Article
Kyungpook Mathematical Journal 2022; 62(1): 29-41
Published online March 31, 2022
Copyright © Kyungpook Mathematical Journal.
Left Regular and Left Weakly Regular n-ary Semigroups
Patchara Pornsurat and Bundit Pibaljommee*
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
e-mail : p.patchara@kkumail.com and banpib@kku.ac.th
Received: July 13, 2021; Revised: November 2, 2021; Accepted: November 15, 2021
Abstract
We study the concept of a quasi-ideal and a generalized bi-ideal of an n-ary semigroup; give a construction of the quasi-ideal of an n-ary semigroup generated by its nonempty subset; and introduce the notions of regularities, namely, a left regularity and a left weakly regularity. Moreover, the notions of a right regularity, a right weak regularity and a complete regularity are given. Finally, characterizations of these regularities are presented.
Keywords: n-ary semigroup, left regular, right regular, quasi-ideal
1. Introduction
The concept of a semigroup has been extensively studied for a long time, as has that of an ordered semigroups, which is a semigroup together with a partially ordered relation. Classical results on ordered semigroups can be found in [1, 2]. Many kinds of regularity of ordered semigroups were studied by Kehayopulu and Tsingelis (see, [12, 13, 14, 15, 16]). In 1991, Kehayopulu [13] introduced the concept of right regular ordered semigroups and considered semigroups in which the right ideals are two-sided. Kehayopulu [14] gave the concept of complete regularity and gave characterizations of completely regular
The definition and theory of ternary semigroups were first introduced by Lehmer [18] in 1932. Santiago [22] investigated the notion of ideals of ternary semigroups and gave some properties of regular ternary semigroups. In 2012, Daddi and Pawar [3] investigated the notion of ordered quasi-ideals and ordered bi-ideals of ordered ternary semigroups and used them to characterize regular ordered ternary semigroups. N. Lekkoksung and P. Jampachon [19] gave the other types of regularities of ordered ternary semigroups, namely, left (right) weakly regular ordered ternary semigroups. Later, several types of regularities of ordered ternary semigroups were characterized in terms of ordered ideals by Pornsurat and Pibaljommee in [7].
The notion of
In this work, we first study the concept of a quasi-ideal and a generalized bi-ideal of an
2. Preliminaries
Let
An
for all
By the associativity of an
for any
For
If
Throughout this paper, we write
Many notations above can be found in [6, 7, 8, 9].
For
Let
As a special case of
Theorem 2.1. Let
Corollary 2.2. Let
(i) and . (ii) Ifand , then
Next, we define the notion of a quasi-ideal of an
Definition 2.3. A nonempty subset
Note that for any quasi-ideal
Proposition 2.4. Let
(i) Everyi -ideal ofS is a quasi-ideal ofS for each. (ii) IfI1, I2,…,In are 1-ideal, 2-ideal,…,n -ideal ofS , respectively, thenis a quasi-ideal of S .
So, every
(2): Let
So,
Let
Example 2.5. Let
Then
Proposition 2.6. A nonempty subset of
For a nonempty subset
Theorem 2.7. The intersection of an arbitrary nonempty family of quasi-ideal of
Theorem 2.8. Let
Therefore,
Definition 2.9. A nonempty subset
Remark 2.10. Every quasi-ideal of
for all
It follows that,
Now, we recall the notion of a regular
An
Many forms of regular
An
An
An
However, Dudek and Groździńska [6] proved that the regularities conditions (2.1), (2.2), (2.3) and (2.4) are equivalent.
3. Left Regular and Right Regular n -ary Semigroups
In this section, we define the notion of left regular, right regular, left weakly regular, right weakly regular and completely regular
Definition 3.1. An element
We note that
Lemma 3.2. The following statements are equivalent.
(i) S is left (resp. right) regular.(ii) (resp. ) for any . (iii) (resp. ) for any a ∈ S .
Definition 3.3. Let
Remark 3.4. A nonempty subset
Theorem 3.5. The following statements are equivalent.
(i) S is left regular.(ii) Everyn -ideal ofS is semiprime.(iii) Jn(a) is a semiprime ofS for anya∈ S .(iv) is a semiprime of S for anya ∈ S .
Next theorem can be similarly proved as Theorem 3.5.
Theorem 3.6. The following statements are equivalent.
(i) S is right regular.-
(ii) Every 1-ideal ofS is semiprime. -
(iii) J1(a) is a semiprime ofS for anya∈ S . -
(iv) is a semiprime of S for anya ∈ S .
Theorem 3.7. Let
Hence,
So, every quasi-ideal of
We consider
We have that
Definition 3.8. An
Lemma 3.9. The following statements are equivalent.
(i) S is completely regular.-
(ii) for any .
Conversely, assume that
For convenience, we write
For
Theorem 3.10. Let
So,
So,
• If
n=2 , then it easy to see that. By assumption, and so . So, S is completely regular.• If
n=3 , then it easy to see that. By assumption, and so . So, S is completely regular.• If
n ≥ 4 , then. By assumption, and so . By Lemma 3.9., S is completely regular.
Definition 3.11. An
Remark 3.12. Let
(i) IfS is left regular, then it is left weakly regular.-
(ii) IfS is right regular, then it is right weakly regular.
Theorem 3.13. The following statements are equivalent.
(i)
S is left weakly regular.-
(ii)
for every n -idealL ofS .
Hence,
The next theorem can be similarly proved as Theorem 3.13.
Theorem 3.14.
(i)
S is right weakly regular.-
(ii)
for every 1-ideal R ofS .
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