mathematician who introduced the concepts of fuzzy and to define and study fuzzy relations, Sanchez [3] and Goguen [4] was adapted this concept.The notion of partial order and lattice order goes back to 19th century investigations in logic.The concepts of fuzzy sublattices and fuzzy ideals of a lattice was introduced by Yuan and Wu [5].Fuzzy lattice was defined as a fuzzy algebra by Ajmal and Thomas [6] and they characterized fuzzy sublattices as a 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 [7].In 2009, fuzzy partial order relation was characterized in terms of its level set by Chon [8].Chon in the same paper defined a fuzzy lattice as a fuzzy relation, developed basic properties and characterized a fuzzy lattice by its level set.As a continuation of these studies, in 2016 Berhanu et al. [9] define an Almost Distributive Fuzzy Lattice as a generalization of Fuzzy Lattice and fuzzify some properties of the classical Almost Distributive Lattice using the fuzzy partial order relation and fuzzy lattice defined by Chon [8].As a contination of Berhanu et al. [9], In this work we introduce a new Mathematical notion, Fuzzy amicable sets and Fuzzy compatible sets of an Almost Distributive Fuzzy Lattice that preserve different properties of classical amicable set and compatible sets of an Almost Distributive Lattice.Lemma 9 ([9]): Let (R,A) be an ADFL.Then for any a and b in R we have 1) A(a, a∧(a∨b)) = 1; 2) A(a∨(a∧b), a) = 1; 3) A(a∨(a∧b), a∧(a∨b)) = 1; 4) A(a, (a∨b)∧a) = 1; 5) A(a∨(b∧a), a) = 1; 6) A(a∨(b∧a), (a∨b)∧a) = 1.
Corollary 10 ( [9]): Let (R, A) be an ADFL.Then for any a and b in R, 1) A(a∨b, a) > 0 if and only if A(b, a∧b) > 0.
Theorem 11 ( [9]): Let (R, A) be an ADFL.Then for any a,b in R the following are equivalent.
3) A(a∨b, b∨a) > 0 and A(b∨a, a∨b) > 0. 4) The infimum of a and b exists in R and equals a∧b.5) A(a∧b, b∧a) > 0 and A(b∧a, a∧b) > 0.
6) The supremum of a and b exists in R and equals a∨b.
2) The fuzzy poset (R, A) is directed above.

III. FUZZY AMICABLE SET
In this section we introduce a new mathematical notion , Fuzzy Amicable Sets of an Almost Distributive Fuzzy Lattice and we investigate and prove some results.

A. Fuzzy Compatible Set of An Almost Distributive Fuzzy Lattice
Definition 19: Let (R,A) be an ADFL.For any a , b ∈ R, we say that a is fuzzy compatible with b ( written a ∼ A b) if A(a∧b, b∧a) > 0 and A(b∧a, a∧b) > 0 or equivalently, A(a∨b, b∨a) > 0 and A(b∨a, a∨b) > 0. For a sub set S of R, if for all a , b ∈ S a ∼ A b, then S is said to be fuzzy compatible.Note that a maximal set in this paper is defined in the usual sense, and next we give the definition of Fuzzy amicable sets.
Definition 20: Let (R, A) be an ADFL and let (M, A) be a fuzzy maximal compatible set.Then an element a of R is said to be M-fuzzy amicable if there is an element d of M such that A(a, d∧a) > 0. We call (M,A) fuzzy amicable set, if every elements of R is M-fuzzy amicable.
Example: Let R = {0, a, b, c } and M ={0, a, b}.Define two binary operations ∨ and ∧ in R as follow:  (R,A) is an ADFL.Now, consider M ⊆ R. For any x, y ∈ M, A(x ∧ y, y ∧ x)> 0 and A(y ∧ x, x ∧ y)> 0. Hence (M, A) is a fuzzy compatible set, and it is Fuzzy maximal compatible set.Thus, (M, A) is a fuzzy maximal compatible set.Now, let x ∈ R. Then there exists y in M such that A(x, y ∧ x)>0.Hence, every elements of R is M-fuzzy amicable.Therefore, (M, A) is fuzzy amicable set.
