Nil extensions of Clifford ordered semigroup
Anjan Kr Bhuniya, Kalyan Hansda

TL;DR
This paper characterizes all ordered semigroups that are nil extensions of various classes such as Clifford, left Clifford, group-like, and left group-like semigroups, providing a comprehensive classification.
Contribution
It provides a complete description of ordered semigroups that are nil extensions of specific well-known classes, expanding the understanding of their structure.
Findings
Characterization of nil extensions of Clifford ordered semigroups
Description of nil extensions of left Clifford ordered semigroups
Classification of nil extensions of group-like and left group-like ordered semigroups
Abstract
In this paper we describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups.
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Taxonomy
Topicssemigroups and automata theory · Advanced Algebra and Logic · Computability, Logic, AI Algorithms
Nil extensions of Clifford ordered semigroup
A. K. Bhuniya and K. Hansda
Department of Mathematics, Visva-Bharati University,
Santiniketan-731235, West Bengal, India
[email protected], [email protected]
Abstract
In this paper we describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups.
Key Words and phrases: nil extension, regular ordered semigroups, group like, left group like, Clifford and left Clifford ordered semigroups.
2010 Mathematics subject Classification: 20M10; 06F05.
1 Introduction:
Nil extensions of a semigroup (without order), are precisely the ideal extensions by a nil semigroup. In 1984, S. Bogdanovi and S. Mili [2] characterized the semigroups (without order) which are nil extensions of completely simple semigroups, where as, a similar work was done by J. L Galbiati and M.L Veronesi [14] in 1980. Authors like S. Bogdanovi, M. Ciri, Beograd have investigated this type extensions for regular semigroup, group, periodic semigroup as well as completely regular semigroup(see [4], [5]).
The notion of ideal extensions in ordered semigroups is actually introduced by N. Kehayopulu and M. Tsingelis [8]. In [11], they have worked on ordered semigroups which are nil extensions of Archimedean ordered semigroups. The concepts of nil extensions have been extended to ordered semigroups by Y. Cao [7], with characterizing ordered semigroups which are nil extensions of simple, left simple, t-simple ordered semigroups. Further he described complete semilattices of nil extensions of such ordered semigroups.
The aim of this paper is to describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups. This research originates from the research papers [4], [5].
Our paper organized as follows. The basic definitions and properties of ordered semigroups are presented in Section 2. Section 3 is devoted to characterizing the nil extensions of Clifford and left Clifford ordered semigroups.
2 Preliminaries:
In this paper will provide the set of all natural numbers. An ordered semigroup is a partiality ordered set , and at the same time a semigroup such that . It is denoted by . For an ordered semigroup and , denote . An element in is called zero of if and for every .
Let be a nonempty subset of an ordered semigroup . is a left (right) ideal of , if and . is an ideal of if is both a left and a right ideal of . An (left, right) ideal is proper if . The intersection of all ideals of an ordered semigroup , if nonempty, is called the kernel of and is denoted by .
An ordered semigroup is called a group like ordered semigroup if for all [1] and left group like if for all such that [1]. Kehayopulu [9] defined Greens relations on an ordered semigroup as follows: . These four relations are equivalence relations. In an ordered semigroup , an equivalence relation is called left (right) congruence if for . is congruence if it is both left and right congruence. A congruence on is called semilattice if for all . A semilattice congruence on is called complete if implies . The ordered semigroup is called complete semilattice of subsemigroup of type if there exists a complete semilattice congruence such that is a type subsemigroup of . Equivalently: There exists a semilattice and a family of subsemigroups of type of such that:
for any 2. 2.
3. 3.
for any 4. 4.
implies where is the order of the semilattice defined by
[10].
Let be an ordered semigroup with the zero [math]. An element is called a nilpotent if for some . The set of all nilpotents of is denoted by . is called nil ordered semigroup (nilpotent) [7] if .
Due to Cao [7] the definition of nil extension of ordered semigroup is as follows.
Definition 2.1**.**
[7]** Let be an ideal of an ordered semigroup . Then is called the Rees factor ordered semigroup of , and is called an ideal extension of by ordered semigroup . An ideal extension is called a nil-extension of if is a nil ordered semigroup.
Lemma 2.2**.**
[7]** Let be an ordered semigroup and an ideal of . Then the following are equivalent:
- (i)
* is a nil-extension of ;*
- (ii)
.
In [1], we have introduced the notion of Clifford and left Clifford ordered semigroups and characterized their structural representation. For the correspondences of the results of this paper we are stating some of them.
Definition 2.3**.**
[1]** A regular ordered semigroup is called Clifford ordered (left Clifford ordered ) semigroup if for all and .
Theorem 2.4**.**
[1]** Let be a regular ordered semigroup. Then followings hold in :
* is Clifford if and only if .* 2. 2.
* is a complete semilattice congruence if is Clifford.* 3. 3.
* is Clifford ordered semigroup if and only if it is a complete semilattice of group like ordered semigroups.*
Theorem 2.5**.**
[1]** Let be a regular ordered semigroup. Then followings hold in :
* is a complete semilattice congruence if is left Clifford.* 2. 2.
* is left Clifford ordered semigroup if and only if it is a complete semilattice of left group like ordered semigroups.*
3 Main Results:
Now we describe all those ordered semigroups which are the nil extensions of Clifford, left Clifford, group like, left group like ordered semigroups. We omit the proof of the following lemma as it is straightforward.
Lemma 3.1**.**
An ordered semigroup is left group like ordered semigroup if and only if if for all .
