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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- OPERATIONS ON AN ORDERED

Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen, 40002, THAILAND

Abstract: In this paper, a 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, , 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 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 . A mapping U:Su(S) → Su(S) (The symbol Su(S) stands for the set of all 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 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 ). 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- (S, ·, ≤):

(1) L̺L;

(2) L̺M;

(3) S is left regular and R ≤ L.

Proof. (1) ⇒ (2). Since L ≤ M, it follows by Lemma 2 that L̺M. (2) ⇒ (3). Assume L̺M. Then R ≤ L by Theorem 4; hence M = L. For any x in S, we have

x ∈ (L(x)M(x)] = (L(x)L(x)] = ((x ∪ Sx](x ∪ Sx]] = ((x ∪ Sx)(x ∪ Sx)] ⊆ (x2 ∪ xSx ∪ Sx2 ∪ SxSx] ON CERTAIN RELATIONS FOR C-CLOSURE... 57

= ((x ∪ xS)x ∪ S(x ∪ xS)x] ⊆ (R(x)x ∪ SR(x)x] ⊆ (L(x)x ∪ SL(x)x] ⊆ (x2 ∪ Sx2].

This shows that S is left regular. (3) ⇒ (1). Assume that S is left regular and R ≤ L. Then, for any x in S, we have x ∈ (Sx2] = ((Sx](x]] ⊆ (L(x)L(x)]. By Theorem 4, L̺L.

Theorem 7. The following statements are equivalent on an ordered semi- group (S, ·, ≤):

(1) R̺R;

(2) M̺R;

(3) S is right regular and L ≤ R.

Proof. The proof is left-right dual of Theorem 6.

An ordered semigroup (S, ·, ≤) is said to be left simple (right simple) if it contains no proper left (right) ideal, and S is said to be simple if it contains no proper ideal. Lemma 8. Let (S, ·, ≤) be an ordered semigroup. Then:

(1) S is left simple if and only if L = I;

(2) S is right simple if and only if R = I;

(3) S is simple if and only if M = I.

Proof. Assume that S is a left simple. If A is a nonempty subset of S, then L(A) = (A ∪ SA] = S = I(A); hence L = I. Conversely, if A is a left ideal of S, then S = I(A) = L(A) = (A ∪ SA] ⊆ A ⊆ S, i.e., A = S. This proves (1). The statements (2) and (3) are proved similarly. 58 T. Changphas

Lemma 9. L ∨ R∗ = I = L∗ ∨ R.

Proof. Let A be an element of F(L∨R∗); then A ∈ F(L) and S\A ∈ F(R). If A is a proper subset of S, then (S \A)A ⊆ A∩(S \A). This is a contradiction. Thus A = S, and hence L ∨ R∗ = I. Similarly, L∗ ∨ R = I.

Theorem 10. The following statements are equivalent on an ordered semigroup (S, ·, ≤):

(1) S is simple;

(2) L ≤ M∗;

(3) R ≤ M∗;

(4) M∗̺I;

(5) I̺M∗.

Proof. If S is simple, then M = I; hence L ≤ I = I∗ = M∗. Conversely, assume that L ≤ M∗. Since R ≤ M, we have R∗ ≤ M∗. By Lemma 9, I = L ∨ R∗ ≤ M∗. Thus M∗ = I, M = I. This proves (1) ⇔ (2). For (1) ⇔ (3) can be proved similarly. By Theorem 4, (5) ⇒ (1) and (4) ⇒ (1). If S is simple, then M = I = I∗. Since (S2] = S, it follows that I̺I. Thus (4) and (5) hold.

Lemma 11. An ordered semigroup (S, ·, ≤) is left simple if and only if R ≤ L∗.

Proof. If S is left simple, then L = I; hence R ≤ I = I∗ = L∗. Conversely, assume that R ≤ L∗. By Lemma 9, I = L∗ ∨ R ≤ L∗ ≤ I. Hence L∗ = I, and so L = I.

Theorem 12. The following are equivalent for an ordered semigroup (S, ·, ≤): ON CERTAIN RELATIONS FOR C-CLOSURE... 59

(1) S is left simple;

(2) L∗̺I;

(3) L̺L∗.

Proof. This can be proved as Theorem 10.

Lemma 13. An ordered semigroup (S, ·, ≤) is right simple if and only if L ≤ R∗.

Proof. This can be proved as Lemma 11.

Theorem 14. The following are equivalent for an ordered semigroup (S, ·, ≤):

(1) S is right simple;

(2) I̺R∗;

(3) R∗̺R.

Proof. This can be proved as Theorem 10.

References

[1] G. Birkhoff, Lattice , 25, Rhode Island, American Mathematical Society Colloquium Pub., Am. Math. Soc., Providence, 1984.

[2] B. Pondˇel´iˇcek, On a certain relation for closure operations on a semigroup, Czechkoslovak Mathematical Journal, 20 (1970), no. 2, 220-231.

[3] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Mathemat- ica Japonica, 35 (1990), no. 6, 1051-1056.

[4] N. Kehayopulu, M. Tsingelis, On left regular ordered semigroups, Southeast Asian Bulletin of Mathematics, 25 (2002), 609-615. 60