Lemma 21: Let (R, A) be an ADFL, and let (M, A) be a fuzzy maximal set in (R, A) and x ∈ R be such that x ∼ A a for all a ∈ M. Then x ∈ M.
Proof: Let (M, A) be a fuzzy maximal set in (R, A), and Proposition 22: Let (R, A) be an ADFL and let (M, A) be a fuzzy maximal set in (R, A).Let a ∈ M. Then for any x ∈ R, x ∧ a ∈ M.
Proof: Let (M, A) be a fuzzy maximal set in an ADFL (R, A) and let a ∈ M. Let x ∈ R and let b be any arbitrary elements of M. Then A((x ∧a)∧b, b∧(x ∧a) Corollary 23: Let (M, A) be a fuzzy maximal compatible set in an ADFL (R, A).Then, for any x ∈ R and a ∈ M, A(x, a) > 0 implies A(x ∧ b, b ∧ x)> 0 and A(b ∧ x, x ∧ b)> 0 for every b ∈ M.
Proof: Let (M, A) be a fuzzy maximal compatible set in an ADFL (R, A).Suppose A(x, a) > 0, where x ∈ R and a Lemma 24: Let (M, A) be a fuzzy maximal compatible set in an ADFL (R, A) and a be an M-fuzzy amicable element.Then, there exists an element d of M such that A(a, d ∧ a) > 0 and if e ∼ A d and A(a, e ∧ a) > 0, then A(d, e) > 0. Thus, if (M, A) is a fuzzy amicable set, then to every a ∈ R there exists a A ∈ M such that, for every x ∈ M, A(x, a A ) > 0 and A(a A , x) > 0 if and only if A(a, x ∧ a) > 0 and A(x, a ∧ x) > 0 and hence, given a and M, such that a A is unique.
Proof: Let (M, A) be a fuzzy maximal set in an ADFL (R, A) and let a be an M-fuzzy amicable element.since a is an M-fuzzy amicable, there exists be since, a is M-fuzzy amicable, there exists a A ∈ M such that A(a, a A ∧ a) > 0 and if e ∼ A a A and A(a, e ∧ a) > 0, then A(a A , e) > 0. Suppose (M, A) is a fuzzy amicable set.Hence, every elements of R is M-fuzzy amicable.Let a ∈ R.. Then there exists a smallest a A ∈ M such that A(a, a A ∧ a) > 0. Let x ∈ M. Suppose A(x, a A ) > 0 and A(a A , x) > 0. Now, we show that Since a A and x are both elements of M, a A ∼ A x and by assumption we have A(a, x ∧ a) > 0. Hence by the first argument of this Lemma, we have Therefore, for any a ∈ R, there exists a unique a A ∈ M such that A(a, a A ∧ a) > 0 and if e ∼ A a A and A(a, e ∧ a) > 0, A(a A , e) > 0, Theorem 25: Let (M, A) be a fuzzy amicable set of an ADFL (R, A).Then for any x, y ∈ R, 1) Proof: let (M, A) be a fuzzy amicable set in an ADFL (R, A) and let x, y ∈ R.
1. Since (M, A) is a fuzzy amicable set and (x ∨ y) ∈ R, by Lemma 24, there exists a unique (x∨y) A in M such that (A(x∨ y, (x ∨ y) A ∧ (x ∨ y)) > 0 and A((x ∨ y) A , (x ∨ y) ∧ (x ∨ y) A ) > 0. On the other hand, (A(x ∨ y, (x A ∨ y A ) ∧ (x ∨ y)) > 0 and Thus, by Lemma 24, A(x A ∨ y A , (x ∨ y) A ) > 0 and A((x ∨ y) A , x A ∨ y A ) > 0. 2. Since (M, A) is a fuzzy amicable set and (x ∧ y) ∈ R, by Lemma 24, there exists a unique (x ∧ y) A ∈ M such that A((x ∧ y), (x ∧ y) A ∧ (x ∧ y)) > 0 and A((x ∧ y) A , (x ∧ y) ∧ (x ∧ y) A ) > 0. On the other hand, A(x ∧ y, Proposition 26: Let (M, A) be a fuzzy maximal set in an ADFL (R, A) and let x, y ∈ R be M-fuzzy amicable and x ∼ A y.Then, A(x A , y A ) > 0 and A(y A , x A ) > 0 if and only if A(x, y) > 0 and A(y, x) > 0.

