Epimorphisms, Dominions, and Various Classes of Saturated Semigroups

Alam, N.; Khan, N.M.; Obeidat, S.; Kirmani, S.A.K.; Ahmad, Jawed

doi:https://doi.org/10.1155/2022/2684474

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Computational and Theoretical Characteristics of Chemical Graph Theory and Applications

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 2684474 | https://doi.org/10.1155/2022/2684474

Epimorphisms, Dominions, and Various Classes of Saturated Semigroups

N. Alam,¹N.M. Khan,²S. Obeidat,¹S.A.K. Kirmani,³and Jawed Ahmad⁴

Academic Editor: Hassan Raza

Received17 Jun 2022

Accepted25 Jul 2022

Published31 Aug 2022

Abstract

In this paper, we discussed some saturated classes of -commutative semigroups, left (right) regular semigroups, medial semigroups, and paramedial semigroups. The results of this paper significantly extend the long standing result about normal bands that normal bands were saturated and, thus, significantly broaden the class of saturated semigroups.

1. Introduction

Epimorphic and dominion-related properties for the various classes of semigroups are internally linked to zigzags, amalgams, and semigroup embeddings. This trend of studying the properties of a semigroup class has been one of the hot pursuits areas in semigroup theory and other universal algebras worldwide, and a lot of original work has been conducted and is still being explored. In general, the theory of semigroups has a significant influence on the theory of computers, languages, and automata where the equation of zigzags can be represented in terms of zigzag languages. Now we come to the basic notion and fundamental concept of our study which are as follows: Suppose that a semigroup has a subsemigroup named . Using the definition from Isbell [17], dominates an element of if for all semigroups and homomorphisms , for every , implies . The of in is denoted by and includes all elements of dominated by . It is simple to prove that is a subsemigroup of that contains . It is said that if , then is in and is if for all generally comprising semigroups . If for all generally comprising semigroup , then the semigroup is said to be . Saturation of a semigroup class occurs when all of its members are saturated.

A morphism from is referred to as a (in short epi) in a category if for all and for all morphisms , implies . It is very clear that a morphism is epi if and only if the inclusion is epi. It is also simple to check that if , then the inclusion map is epi for any subsemigroup of a semigroup . This scenario also implies that is or in . This also means that every onto morphism is an epimorphism.

Surjective morphisms are epimorphisms in the case of category theory, while the opposite is not always true in general. For example, in the categories of Sets, Abelian Groups, and Groups, epis are onto, but generally it is not true in the categories of semigroups and rings.

Dominions have a great connection with epimorphisms as it follows from the above definitions that all epis from a semigroup are onto states that all morphic images of are saturated.

The structure of the paper is as follows: in Section 3, we have extended Khan’s result from commutativity to -commutativity and shown that any -commutative semigroup satisfying a nontrivial identity of which at least one side has no repeated variable is saturated. In Section 4, it has been shown that all right [left] regular medial semigroups are saturated, while in Section 5, we have proved that all medial semigroups satisfying the identities and are, respectively, saturated; and all paramedial semigroups satisfying the identity are saturated.

2. Preliminaries

Isbell’s Zigzag theorem gives a highly useful tool for describing semigroup dominions.

Result 1 (see [17, Theorem 2.3] or [15, theorem VII.2.13]). Let be a subsemigroup of a semigroup and let , then if and only if or there exists a series of factorizations of as follows:where , , andWhen factoring over , this is known as a zigzag with the value , length and spine .
Results 1’s equations are referred to throughout the remainder of the paper as “the zigzag equations.”

Result 2. ([19, Result 3]). Let be any subsemigroup of a semigroup and let . If (1) is a zigzag of minimum length over with value , then for .
In the following results, let and be any semigroups with dense in .

