ON FUZZY POINTS IN SEMIGROUPS

We consider the semigroup S of the fuzzy points of a semigroup S, and discuss the relation between the fuzzy interior ideals and the subsets of S in an (intra-regular) semigroup S. 2000 Mathematics Subject Classification. 03E72, 20M12.


Introduction.
After the introduction of the concept of fuzzy sets by Zadeh [8], several researches were conducted on the generalizations of the notion of fuzzy sets.Pu and Liu [5] introduced the notion of fuzzy points.In [6,7,8], authors characterized fuzzy ideals as fuzzy points of semigroups.In [1,2,3], Kuroki discussed the properties of fuzzy ideals and fuzzy bi-ideals in a semigroup and a regular semigroup.In this paper, we consider the semigroup S of the fuzzy points of a semigroup S, and discuss the relation between the fuzzy interior ideals and the subsets of S in an (intra-regular) semigroup S.

Preliminaries.
Let S be a semigroup with a binary operation "•".A nonempty subset A of S is called a subsemigroup of S if A 2 ⊆ A, a left (resp., right ) ideal of S if SA ⊆ A (resp., AS ⊆ A), and a two-sided ideal (or simply ideal) of S if A is both a left and a right ideal of S. A subsemigroup is a complete lattice with two binary operations "∨" and "∧", where α∨β = sup{α, β} and for each y ∈ X.If f is a fuzzy subset of X, then a fuzzy point x α is said to be contained in f , denoted by 3. Interior ideals of fuzzy points.Let Ᏺ(S) be the set of all fuzzy subsets of a semigroup S. For each f ,g ∈ Ᏺ(S), the product of f and g is a fuzzy subset f • g defined as follows: , g, and h ∈ Ᏺ(S).Thus Ᏺ(S) is a semigroup with the product "•".Let S be the set of all fuzzy points in a semigroup S. Then For any f ∈ Ᏺ(S), f denotes the set of all fuzzy points contained in f , that is, For any A, B ⊆ S, we define the product of two sets A and B as A Lemma 3.1 (see [7,Lemma 4.1]).Let f be a nonzero fuzzy subset of a semigroup S. Then the following conditions are equivalent: (1) f is a fuzzy left (right, two-sided) ideal of S.
(2) f is a left (right, two-sided) ideal of S.
Lemma 3.2 (see [7,Lemma 4.2]).Let f and g be two fuzzy subsets of a semigroup S.
A fuzzy subsemigroup f of a semigroup S is called a fuzzy interior ideal of S if f (xay) ≥ f (a) for all x, a, y ∈ S. Lemma 3.3.Let f be a nonzero fuzzy subset of a semigroup S. Then the following conditions are equivalent: (1) f is a fuzzy interior ideal of S.
(2) f is an interior ideal of S.
Proof.Let f be a fuzzy interior ideal of S, and let x α ,z γ ∈ S and y β ∈ f .Then since α > 0, γ > 0, and 0 < β ≤ f (y), we have Since f is an interior ideal of S, we have This implies that f (xyz) ≥ f (y), and hence f is a fuzzy interior ideal of S.
It is clear that any ideal of a semigroup S is an interior ideal of S. It is also clear that any fuzzy ideal of S is a fuzzy interior ideal of S. A semigroup S is called regular if, for each element a of S, there exists an element x in S such that a = axa.Theorem 3.4.Let f be any fuzzy set in a regular semigroup S. Then the following conditions are equivalent: (1) f is a fuzzy right (resp., left) ideal of S.
(2) f is an interior ideal of S.
Proof.It suffices to show that ( 2) implies (1).Assume that (2) holds.Let x be any element in S. Then since S is regular, there exists element a in S such that This implies that f (xy) ≥ f (x), and hence f is a fuzzy right ideal of S. Theorem 3.5 (see [7,Theorem 3.3]).Let S be a semigroup.If for a fixed α ∈ (0, 1], f α : S → S is a function defined by f α (x) = x α , then f α is a one-to-one homomorphism of semigroups.
From Theorem 3.5, we can consider S as an extension of a semigroup S. Let f be a fuzzy subset of a semigroup S. If f is the subset of S × S given as following: then the set f is an equivalence relation on S. We can consider the quotient set S/ f , with the equivalence classes x α for each x ∈ S. We will denote the subset Let f be a fuzzy subsemigroup of S. If the product " * " on E(f ) is defined by x α * y β = (xy) α∧β for each x α , y β ∈ E(f ), then E(f ) is a semigroup under the operation " * ".
Theorem 3.6.Let f be a fuzzy interior ideal of S. Then E(f ) is an interior ideal of (S/ f , * ).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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