B. Homomorphism on Fuzzy Amicable Sets
In this section, we define homomorphism of ADFLs as follows and we also prove the isomorphism of Fuzzy amicable sets.
Definition 27: Let L = (R, A) and K = (M, B) be two ADFL's and let f be a map from L to K. Then f is side to be homomorphism from an ADFL L to an ADFL K if the following axiom holds true: 3) f (0 R ) = 0 M where 0 R and 0 M are the zeros of R and M, respectively.A homomorphism f from L to K is called epimorphism, if f is an on-to map from L to K. A homomorphism f from L to K is called monomorphism, if f is a one-to-one map from L to K. A homomorphism f from L to K is called isomorphism, if f is both on -to and one-to-one map from L to K. A homomorphism f is called automorphism, if f is isomorphism on L.
Definition 28: Let L = (R, A) and K = (M, B) be two ADFL's and let f be a homomorphism from L to K. The kernel of f is defined as follow: Lemma 29: Let f be a homomorphism from an ADFL L = (R, A) to an ADFL K = (M, B).For any x, y ∈ R, B(f(x), f(y)) >0 whenever A(x, y) > 0. If B(f(x), f(y)) > 0 and f is monomorphism, then A(x, y) > 0 for x, y ∈ R.
Proof: Let f be homomorphism from L to K and let x, y ∈ R. suppose A(x, y) >0.Then x∧ R y = x and it follows that , and it follows x ∧ R y = x.Hence, A(x, y) > 0, since x ≤ y.
Corollary 30: Let f be an on to map from an ADFL (R, A) to an ADFL (H, B).If f preserves order, then f is a homomorphism.
Proof: Suppose f is an on to map which preserves order.Let x, y ∈ R such that x ≤ y.Then f (x) ≤ f (y) and hence f (x) ∧ H f (y) = f (x) and f (x) ∨ H f (y) = f (y).Since f is well defined and x ≤ y, f Finally, let 0 R and 0 H are the zero elements of (M, A) and (H, B) respectively.Since f is an onto map, there exists t ∈ R such that f (t) = 0 H . Since 0 R is the zero element of (R, A), 0 R ≤ t and since f preserves order, f (0 R ) ≤ f (t) = 0 H . on the other hand, since 0 H is the zero element of (H, B), 0 H ≤ f (0 R .Hence, f is homomorphism.
Next we prove the existence of Fuzzy maximal compatible set isomorphic with a given fuzzy maximal compatible set in an ADFL.
Theorem 31: Let (M, A) be a fuzzy maximal compatible set of an ADFL L = (R, A) and a ∈ R be M-fuzzy amicable.Then, there exists a fuzzy maximal compatible set (M , A) in L such that a ∈ M and the fuzzy lattice (M , A) is isomorphic with the fuzzy lattice (M, A).
Proof: Let (M, A) be a fuzzy maximal compatible set and let a ∈ R be an M-fuzzy amicable.Define M = {x∧(a∨x)/x ∈ M}.Let b, c ∈ M .Then, there exist x and y in M such that b = x∧(a∨ x) and c = y∧(a∨ y).
Next we prove that any Fuzzy amicable set is isomorphic with a Fuzzy maximal set in ADFL.
Hence, a ∈ M Since (M , A) and (M, A) are fuzzy maximal sets, for any a, b ∈ M and c, d ∈ M, A(a ∧ b, b ∧ a) = A(b ∧ a, a ∧ b) = 1 and A(c ∧ d, d ∧ c) = A(d ∧ c, c ∧ d) = 1.Thus, there exist sup's and inf's of { a, b } and { c, d}.Hence,(M , A) and (M, A) are fuzzy lattices.Finally, let's define a map from (M, A)in to (M , A) by f(x) = x∧ (a∨ x) for all x ∈ M, and let x, y ∈ M such that A(x, y) > 0 and A