Result 3. ([19, Result 4]). For any and any positive integer, if (1) is a zigzag of minimum length over with value , then there exists and such that .
If there are any symbols or terms that are not explained, we refer the readers to Clifford, Preston, and Howie [8, 15]. Furthermore, bracketed assertions or conceptions are dual to the other claims or notions in what follows.
In semigroup theory, ring theory, and elsewhere, there have been several efforts to find the classes of algebras which are saturated [7]. It was proved by Gardner [9, Theorem 2.10] that, in the class of all rings, any regular ring is saturated but Higgins [12, Corollary 4] established that not every regular semigroup is saturated. Any class of generalized inverse semigroups, on the other hand, is saturated and had been shown by Higgins in [8]. There must be at least one side of any identity defining a semigroup variety that has no repeating variables in order for it to be saturated [13, Theorem 6]. Commutative and heterotypical varieties of semigroups have already been dealt with, however, it remains an unanswered question how to identify all saturated semigroups (see [13, 14, 18]). The following semigroups are not saturated: commutative cancellative semigroups, subsemigroups of finite inverse semigroups [17], commutative periodic semigroups [14], and bands, since Trotter [24] has produced a band with a correctly epimorphically embedded subband. In this direction, a very recent significant and remarkable work have been made by Ahanger and Shah on partially ordered semigroups (posemigroups), and commutative posemigroups (see [1–3], [23]).
Now, we begin with the class of -commutative semigroups whose concept was first developed by Tully [25]. In [19], Nagy presented a new concept of -commutativity, i.e., for all , there exists such that . He also found that the two characterizations coincide (Theorem 5.1, 18). Recently, the structure of semigroups of this class has been explored by Alam, Higgins, and Khan [5].
A semigroup is known as left (right) quasi commutative if, for all , there exists a positive integer such that . A semigroup is called quasi commutative semigroup if it is both left [right] quasi commutative semigroup. It can be easily seen that all quasi commutative semigroups are -commutative, but this is not always be true for the converse case [see Ch.8, 19].
An element of a semigroup is called left [right] regular if for some , or in other words, . If all elements of is left [right] regular, then is called a left [right] regular semigroup, see [20]. A medial semigroup is a semigroup which fulfills the identity , as shown in [Ch. 9, 19].
Protic [21] introduced the concept of paramedial semigroups as a generalization of externally commutative semigroups. A semigroup is called paramedial semigroup if it satisfies the paramedial law: for all . For further information and related results, see [10, 11, 16, 19].

3. Saturated Class of -Commutative Semigroups

The general question of identifying all saturated varieties of semigroups has been open for long, though lot of efforts had been made over the last four decades. For example, there was an answer to the question for commutative varieties (Higgins [14], Khan [18]) and heterotypical varieties (Higgins [13]). Readers may refer [10, 11] for the related notions, results, and materials on the topic. Our first finding extends Khan’s result [18, Theorem 3.4] providing an essential condition for the saturation of commutative varieties. Authors had previously extended this result to quasi commutative semigroups [6, Theorem 2.5]. Since the class of -commutative semigroups contains the class of quasi commutative semigroups, it is worth to explore whether this result may further be extended to the class of -commutative semigroups. Here, we have answered the above question and generalized authors’s result for the class of -commutative semigroups. This class, infact, also contains the class of commutative semigroups.

If the semigroup is commutative and at least one side has no repeating variables, Khan [18] established that it was saturated. This conclusion was extended in [6] for quasi commutative semigroups. Saturation of a -commutative semigroup fulfilling a nontrivial identity of which at least one side has no repeating variable is further shown in the current article.

Theorem 1. If a -commutative semigroup satisfies a nontrivial identity of which at least one side has no repeated variable, then is saturated.

Proof. Since one side of the identity has no repeated variable, the identity has the following form:

Proposition 1. In any -commutative semigroup , for any and positive integer , there exists such that

Proof. We shall use induction on to prove the proposition. For , by repeated application of -commutativity of and for some , we haveSo the result holds for .
Similarly, for , by repeated application of -commutativity of , the case for and for some , we haveThus the result is true for .
For all positive integers less than or equal to , suppose inductively that the conclusion is true. Thus, we havefor some .
Now for positive integer , we repeat the application of -commutativity of for some . Therefore, we havewhere .
Returning to the proof of the theorem, we assume to the contrary that is not saturated, so for some semigroup containing properly.

Lemma 1. for some and, all , .

Proof. By Result 3, as , we havefor some and .
Since satisfies (3), we have the two cases: Case (1): identity (3) is heterotypical and Case (2): identity (3) is homotypical.

Case 1. Suppose identity (3) is heterotypical. Then for some (, for any word , denotes how many times the variable appears in the word ). Now, we havefor any .
Since is -commutative and contains the element , by using equality (9) and for some , we havewhere .

Case 2. Suppose identity (3) is homotypical. According to the nontriviality of the identity (3), we may consider that (3) is in the following form:where for all and for some . Since is -commutative and by using equality (9) and (12) for some , we haveas required, where .
Returning back to the proof of the theorem, let be any element. As , let (1) be a zigzag for in over of minimal length . Now, by using equalities (1), Lemma 1 for some , and -commutativity of for some This is a contradiction which shows that , where is saturated.
As a corollary, we have the following interesting result:

Corollary 1. Classes of all quasi commutative semigroups satisfies a nontrivial identity of which at least one side has no repeated variable are saturated.
Since the class of weakly commutative semigroups is a wider class than the class of -commutative semigroups, so one may arise the open problem as follows:

Open Problem 1. Whether the class of all weakly commutative semigroups satisfying a nontrivial identity of which at least one side has no repeated variable is saturated. If not, then under what condition this class may be saturated?

