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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 676020, 8 pages
http://dx.doi.org/10.1155/2011/676020
Research Article

On -Semigroups

1Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea
2Department of Mathematics, Chungbuk National University, Chongju 361-763, Republic of Korea

Received 26 January 2011; Accepted 2 March 2011

Academic Editor: Young Bae Jun

Copyright © 2011 Sun Shin Ahn and Young Hee Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The notion of a -semigroup is introduced, and related properties are investigated. The concept of left (resp., right) deductive systems of a -semigroup is also introduced.

1. Introduction

Hu and Li, Iséki and Tanaka, respectively, introduced two classes of abstract algebras: -algebras and -algebras [13]. It is known that the class of -algebras is a proper subclass of the class of -algebras. In [1, 4] Hu and Li introduced a wide class of abstract algebras: -algebras. They have shown that the class of -algebras is a proper subclass of the class of -algebras. We refer to [5] for general information on -algebras. Neggers and Kim [6] introduced the notion of a -algebra which is a generalization of -algebras, and also they introduced the notion of a -algebra [7, 8], that is, (I) , (II) , (III) , for any , which is equivalent to the idea of groups. Moreover, Jun et al. [9] introduced a new notion, called an - algebra, which is another generalization of -algebras, that is, (I), (II), and (IV) and imply that for any . Walendziak obtained other equivalent set of axioms for a -algebra [10]. Kim et al. [11] introduced the notion of a (pre-) Coxeter algebra and showed that a Coxeter algebra is equivalent to an abelian group all of whose elements have order 2, that is, a Boolean group. C. B. Kim and H. S. Kim [12] introduced the notion of a -algebra which is a specialization of -algebras. They proved that the class of -algebras is a proper subclass of -algebras and also showed that a -algebra is equivalent to a 0-commutative -algebra. In [13], H. S. Kim and Y. H. Kim introduced the notion of a -algebra as a generalization of a -algebra. Using the notion of upper sets, they gave an equivalent condition of the filter in -algebras. In [14, 15], Ahn and So introduced the notion of ideals in -algebras and proved several characterizations of such ideals.

In this paper, by combining -algebras and semigroups, we introduce the notion of -semigroups. We define left (resp., right) deductive systems (LDS (resp., RDS) for short) of a -semigroup, and then we describe LDS generated by a nonempty subset in a -semigroup as a simple form.

2. Preliminaries

We recall some definitions and results discussed in [13].

Definition 2.1 (see [13]). An algebra of type (2, 0) is called a -algebra if (BE1) for all , (BE2) for all , (BE3) for all , (BE4) for all (exchange). We introduce a relation “≤" on by if and only if .

Proposition 2.2 (see [13]). If is a -algebra, then for any .

Example 2.3 (see [13]). Let be a set with the following table: Then is a -algebra.

Definition 2.4 (see [13]). A -algebra is said to be self-distributive if for all .

Example 2.5 (see [13]). Let be a set with the following table: Then it is easy to see that is a self-distributive -algebra.

Note that the -algebra in Example 2.3 is not self-distributive, since , while .

Proposition 2.6. Let be a self-distributive -algebra. If , then and for any .

Proof. The proof is straightforward.

3. -Semigroups

Definition 3.1. An algebraic system is called a -semigroup if it satisfies the following: (i) is a semigroup, (ii) is a -algebra, (iii)the operation “” is distributive (on both sides) over the operation “”.

Example 3.2. (1) Define two operations “” and “” on a set as follows: It is easy to see that is a -semigroup.
(2) Define two binary operations “” and “” on a set as follows: It is easy to show that is a -semigroup.

Proposition 3.3. Let be a -semigroup. Then (i), (ii), .

Proof. (i) For all , we have that and .
(ii) Let be such that . Then Hence and .

Definition 3.4. An element in a -semigroup is said to be a left (resp., right) unit divisor if A unit divisor is an element of which is both a left and a right unit divisors.

Theorem 3.5. Let be a -semigroup. If it satisfies the left (resp., right ) cancellation law for the operation , that is, then contains no left (resp., right) unit divisors.

Proof. Let satisfy the left cancellation law for the operation and assume that where . Then by Proposition 3.3(i), which implies . Similarly it holds for the right case. Hence there is no left (resp., right) unit divisors in .

Now we consider the converse of Theorem 3.5.

Theorem 3.6. Let be a -semigroup in which there are no left (resp., right ) unit divisors. Then it satisfies the left (resp., right) cancellation law for the operation .

Proof. Let be such that and . Then Since has no left unit divisor, it follows that so that . The argument is the same for the right case.

Definition 3.7. Let be a -semigroup. A nonempty subset of is called a left (resp., right) deductive system (LDS (resp., RDS), for short) if it satisfies (ds1) (resp., ), (ds2).

Example 3.8. Let be a set with the following Cayley tables: It is easy to show that is a -semigroup. We know that is an LDS of , but is not an LDS of , since and/or , but .

Let be a -algebra, and let . Then the set is nonempty, since .

Proposition 3.9. If is an LDS of a -semigroup , then

Proof. Let where . Then and so by (ds2). Therefore .

Theorem 3.10. Let be an arbitrary collection of LDSs of a -semigroup , where ranges over some index set . Then is also an LDS of .

Proof. The proof is straightforward.

Let be a -semigroup. For any subset of , the intersection of all LDSs (resp., RDSs) of containing is called the LDSs (resp., RDSs) generated by , and is denoted by (resp., ). It is clear that if and are subsets of a -semigroup satisfying , then (resp., ), and if is an LDS (resp., RDS) of , then (resp., ).

A -semigroup is said to be self-distributive if is a self-distributive -algebra.

Theorem 3.11. Let be a self-distributive -semigroup and let be a nonempty subset of such that . Then for some .

Proof. Denote Let and . Then there exist such that . It follows that Since for , we have that . Let be such that and . Then there exist such that Using (BE4), it follows from (3.12) that , that is, , and so from (3.13) and Proposition 2.6 it follows that Thus , which implies . Therefore is an LDS of . Obviously . Let be an LDS containing . To show , let be any element of . Then there exist such that . It follows from (ds2) that so that . Consequently, we have that .

In the following example, we know that the union of any LDSs (resp., RDSs) and may not be an LDS (resp., RDS) of a self-distributive -semigroup .

Example 3.12. Let be a set with the following Cayley tables: It is easy to check that is a self-distributive -semigroup. We know that and are LDSs of , but is not an LDS of , since , .

Theorem 3.13. Let and be LDSs of a self-distributive -semigroup . Then

Proof. Denote Obviously, . Let . Then there exist such that by Theorem 3.11. If (resp., ) for all , then (resp., ). Hence since (resp., ). If some of belong to and others belong to , then we may assume that and for , without loss of generality. Let . Then and so . Now let . Then which implies that . Since , it follows that so that . This completes the proof.

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