SOFT VALUATION ON A GENERALIZED SOFT LATTICE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Lattice theory play an important role in mathematics as well as in other disciplines such as computer science, engineering, cryptography, etc. In this paper, we introduce the concept of generalized soft lattice (gs lattice) and investigate some of its fundamental properties. Further we define soft valuation on a generalized soft lattice (gs lattice) and study its major properties. In the last section we discuss the notion of soft distance function and express it in terms of soft valuation. Here we discuss the notions of soft pseudo metric lattice and soft metric lattice.


INTRODUCTION
D Molodstov in [10] introduced the concept of soft sets and is defined as a parameterized family of subsets of an initial universal set. He defined some basic notions and showed that it can be applied to more fields of mathematics as well as various fields which contain uncertain data.
The theory of lattices were developed by R. Dedekind in early 1980's. In every lattice, meet and join are commutative and associative binary operations. So it can be considered as commutative semigroup with the binary algebraic operations meet and join. In [2], Birkhoff present deeper ideas of lattice theory. Presently lattice theory is developing rapidly. Applications of soft sets to lattices is described in [11]. The concept of soft elements of a soft set was introduced by S. Das and S.K. Samanta in [6] and they introduced soft metric space in terms of soft elements in [5]. Also, S. Das, P. Majumdar and S.K. Samanta introduced the notion of soft linear space in terms of soft elements in [4], M. Chiney and S.K. Samanta introduced a new version of soft topology in [3]. Based on this,we would like to express certain concepts in lattice theory to soft sets in terms of soft elements. This paper is organized as follows. In section 2 we recall some preliminaries. In section 3 we introduce generalized soft lattices (in short gs lattices) and investigate its properties. Here we observe that any gs lattice satisfy the distributive inequality and soft modular identity. Section 4 deals with the notion of soft valuation on a gs lattice. Here we prove that any gs lattice with positive soft valuation is soft modular. In the last section we define soft distance function and express in terms of soft valuation and verify that the soft distance function satisfies the axioms of soft pseudo metric space. We conclude this section by defining soft pseudometric lattice and soft metric lattice with some properties.

PRELIMINARIES
Definition 2.1. [2] A poset is a set P in which a binary relation x ≤ y is defined which satisfies the following conditions: (i)x ≤ x, ∀x ∈ P(Reflexive) [10] Let X be an initial universe and E be a set of parameters. Let P(X) denotes the power set of X and A ⊆ E. Then a soft set over X is a pair (F, A), where F is a mapping from A to P(X).
Definition 2.4. [8] Let E be the set of parameters and A ⊆ E. Then for each soft set (F, A) over an initial universal set X, a soft set (H, E) is constructed over X, where for all λ ∈ E, We write (F, A)⊂(G, B). Definition 2.6.
[9] Let X be a nonempty universal set and A be a nonempty parameter set. Then a soft set (F, A) over X is said to be a null soft set if F(λ ) = φ , ∀λ ∈ A and absolute soft set if The absolute soft set over X with parameter set A is denoted byX or (X, A) and the null soft set is denoted byΦ or (Φ, A). Definition 2.8. [11] Let (F, A) be a soft set over a lattice L. Then (F, A) is said to be a soft lattice [6] Let X be a nonempty universal set and A be a nonempty parameter set. Then a functionx : A → X is called a soft element ofX and we writex∈X.
A soft elementx is said to be belongs to a soft set (F, A) over X ifx(λ ) ∈ F(λ ), ∀λ ∈ A and we writex∈(F, A).
Thus for a soft set (F, A) over X with the parameter set A with F(λ ) = φ , ∀λ ∈ A, we have For any non null soft set in S(X), the collection of all soft elements of (F, A) is denoted by

SE((F, A).
Let B be a collection of soft elements of (F, A). Then the soft set generated by B is given by (1)r ≤s ifr(λ ) ≤s(λ ), ∀λ ∈ A.
Definition 2.13. [2] A valuation on a lattice (L, ≤, ∨, ∧) is a real valued function v on L such Result. [2] In any modular lattice of finite length, the height function is a positive valuation.
Definition 2.14. [3] Let X and Y be two nonempty sets and { f λ : λ ∈ A} be a collection of functions from X to Y. Then a function f : is called a soft function.
on the soft setX if d satisfies the following axioms.