4. Saturated Classes of Medial Semigroups

Normal bands have long been known to be closed [22], and the class of generalized inverse semigroups (regular semigroups whose set of idempotents form normal bands) has been saturated (see [11]). So, it is natural to ask whether this result can be extended to the class of left [right] regular semigroups.

In the present section, we prove that the class of left [right] regular medial semigroups are saturated. We also show that medial semigroups satisfying the identities , and are saturated and consequently deduce the known fact that normal bands are saturated.

Lemma 2. Let be any semigroup with a medial subsemigroup and such that . Then for all and .

Proof. When , then there is nothing to prove. So assume that . As , by Result 3 and based on property , we obtainNow, if , thenas required.
In the other case, i.e., when , thenSoThis completes the proof of the lemma.

Theorem 2. All right regular medial semigroups are saturated.

Proof. Suppose all right regular medial semigroups are not saturated. So there is a right regular medial semigroup and a semigroup containing as a proper subsemigroup and such that . For any , if (1) is a zigzag for in over of minimal length , then by using equalities (1), Lemma 2 and based on property for some , we haveThis is a contradiction and, so, . Thus is saturated.
Here, we generalize the result that the variety of normal bands was saturated [11] to medial semigroups fulfilling the identity .

Theorem 3. Any medial semigroup satisfying the identity is saturated.

Proof. Assume that is a nonsaturated medial semigroup fulfilling the identity . So there is a semigroup containing as a proper subsemigroup and such that . For any , if (1) is a zigzag for in over of least length , then by using equalities (1), Lemma 2 and based on property , it implies thatThis is a contradiction implying that . Therefore is saturated.

Classes of left [right] commutative semigroups and externally commutative semigroups satisfying the identity were saturated was shown in [4]. Here, we further extend this result for a class of medial semigroups.

Lemma 3. Let be a semigroup fulfilling the identity and let be a semigroup containing as a proper subsemigroup and such that . Then for all and .

Proof. If , then the proof follows trivially. In the other case, i.e., when , then, by Result 3, for some and . Now, based on property , we obtain .

Theorem 4. Medial semigroups satisfying the identity are saturated.

Proof. Take any semigroup fulfilling the identities and . If, by contradictory is nonsaturated, then there is a semigroup containing properly and such that . For any , if (1) is a zigzag for in over of minimal length , then by using equalities (1), Lemma 2, Lemma 3, and by the property of , we haveA contradiction and, so, . Hence is saturated.

Corollary 2. The variety of semigroups is saturated.

Corollary 3. Varieties of left [right] normal bands and normal bands are saturated.

5. Saturated Class of Paramedial Semigroups

It was shown, in [4], that a medial semigroup fulfilling the identity was saturated. In the following, this result is extended to paramedial semigroups by showing that a paramedial semigroup with the identity is saturated.

Proposition 2. For any proper subsemigroup of a semigroup , if and the identity is satisfied by , then and for all and , where for some and .

Proof. Now, as , let for some and . So, based on property , we obtainas required.

Theorem 5. Any paramedial semigroup satisfying the identity is saturated.

Proof. If to the contrary, a paramedial semigroup satisfying the identity is nonsaturated, then there is a semigroup with proper containment of such that . Now, for any , if (1) is a zigzag for in over of minimal length , then by using equalities (1), Proposition 2, and by the property of , we haveThus , which is a contradiction. Consequently, and, thus, is saturated and the proof of the theorem is completed.

6. Conclusion

In the present paper, authors have successfully proved that any -commutative semigroup satisfying a nontrivial identity of which at least one side has no repeated variable is saturated. Then it has been shown that all right [left] regular medial semigroups are saturated. Finally, we have proved that all medial semigroups satisfying the identities and are, respectively, saturated; and all paramedial semigroups satisfying the identity are saturated.

The results obtained in the paper have their immense utility as they imply that all epis from these classes are onto and open avenues and hope to explore further classes of semigroups for which epis are onto; for example, we list a few open problems in this direction to look into by researchers:(i)Is it possible to extend the results proved in the paper for semigroups satisfying identities other than used in the paper?(ii)The determination of all saturated classes of bands has been unanswered for long and an effort may be made in this direction.(iii)To explore whether the extension of Theorem 1 for the class of weakly commutative semigroups which is wider and larger class of H-commutative semigroups is possible or not?

Data Availability

No data were used to support the study

Conflicts of Interest

The authors declare that they have no conflict of interests.

Acknowledgments

This research has been funded by Research Deanship at the University of Ha’il, Saudi Arabia, through project number RG-20 189.

