L-Fuzzy Meet Semi Almost Ideals

Algebraic structures like fuzzy sub rings of L-Fuzzy sets have been well studied in the case where the lattice L is distributive. In this paper axioms the concepts can be generalised to the case where the lattice is not distributive. The aim of this current paper is to define the concept of L-fuzzy meet semi almost ideals and also Intutionistic L-fuzzy meet semi almost ideals . Investigate some theorems and examples.


Introduction
Zadeh [1] introduced the concept of fuzzy sets in 1965.Intuitionistic Fuzzy sets was initiated by K.T.Akanassov [2].In our previous papers [5], [6] and [7] the concept of L-Fuzzy Almost Ideal (LFAI) was introduced and concept of primality in L-Fuzzy Almost Ideals and Intutionistic L-Fuzzy Almost Ideals was studied.In [10] , [11] Chellappa.B and Anand .B discussed Fuzzy join semi L-ideal and Fuzzy join subsemilattices.A.Kavitha and B.Chellappa studied A study on fuzzy meet semi L-ideal and Fuzzy meet semi L-Filter in 2013.In [15] R. Arimalar and Dr. B. Anandh investigated An Intuitionstic Fuzzy Meet Semi L-Filter.

Preliminaries
Let X be a nonempty subset, (L ,≤, ∨, ⋀) be a complete distributive lattice, which has least and greatest elements, say 0 and 1 respectively.Relevant definitions are recalled in this section.
Definition 2.1 Let X be a nonempty set.A mapping by μ: X → [0,1] is called a fuzzy subset of X. Definition 2.2 Let X be any non-empty set.A mapping μ: X → L with μ(0)=1 and μ(1)=0 is called a L-fuzzy subset of X.
Definition 2.3 Let A be a fuzzy meet semi Lattice.A fuzzy meet subsemi Lattice μ: X → [0,1] is called fuzzy meet semi L-ideal of A if ∀x,y∈A, μ(x ⋀ y) ≥ max{μ(x), μ(y)} Definition 2.4 Let A be a fuzzy join semi Lattice.A fuzzy join subsemi Lattice μ: X → [0,1] is called fuzzy join semi L-ideal of A if ∀x,y∈A, μ(x ⋀ y)≤min{μ(x ), μ(y)} Definition 2.5 Let R be a ring with unity.Let L be a lattice (L ,≤, ∨, ⋀) not necessarily distributive with least and greatest element 0 and 1 respectively.μ: R → L with μ(0) =1 and μ(1) = 0 is said to be L-fuzzy almost ideal if ∀x,y∈R Definition 2.6 Let (L ,≤) be any lattice with an involutive order reversing operation N: L→ L. Let X be any non-empty set.An Intuitionistic L-Fuzzy Set (ILFS) A in X is defined as an object of the form A = { < x , μ(x) , ν (x) > / x∈X } , where the functions μ : X→L and ν: X→ L define the degree of membership and the degree of non-membership respectively and for every x∈X satisfy μ(x)≤ N( ν(x)).

L-Fuzzy Meet Semi Almost Ideals
Definition 3.1 Let R be a ring with unity.Let L be a lattice (L ,≤,⋁ ,⋀) not necessarily distributive with least and greatest element 0 and 1 respectively.μ: R→L with μ(0) =1 and μ(1) = 0 is said to be L-fuzzy semi almost ideal if μ(x y)≮ μ(x) ⋁ μ(y), for all x,y ∈R.Definition 3.2 Let R be a ring with unity.Let L be a lattice (L ,≤,⋁ ,⋀) not necessarily distributive with least and greatest element 0 and 1 respectively.μ : R→ L with μ (0) =1 and μ (1) = 0 is said to be L-fuzzy meet semi almost ideal if μ(x ⋀ y)≮ μ(x) ⋁ μ(y), ∀ x,y ∈R.Definition 3.3 Let R be a ring with unity.Let L be a lattice (L ,≤,⋁ ,⋀) not necessarily distributive with least and greatest element 0 and 1 respectively.μ : R→ L with μ (0) =1 and μ (1) = 0 is said to be Intuitionistic L-fuzzy meet semi almost ideal if ∀ x,y ∈R

y) Example 3 . 4 L
The following is an example of a L-fuzzy meet semi almost ideal.Let R= {0,a,b,c,1}.Let L be a lattice (L ,≤,⋁,⋀) defined by below Hasse diagram.Note that L is not distributive.Define SNC JOURNAL OF ACADEMIC RESEARCH IN HUMANITIES AND SCIENCES μ: R→ L with μ(x) =1 as μ(x) ={1 if x = 0.4 a if x=0.5 b if x=0.6 c if x=0.7 0 if x=0.8))Verification that μ is a L-fuzzy meet semi almost ideal can be summarized in the form of a table as follows

c a Example 3 . 5
Figure I