On the Equational Base of SMB Algebras

Round 1

Reviewer 1 Report

Hello Dear Authors!   An article submitted for review on algebraic topics.   However, in our opinion, the material presented is not entirely satisfactory. There are the following remarks:   1. The abstract contains abbreviations not defined in the text. In the text, instead of a definition, a link to the textbook is given. This does not meet the standards for presenting material related to new and not publicly known research.   2. There is a link in the abstract. Moreover, this is a link to an unpublished work [1].   3. The specified work [1] is on the Internet. We see the problem in that the Introduction in the peer-reviewed article is shorter and less informative than the indicated preprint [1].   4. The introduction does not allow us to understand the significance of further calculations, to assess their novelty, fundamentality, and further prospects for the results.   5. The article does not have a section "Conclusions". Besides the fact that this does not meet the standards for presenting scientific work, it makes it extremely difficult to understand: what the array of calculations was done for.   In our opinion, the proposed work is in need of significant revision. As presented, it cannot be published. But it is possible that this issue can be resolved differently, after processing the material.   Best regards, your Reviewer

 

Comments for author File: Comments.pdf

Author Response

We thank Reviewer 1 for detailed and constructive remarks.
1. We have expanded the abstract to explain what the abbreviation “SMB algebra” stands for.
2. We have removed the link to the reference from the abstract. We describe both that paper and another paper which we mention in the new abstract, instead.
3. We have expanded the introduction, hopefully it is now informative enough. We could say a lot more if you want us to, but our paper is pretty short so we thought it should not have a very long introduction. Moreover, the preprint which was labeled [1] in the previous version is essentially equal to the final form of an accepted paper, the only thing missing is the journal formatting, so its text is also that of a peer-reviewed journal paper.
4. We have expanded the introduction, connecting our results to major (even award-winning) recent papers. We do not pretend our paper is at that level, but we hope it may be useful in further results of that kind.
5. We have added the Conclusion section, pointing out how our results can be used in computation. We also indicated a potential area of application towards Park’s Conjecture, a major open problem.

Reviewer 2 Report

The definition of semilattices of Mal’cev blocks (SMB) algebras  was first given in the paper by A. Bulatov [1]. In [2] it was proved that they are a finitely axiomatizable variety with various interesting properties. However, no finite equational base was explicitly found in that paper. In a recent lecture, P. Markovi´c mentioned that it may be useful to find such a finite base.
As the variety of SMB algebras will be further investigated in the near future, a compact set of equations axiomatizing it may prove useful for computer-assisted research. Moreover, the subvariety of regular SMB algebras was defined and a finite equational base for regular SMB algebras was given
in [2]. In this work the authors found an equational base for the variety of SMB algebras, simplify the equational base for regular SMB algebras and proved that one of our bases is minimal in the sense of inclusion order, while the other base might have one superfluous equation.

The work provided in this manuscript is new and it will be useful for the researcher working in this area of research. The new results are clearly stated and the reader should have no problem in understanding the reasoning. I therefore accept the publication of this manuscript.



[1] Bulatov, A. Constraint Satisfaction Problems over semilattice block Mal’tsev algebras. Information and Computation 2019, 268, 233 Article no. 104437, 14 pp.

2. Proki´c, A.; -Dapi´c, P.; Markovi´c, P.; McKenzie, R. SMB algebras I: on the variety of SMB algebras. Filomat 2022, accepted. 

Author Response

We thank Reviewer 2 for kind remarks. We note that, in this new version, we expanded the abstract, the introduction, added a Conclusions section and a few references in order to comply with other reviewer requests, but we have not changed the essential part of the paper. No statements of results, nor any of the proofs, have been changed.

Reviewer 3 Report

Title: On the equational base of SMB algebras

I think that the manuscript contain few new results and cannot be published in a reputed journal like Mathamtics (which had a high Impact Factor in 2021 and seems to be quite selective).

In general, the work needs better presentation and restructuring. 

The text is difficult to read, the objectives are not clear and the real contributions of the work are not presented.

The abstract is poor and it needs more information and contextualization.

The introduction must be improved to provide sufficient background and include all relevant references.

The adopted notation should also be explained.

The authors must comment and explain the development of the demonstration so that the reading is facilitated.

In addition, authors should present more relevant examples throughout the text.

Conclusion must describe the limitations of the work, the future prospects and the main contributions achieved.

 

Author Response

We thank Reviewer 3 for constructive remarks. Here are some changes we made in order to address some of his/her remarks:
1. We have not added any new results, so the overall level is as it was. We regret that Reviewer 3 believes it to be insufficient for the journal, but we have made the effort to address all other concerns he/she expressed.

2. We have added a Conclusions section, and expanded the abstract, the introduction and the literature. We hope this improves the presentation of our work. This should also clear up what are our contributions and what motivates the work we have done, putting it into the context of relevant recent research. The new introduction is also expanded to provide more background and we give more references.

3. Regarding the request for more explanation of the notation, we have added the definitions of semilattices and of Mal’cev algebras, and explained the notation of an identity and that the boldface stands for an algebra. These are all familiar to any researcher working in Universal Algebra since they have been adopted by major textbooks, but we realize we should have made more explanations in a general-audience journal. 

4. We added a few sentences before lemmas and theorems stating what will be achieved next. We hope that this sufficiently demonstrates the development of our proof.

5.  Examples are included in a sense. When we mention in the newly-expanded introduction that all semilattices of groups, i.e. all inverse semigroups which are unions of groups, have term reducts which are SMB algebras, we automatically give a large and well-studied class of examples. There are also interesting examples in Bulatov’s papers and in the notes by Marković, McKenzie and Maroti, but we refrained from including them. Our paper is syntactical in nature, providing semantical examples would not greatly increase the understanding of our results.

6. We have added a concluding section, which presents our hopes for future developments in the area. 

Reviewer 4 Report

In this paper, the authors find an equational base for the variety of SMB algebras, simplify the equational base for regular SMB algebras and prove that one of our bases is minimal in the sense of inclusion order.The variety of semilattices of Mal’cev blocks, for short SMB algebras, first appeared in 2009-2010 in some unpublished notes by P. Markovi´c and R. McKenzie and by M. Maróti.A. Bulatov first defined SMB algebras in (Bulatov, A. ”Constraint Satisfaction Problems over semilattice block Mal’tsev algebras.” Information and Computation 2019, 268, 233 Article no. 104437, 14 pp.) The work is exciting and deserves all the attention. However, a short chapter of conclusions would be necessary.

Author Response

We thank Reviewer 4 for kind remarks. We added a Conclusions section, as requested by this reviewer and others. We note that, in this new version, we also expanded the abstract, the introduction, and added a few references in order to comply with other reviewer requests, but we have not changed the essential part of the paper. No statements of results, nor any of the proofs, have been changed.

Round 2

Reviewer 3 Report

I think that the authors make efforts to improve the previous version. So, I recommend the paper for publication.