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On Certain Relations for C-Closure Operations on an Ordered Semigroup

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On Certain Relations for C-Closure Operations on an Ordered Semigroup International Journal of Pure and Applied Mathematics Volume 92 No. 1 2014, 51-59 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v92i1.4 ijpam.eu ON CERTAIN RELATIONS FOR C-CLOSURE OPERATIONS ON AN ORDERED SEMIGROUP Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen, 40002, THAILAND Abstract: In this paper, a relation for C-closure operations on an ordered semigroup is introduced, using this relation regular and simple ordered semi- groups are characterized. AMS Subject Classification: 06F05 Key Words: semigroup, ordered semigroup, ideal, regular ordered semigroup, simple ordered semigroup, C-closure operation 1. 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, Pondˇel´iˇcek [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 Pondˇel´iˇcek’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]. c 2014 Academic Publications, Ltd. Received: November 7, 2013 url: www.acadpubl.eu 52 T. Changphas 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: (i) U(∅) = ∅; (ii) A ⊆ B ⇒ U(A) ⊆ U(B); (iii) A ⊆ U(A); (iv) U(U(A)) = U(A). For an element x in S we write U(x) instead of U({x}). A subset A of S is said to be U-closed if U(A) = A; and A is said to be U-open if S \ A is U-closed. Let F(U) 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. (2) U ≤ V if and only if F(V) ⊆ F(U). (3) U ∨ V, U ∧ V exist, and (3.1) F(U ∨ V) = F(U) ∩ F(V), (3.2) F(U ∧ V) = {A ∩ B | A ∈ F(U),B ∈ F(V)}. A C-closure operation U on a nonempty set S is said to be a D-closure operation if, for any indexed family {Ai | i ∈ I} of subsets of S, it satisfies: U( Ai∈I ) = i∈I U(Ai). Note that if U and V are D-closureS operationsS on S, then U∨ V is a D-closure operation on S. For each C-closure operation U on a nonempty set S, a D-closure operation U∗ is defined on S by U∗(A) = {x | U(x) ∩ A 6= ∅} ON CERTAIN RELATIONS FOR C-CLOSURE... 53 for any A in Su(S). It is known that: (1) I∗ = I, O∗ = O. (2) For any U, V in C(S), U ≤ V implies U∗ ≤ V∗. (3) For any U in C(S), the following conditions are equivalent: (3.1) U is a D-closure operation; (3.2) U = U∗∗; (3.3) F(U) = O(U∗); (3.4) O(U) = F(U∗). (4) For a D-closure operation U on S, U ≤ U∗ or U∗ ≤ U implies U = U∗. 2. 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 (A] = {x ∈ S | x ≤ a for some a ∈ A}. 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]; (3) (A](B] ⊆ (AB]; (4) ((A](B]] = (AB]; (5) (A ∪ B] = (A] ∪ (B]. 54 T. Changphas 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. Definition 1. Let (S, ·, ≤) be an ordered semigroup. Define a relation ̺ on C(S) by U̺V if and only if A ∩ B = (AB] 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. If U ≤ U′ and V ≤ V′, then U′̺V′. Proof. This follows directly from the definition of ̺. Let (S, ·, ≤) be an ordered semigroup. Define a mapping L on Su(S) by L(∅) = ∅, and L(A) = (A ∪ SA] 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 L ⊆ L(L) = (L ∪ SL] ⊆ (L] = L; hence L ∈ F(L). Conversely, if L ∈ F(L), then L = L(L) = (L ∪ SL], 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). ON CERTAIN RELATIONS FOR C-CLOSURE... 55 Lemma 3. Let (S, ·, ≤) be an ordered semigroup. Then L and R are D-closure operations on S. Proof. Let {Ai | i ∈ I} be an indexed family of subsets of S. We have L Ai = Ai S Ai i∈I ! i∈I ! i∈I !# [ [ [ [ = Ai (SAi) i∈I ! i∈I !# [ [ [ = (Ai ∪ SAi) i∈I # [ = (Ai ∪ SAi] i∈I [ = L(Ai) i∈I [ Then L is a D-closure operation on S. Similarly, R is a D-closure operation on S. Theorem 4. Let (S, ·, ≤) be an ordered semigroup, and let U, V be C-closure operations on S. Then U̺V if and only if R ≤ U, L ≤ V and x ∈ (U(x)V(x)] for all x in S. Proof. Assume U̺V. Clearly, S ∈ F(V). If A ∈ F(U), then A = A ∩ S = (AS]; hence R(A) = (A ∪ AS] = ((AS] ∪ AS] = ((AS]] = (AS] = A. This shows that A ∈ F(R); thus R ≤ U. Similarly, L ≤ V. Let x be an element of S. Since U(x) ∈ F(U) and V(x) ∈ F(V), we have x ∈ U(x) ∩ V(x) ⊆ (U(x)V(x)] as required. Conversely, assume that R ≤ U, L ≤ V and x ∈ (U(x)V(x)] for all x in S. To show that U̺V, let A ∈ F(U) and B ∈ F(V) be nonempty. Then A ∈ F(R) and B ∈ F(L). We have (AB] ⊆ (AS] ⊆ (A] = A and (AB] ⊆ (SB] ⊆ (B] = B. This shows that (AB] ⊆ A ∩ B. For the reverse inclusion, let x be an element of A ∩ B. Then U(x) ⊆ U(A) = A. Similarly, V(x) ⊆ B. By x ∈ U(x)V(x) ⊆ AB ⊆ (AB] 56 T. Changphas follows A ∩ B ⊆ (AB]. An ordered semigroup (S, ·, ≤) is said to be left regular if x ∈ (Sx2] for every x in S, or equivalently, x ∈ (x2 ∪ Sx2] 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 ∈ (x2 ∪ 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 x ∈ (xSx] = ((x](Sx]] ⊆ (R(x)L(x)]; hence R̺L by Theorem 4. Conversely, R̺L implies S is regular since, for any x in S, we have x ∈ (R(x)L(x)] = ((x ∪ xS](x ∪ Sx]] = ((x ∪ xS)(x ∪ Sx)] ⊆ (x2 ∪ xSx]. 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 semi- group (S, ·, ≤): (1) L̺L; (2) L̺M; (3) S is left regular and R ≤ L.
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