References

S. A. Ahanger, “Finite monogenic semigroups and saturated varieties of semigroups,” Semigroup Forum, vol. 101, no. 1, pp. 1–10, 2020.
View at: Publisher Site | Google Scholar
S. A. Ahanger, A. H. Shah, and N. M. Khan, “On saturated varieties of posemigroups,” Algebra Universalis, vol. 81, no. 4, p. 48, 2020.
View at: Publisher Site | Google Scholar
S. A. Ahanger and A. H. Shah, “Epis, dominions and varieties of commutative posemigroups,” Asian-European Journal of Mathematics, vol. 14, no. 04, Article ID 2150048, 2021.
View at: Publisher Site | Google Scholar
N. Alam, “On saturated semigroups,” Journal of Abstract and Computational Mathematics, vol. 3, pp. 110–115, 2019.
View at: Google Scholar
N. Alam, P. M. Higgins, and N. M. Khan, “Epimorphisms, dominions and $$\mathcal {H}$$-commutative semigroups,” Semigroup Forum, vol. 100, no. 2, pp. 349–363, 2020.
View at: Publisher Site | Google Scholar
N. Alam, N. M. Khan, and A. H. Shah, Epimorphisms, Dominions and Inflations of Clifford Semigroups, Algebra and Analysis Theory and its Applications, Narosa Publishing House Pvt. Ltd, New Delhi, India, 2015.
W. Burgess, “The meaning of mono and epi in some familiar categories,” Canadian Mathematical Bulletin, vol. 8, no. 6, pp. 759–769, 1965.
View at: Publisher Site | Google Scholar
A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, American Mathematical Society, RI, USA, 1961.
B. J. Gardner, “Epimorphisms of regular rings,” Commentationes Mathematicae Universitatis Carolinae, vol. 161, pp. 151–160, 1975.
View at: Google Scholar
T. E. Hall and P. R. Jones, “Epis are onto for finite regular semigroups,” Proceedings of the Edinburgh Mathematical Society, vol. 26, no. 2, pp. 151–162, 1983.
View at: Publisher Site | Google Scholar
P. M. Higgins, “Epis are onto for generalised inverse semigroups,” Semigroup Forum, vol. 23, no. 1, pp. 255–259, 1981.
View at: Publisher Site | Google Scholar
P. M. Higgins, “A semigroup with an epimorphically embedded subband,” Bulletin of the Australian Mathematical Society, vol. 27, no. 2, pp. 231–242, 1983.
View at: Publisher Site | Google Scholar
P. M. Higgins, “Saturated and epimorphically closed varieties of semigroups,” Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, vol. 36, no. 2, pp. 153–175, 1984.
View at: Publisher Site | Google Scholar
P. M. Higgins, “The varieties of commutative semigroups for which epis are onto,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 94, no. 1-2, pp. 1–7, 1983.
View at: Publisher Site | Google Scholar
J. M. Howie, An Introduction to Semigroup Theory, Academic Press, San Diego, CA, USA, vol. 7, 1976, London Mathematical Society Monographs.
J. M. Howie and J. R. Isbell, “Epimorphisms and dominions II,” Journal of Algebra, vol. 6, no. 1, pp. 7–21, 1967.
View at: Publisher Site | Google Scholar
J. R. Isbell, “Epimorphisms and Dominions,” in Proceedings of the Conference on Categorical Algebra, pp. 232–246, LaJolla, 1965.
View at: Publisher Site | Google Scholar
N. M. Khan, “Epimorphisms, dominions and varieties of semigroups,” Semigroup Forum, vol. 25, no. 1, pp. 331–337, 1982.
View at: Publisher Site | Google Scholar
A. Nagy, “Special classes of semigroups,” Advances in Mathematics, Kluwer Academic Publishers, The Netherlands, 2001.
View at: Publisher Site | Google Scholar
M. Petrich and N. Reilly, Completely Regular Semigroups, John Wiley & Sons, Canada, 1999, Canadian Mathematical Society Monograph.
V. Petar, “Protic: some remarks on paramedial semigroups,” Matematicki Vesnik, vol. 67, 2015.
View at: Google Scholar
H. E. Scheiblich, “On epics and dominions of bands,” Semigroup Forum, vol. 13, no. 1, pp. 103–114, 1976.
View at: Publisher Site | Google Scholar
A. H. Shah, S. Bano, S. A. Ahanger, and W. Ashraf, “On epimorphisms and structurally regular semigroups,” Categories and General Algebraic Structures with Applications, vol. 15, no. 1, pp. 231–253, 2021.
View at: Publisher Site | Google Scholar
P. G. Trotter, “A non-surjective epimorphism of bands,” Algebra Universalis, vol. 22, no. 2-3, pp. 109–116, 1986.
View at: Publisher Site | Google Scholar
E. J. Tully jr, “ℋ-commutative semigroups in which each homomorphism is uniquely determined by its kernel,” Pacific Journal of Mathematics, vol. 45, no. 2, pp. 669–679, 1973.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 N. Alam et al. 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.

PDF Download Citation

Download other formats

Order printed copies