GENERALIZED SOFT LATTICES(GS LATTICES)
In this section we introduce generalized soft lattice (gs lattice) and study its properties. We first define gs poset in terms of soft elements.   Proof. (F, A) and (G, B) be two gs posets with the soft partial orderings ≤ 1 and ≤ 2 respectively. .
Hence the soft relation ≤ is soft symmetric. .
Hence the soft relation ≤ is soft transitive. Consequently it is a soft partial ordering on (F, A)(G, B) and the direct product (F, A)(G, B) is a gs poset.   i.e, the soft partial ordering onL is given by forx,ỹ∈L, we havex ≤ỹ if and only ifx(λ ) ≤ λỹ (λ ), ∀λ ∈ A.
∴x ∨ỹ exists and it is the least upper bound ofx andỹ.
So it is the soft join ofx andỹ.
Note. The soft meet and soft join defined in the above proof are respectively called the soft meet and the soft join generated by meet(∧ L ) and the join(∨ L ) on L. Also,L is called the absolute gs-lattice generated by the lattice (L, ≤, ∨, ∧).
Theorem 3.11. Any soft lattice is a gs-lattice.
Proof. By definition of soft lattice, any soft lattice is a soft set over a lattice. Let (F, A) be a soft lattice over a lattice (L, ≤ L , ∨ L , ∧ L ). Consider the absolute gs-lattice (L, ≤, ∨, ∧) generated by the lattice L.
Then since (F, A) is a soft subset of (L, A), it is a gs poset.
Since (F, A) is a soft lattice, F(λ ) is a sublattice of L,∀λ ∈ A.
Hence (F, A) is a gs-lattice.
Remark. Any non null soft subset of an absolute gs-lattice need not be a gs-lattice.
Proof. Consider Z + which is a lattice with the partial ordering |'. ThenZ + is a gs lat-   Also any other upper bound (p,q) of (x 1 ,ỹ 1 ) and (x 2 ,ỹ 2 ) is such thatp ≥x i andq ≥ỹ i for i = 1, 2.
Hence the direct product of gs lattices (F, A)(G, B) is also a gs lattice. (1)x ∨x =x,x ∧x =x(idempotent) Proof. If the above relations exist, we have  By duality principle, we getx ∨ỹ ≤x ∨z.
Theorem 3.16. In any gs-lattice, the distributive inequalities hold.
Definition 3.18. A gs-lattice is said to be soft modular if it satisfy the soft modular identity.

SOFT VALUATION ON A GS LATTICE
Here we define soft valuation on a gs lattice. We verify that any gs lattice with positive soft valuation is a modular gs lattice.      Then Hence V is a soft valuation onL.  Proof. Let V F and V G be two soft valuations on gs lattices (F, A) and (G, B) respectively.
Since V F and V G are two positive soft valuations, we have Hence V F +V G is a soft valuation on (F, A)(G, B). Also Hence V F +V G is positive.
Thus V F +V G defines a positive soft valuation on the direct product. Proof. Let (F, A) be a gs lattice andx,z∈(F, A) withx ≤z. =0.

SOFT METRIC LATTICE
In this section we define soft distance function. We express soft distance function in terms of soft valuation and prove some of its properties. Here we have the main result that any gs lattice with isotone soft valuation is a soft pseudo metric space and any gs lattice with positive soft valuation is a soft metric space. Also, we introduce soft pseudo metric lattice and soft metric lattice.
Proof. Let (F, A) be a gs-lattice with an isotone soft valuation V.

CONCLUSION
In this paper, we introduce the concept of gs-lattice in terms of soft elements and define a soft valuation on it. Then we introduce a soft pseudo metric on the gs-lattice by means of soft valuation. Also we discuss the circumstances under which it becomes a soft metric space.