## Fuzzy subinclines (ideals) of incline algebras

- Young Bae Jun, Sun Shin Ahn, Hee Sik Kim
- Fuzzy Sets and Systems
- 2001

- Published 2004

In this paper, we introduce the concept of doubt fuzzy subinclines(ideals) of incline algebras and some related properties are investigated. We also state the doubt product of doubt fuzzy subinclines(ideals), and the injections of doubt fuzzy subinclines(ideals). Moreover, we discuss the chain condition of subinclines(ideals). 1.Introduction and Preliminaries An incline algebra is a set H with two binary operations denoted by “ + ” and “ ∗ ” satisfies the following axioms for all x, y, z ∈ H: (i) x+ y = y + x (ii) x+ (y + z) = (x+ y) + z (iii) x ∗ (y ∗ z) = (x ∗ y) ∗ z (iv) x ∗ (y + z) = x ∗ y + x ∗ z (v) (y + z) ∗ x = y ∗ x+ z ∗ x (vi) x+ x = x (vii) x+ (x ∗ y) = x (viii) y+ (x ∗ y) = y For convenience, we pronounce “ + ”(resp. “ ∗ ”) as addition (resp. multiplication). Every distributive lattice is an incline. An incline is a distributive lattice(as simiring) if and only if x ∗ x = x for all x ∈ H. Note that x ≤ y if and only if x + y = y for all x, y ∈ H. A subincline of an incline H is a subset M of H closed under addition and multiplication . An ideal in an incline H is a subincline M ⊂ H such that if x ∈ M and y ≤ x then y ∈ M. By a homomorphism of incline H into an incline I such that f (x+ y) = f (x) + f (y) and f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ H. 2. Doubt fuzzy subinclines(ideals) In what follows, F (H) denotes the set of all fuzzy subsets in H, i.e., maps fromH into ([0, 1],∨,∧),where [0, 1] is the set of reals between 0 and 1 and x ∨ y = max{x, y}, x∧ y = min{x, y}. Definition 2.1. A ∈ F (H) is called a doubt fuzzy subincline of H if A(x+ y)∨A(x ∗ y) ≤ A(x) ∨ A(y) for all x, y ∈ H. A fuzzy subset A ∈ F (H) is said to be order preserving if A(x) ≤ A(y) whenever x ≤ y. Definition 2.2. A fuzzy set subincline A is called a doubt fuzzy fuzzy ideal of H if it is order preserving. Example 2.3. Note that for any x ∈ H, the set M = {a|a ≤ x} is an ideal of H. Define A ∈ F (H) by A(x) = { 0.3 if x ∈ M 0.8 otherwise

@inproceedings{Jianming2004DoubtFS,
title={Doubt Fuzzy Subinclines ( Ideals ) of Incline Algebras},
author={Zhan Jianming and Ma Xueling},
year={2004}
}