On fuzzy sublattice of a lattice

Introduction

The notion of fuzzy sets was introduced by Lofti. A. Zadeh in 1965. Zadeh had initiated fuzzy set theory as a modification of the ordinary set theory. The concept of lattice was first defined by Dedekind in 1897, and then developed by Birkhoff G. in 1933. Here we made an attempt to study the algebraic nature of fuzzy subset of lattice especially fuzzy sublattice, by using the ideas of fuzzy theory and lattice theory.

Preliminaries

Some well -known definitions and preliminary results are recalled Definition 1.1 A relation defined on a set S which is reflexive, antisymmetric and transitive is called a partial ordering on S. A set S with a partial ordering  defined on it is called a partially ordered set or a poset and is denoted by (S, ).

Definition 1.2

Let (P, ≤) be a poset. Let A be a non-empty subset of P.An element u  P is called an upper bound of A if a ≤ u for all aA. An element u P is called the least upper bound (l.u.b) of A if 1. u is an upper bound of A. 2. if v is any other upper bound of A, then u ≤ v.

An element l P is called the lower boundof A if l ≤ a for all a A. An element l  P is called the greatest lower bound (g.l.b) of A if 1. l is an lower bound of A. 2. if m is any other lower bound of A, then m ≤ l.

Definition 1.3

A lattice is a poset in which any two elements have a g.l.b and a l.u.b. We denote the l.u.b of a ˅ b and g.l.b by a ˄ b.

Consider the function, µ(x)

where s, t  [0, 1], s > t.

Assume that µ is a fuzzy sublattice of L. Then, 1. µ(x  y) ≥ min {µ(x), µ(y)} 2. µ(x  y) ≥ min {µ(x), µ(y)}  To prove H is a sublattice of L.

 Let a, b  H be arbitrary.  This proves that H is a sublattice of L.

Theorem: 2.6

If H is any sublattice of a lattice L, H  L, then the fuzzy subset µ of L defined by µ(x)

Where s, t[0, 1], s > t is a fuzzy sublattice of L.

Proof

Let H be any sublattice of a lattice L, H  L. Consider the fuzzy subset µ of L, defined by µ(x)

 To prove µ is a fuzzy sublattice of L.

It is enough to prove that

We prove this in three cases

Here, x, y  H That is, ¥H

If a nonempty subset H of a lattice L, is a sublattice of L, then ¥H is a fuzzy sublattice of L.

From theorem 2.5 and theorem 2.6, we have, Theorem: 2.9 Let H be any nonempty subset of a lattice L, H  L. Let µ be any fuzzy subset of L defined by µ(x)

Where s, t[0, 1], s > t. Then µ is a fuzzy sublattice of L if and only if, H is a sublattice of L.

Here, the concept of level sublattice of a lattice is studied. Then the necessary and sufficient condition of a fuzzy subset of a lattice to be a fuzzy sublattice of that lattice is proved. Further, the necessary and sufficient condition of equality of two level sublattices in a family of level sublattices is obtained.

Proposition: 2.10

Let µ be any fuzzy sublattice of L. Then the level subsets µt, t Im µ are sublattices of L.

Theorem: 2.11 [Characterization Theorem]

A fuzzy subset µ of a lattice L, is a fuzzy sublattice of L if and only if, the level subsetsµt, t Im µ are sublattices of L.

Proof

 First part is shown Proposition 2.10.

 Conversely,assume that the level subsets µt, t Im µ are sublattices of L.

 To prove: µ is a fuzzy sublattice of L.

 Let x, y  L be arbitrary.  µ(xy) ≥ r; µ(xy) ≥ r -------- ( 1)

To prove µ(xy) ≥ min {µ(x), µ(y)}.

That is to prove s1 ≥ r Suppose s1< r. ------------- (3)

From ( 2) and ( 3), we have, µ(xy) = s1< r.

 µ(xy) < r, which is a contradiction to (1).

Theorem: 2.16

Let µ be any fuzzy sublattice of a lattice L. Let µt and µs be any two level sublattices of µ. Then µt  µs and µt  µs are also level sublattices of µ.

Proof

Let µ be any fuzzy sublattices of a lattice L. Let µt and µs be any two level sublattices of µ.  µt and µs are sublattices of L.

To prove (i) µt µs and (ii) µt µs are level sublattices of µ. That is to prove:µt µs and µt µs are sublattices of L.

For (i)

Let x, y  µt µs be arbitrary.

 x, y L such that µ(x) ≥ min {t, s} and µ(y) ≥ min {t, s} ------------- ( 1)

 xy, xy  µt, since µt is a sublattice of L.

 xy, xy  L such that µ(xy) ≥ t; µ(xy) ≥ t.

 xy, xy  L such that µ(xy) ≥ min {t, s}; µ(xy) ≥ min {t, s}, by the assumption in this case.

Hence µt µs is a sublattice of L. Thus, µt µs is a level sublattice of µ. Hence µt µs is a sublattice of L.Thus, µt µs is a level sublattice of µ.

For (ii)

Again, let x, y  µt µs be arbitrary.

 x  L such that µ(x) ≥ max {t, s} and y L such that µ(y) ≥ max {t, s}-- ( 2)

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