Theorem 3.2**.**
An ordered semigroup is a nil extension of a left group like ordered semigroup if and only if for every and for every implies .
Proof.
Suppose that is a nil extension of a left group like ordered semigroup and . Then there is such that . Regularity of implies that for some . Further, for ; the left simplicity of yields that . This gives that . Next let and such that . Since such that . Then for some , ; whence . Thus and so . Thus . Since is a left group like ordered semigroup, for it follows that for some , by Lemma 3.1. Thus the given conditions follow.
Conversely, assume that given conditions hold in . Let be arbitrary. Then by given condition we have , for some and . This implies and so . Denote .
Let us choose . Then the definition of implies
[TABLE]
Thus . Now for and such that
[TABLE]
Then implies and hence ; where . So . Also . Let be such that . Then implies by the second condition that
[TABLE]
Denote . Then the definition of implies
[TABLE]
Then using the first condition for we have that . That is , by (2). So from (1) we have , and hence .
Next choose such that . Since there is such that for all . Now for , it follows from the first condition that
[TABLE]
So implies that
[TABLE]
Since , by above we have . Say . Then for yields that for some , by second condition. Therefore
[TABLE]
Clearly , as . Thus , which shows that is an ideal of .
Finally let . Then there is . Now for such that . Since . Hence is left simple. Thus is left group like ordered semigroup such that for every . Hence is nil extension of a left group like ordered semigroup . ∎
In the following result we provide an independent proof of Corollary 5.2 of [7].
Theorem 3.3**.**
An ordered semigroup is a nil extension of a group like ordered semigroup if and only if for all .
Proof.
Suppose that is a nil extension of a group like ordered semigroup and . Then there exists . Since is a group like ordered semigroup, there exists . Also for . This implies . Thus .
Conversely, let us assume that given condition holds in . Choose . Then for some . Thus . Say . So for every such that . Let us consider . Then for all
[TABLE]
Using the given condition for , we obtain . This yields that
[TABLE]
Thus . Similarly .
Next let and be such that . Since there exists such that and hence for all , which implies that
[TABLE]
So . Hence is an ideal of .
Finally, consider . Then there exists such that
[TABLE]
and so by the given condition it follows that for some . This gives that . Similarly there is some such that . This shows that is a group like ordered semigroup. Hence is a nil extension of a group like ordered semigroup . ∎
Theorem 3.4**.**
An ordered semigroup is a nil extension of a Clifford ordered semigroup if and only if for every and such that , implies .
Proof.
First suppose that is a nil extension of a Clifford ordered semigroup . Let . Then there is such that . Since is an ideal of , . Since is a regular, there exists such that
[TABLE]
Now implies that
[TABLE]
Similarly for there is such that
[TABLE]
Therefore
[TABLE]
Thus
[TABLE]
Also, for such that . Then from (8), we obtain that
[TABLE]
Therefore . Similarly .
Now implies that . Consider . Since such that . Since is a nil extension of such that . This gives , which gives and so , since is an ideal of . Since is a Clifford ordered semigroup, by Theorem 2.4(ii) is a congruence on . Since we have and hence .
Conversely, let us assume that given conditions hold in . Let be arbitrary. Then by the first condition there exists such that . Thus . Say . It is now clear that for each such that .
Let and . Then for all and for some which implies that . By first condition there are such that and thus . Also for every ,
[TABLE]
So there is such that
[TABLE]
Thus .
To show , a Clifford ordered semigroup, choose . Then there is such that
[TABLE]
Now for , the first condition yields that
[TABLE]
Therefore from (3)
[TABLE]
Similarly there are and such that . So from (3),
[TABLE]
Hence is Clifford ordered semigroup.
Now let for some and . Then by the second condition, there is such that ,
[TABLE]
Since is Clifford ordered semigroup, for it follows that
[TABLE]
Similarly for , we have .
The last two inequalities together with (11) yields that . Thus and so is an ideal of . Hence is a nil extension of a Clifford ordered semigroup . ∎
Theorem 3.5**.**
An ordered semigroup is a nil extension of a left Clifford ordered semigroup if and only if for every and such that implies .
Proof.
Let be a nil extension of a left Clifford ordered semigroup . Choose . Then there exists such that . Since is an ideal of , . Also the regularity of yields that
[TABLE]
Since is a left Clifford ordered semigroup and , by Theorem 2.5 it follows that for some . Therefore
[TABLE]
Similarly, for there is such that
[TABLE]
and for there is such that
[TABLE]
Thus from (3) we obtain that
[TABLE]
Hence and so , from (12).
To show the second condition choose and be such that . By the regularity of yields that for some and for all . Then there is such that . Since ia an ideal and so . Thus . Since is left Clifford ordered semigroup, is congruence on , by Theorem 2.5(i). Thus . Then are in Theorem 3.1, for some . This proves the necessary condition.
Conversely, suppose that given conditions hold in . Let . Then by the first condition there exists such that . Thus . Say . Now for each such that . Let and . Then for all . This implies .
Then by the first condition there are such that
[TABLE]
Therefore .
We now show that is a left Clifford ordered semigroup. For this let us assume that . Then there is such that
[TABLE]
Then by first condition there are such that
[TABLE]
Therefore Since we have . Hence is left Clifford ordered semigroup.
Next let such that . Using second condition we have , for some . Then for all . So for some . Since is a left Clifford, . So , that is .
Finally to show , an ideal of we need only to show that . The regularity of yields that for all . Also for some , . Then implies that by above. Thus is an ideal of . Hence is a nil extension of a left Clifford ordered semigroup . ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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