ON CERTAIN RELATIONS FOR C-CLOSURE OPERATIONS ON AN ORDERED SEMIGROUP

In this paper, a relation for C-closure operations on an ordered semigroup is introduced, using this relation regular and simple ordered semigroups are characterized. AMS Subject Classification: 06F05


Preliminaries
It is known that a semigroup S is regular if and only if it satisfies: A ∩ B = AB for all right ideals A and for all left ideals B of S. Using this property, Ponděl íček [2] introduced a relation for C-closure operations on S, and studied some types of semigroups using the relation.The purpose of this paper is to extend Ponděl íček's results to ordered semigroups.In fact, we define a relation for C-closure operations on an ordered semigroup, and characterize regular and simple ordered semigroups using the relation.Firstly, let us recall some certain definitions and results which are in [2].
Let S be a nonempty set.A mapping U:Su(S) → Su(S) (The symbol Su(S) stands for the set of all subsets of S) is called a C-closure operation on S if, for any A, B in Su(S), it satisfies: (2) U ≤ V if and only if F(V) ⊆ F(U).
(3) U ∨ V, U ∧ V exist, and A C-closure operation U on a nonempty set S is said to be a D-closure operation if, for any indexed family {A i | i ∈ I} of subsets of S, it satisfies: For each C-closure operation U on a nonempty set S, a D-closure operation U * is defined on S by It is known that: (1) ( (3) For any U in C(S), the following conditions are equivalent:

Main Results
An ordered semigroup [1] is defined to be a semigroup (S, •) together with a partial order ≤ that is compatible with the semigroup operation, meaning that for x, y, z ∈ S, x ≤ y ⇒ zx ≤ zy, xz ≤ yz.
Let (S, •, ≤) be an ordered semigroup.If A, B are nonempty subsets of S, we write AB for the set of all elements xy in S such that x in A and y in B, and write For an element x in S, we write Ax and xA for A{x} and {x}A, respectively.In [4], the following conditions hold: (1) A ⊆ (A]; (2) A ⊆ B ⇒ (A] ⊆ (B]; The following concepts can be found in [3].Let (S, •, ≤) be an ordered semigroup.A nonempty subset A of S is called a left (respectively, right) ideal [3] of S if it satisfies: (i) SA ⊆ A (respectively, AS ⊆ A); (ii) A = (A], that is, for x ∈ A and y ∈ S, y ≤ x implies y ∈ A.
If A is both a left and a right ideal of S, then A is called an ideal of S.
Let (S, •, ≤) be an ordered semigroup.If A is a nonempty subset of S, then (A ∪ SA] (respectively, (A ∪ AS], (A ∪ SA ∪ AS ∪ SAS]) are left (respectively, right, two-sided) ideals of S. for all nonempty subsets A in F(U) and for all nonempty subsets B in F(V).Lemma 2. Let (S, •, ≤) be an ordered semigroup, and let U, U ′ , V, V ′ be C-closure operations on S such that U̺V.
Proof.This follows directly from the definition of ̺.
Let (S, •, ≤) be an ordered semigroup.Define a mapping L on Su(S) by L(∅) = ∅, and for any nonempty subset A of S. It is easy to verify that L is a C-closure operation on S. Note that F(L) is the set of all left ideals of S (including empty set).Indeed, if L is a left ideal of S, then An ordered semigroup (S, •, ≤) is said to be left regular if x ∈ (Sx 2 ] for every x in S, or equivalently, x ∈ (x 2 ∪ Sx 2 ] for every x in S. A right regular ordered semigroup is defined dually.S is said to be regular if x ∈ (xSx] for every x in S, or equivalently, x ∈ (x 2 ∪ xSx] for every x in S.These concepts can be found in [4]. Theorem 5.An ordered semigroup (S, •, ≤) is regular if and only if R̺L.
Proof.Assume that S is regular.Then for any x in S we have hence R̺L by Theorem 4.
Conversely, R̺L implies S is regular since, for any x in S, we have Let (S, •, ≤) be an ordered semigroup.We denote the D-closure operation R ∨ L on S by M. Note that F(M) is the set of all ideals of S (including empty set).Theorem 6.The following statements are equivalent on an ordered semigroup (S, •, ≤) : (1) L̺L; (2) L̺M; (3) S is left regular and R ≤ L.
denote the set of all U-closed subsets of S, and let O(U) denote the set of all U-open subsets of S. Define a relation ≤ on C(S), the set of all C-closure operations on a nonempty set S, by U ≤ V if and only if U(A) ⊆ V(A) for any A in Su(S).A C-closure operation I on S is defined by I(∅) = ∅, and I(A) = S for any nonempty subset A of S. A C-closure operation O on S is defined by O(A) = A for all subsets A of S. For any U and V in C(S) it is known that: (1) O ≤ U ≤ I.

Definition 1 .
Let (S, •, ≤) be an ordered semigroup.Define a relation ̺ on C(S) by U̺V if and only if A ∩ B = (AB] , and thus L is a left ideal of S. Similarly, we define a C-closure operation on S by R(∅) = ∅, and R(A) = (A ∪ AS] for any nonempty subset A of S. F(R) is the set of all right ideals of S (including empty set).follows A ∩ B ⊆ (AB].