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 [1–3]. 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.