A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite

Round 1

Reviewer 1 Report

See my referee's report (ref-axioms.pdf) which is to be passed on to the authors

Comments for author File: Comments.pdf

Author Response

We agree with all Referee's remarks and corrected the text according to them.

Reviewer 2 Report

Dear Authors, 

There are many questions and misunderstandings in the reviewed paper.

Now, point by point.

1. The abstract is written incorrectly and does not reflect the essence of the proposed paper. The Abstract should contain a concise description of the paper without definitions and formulas.

2. Introduction is almost none. There is no prehistory of this theory or problems from this theory in the Introduction. Definitions and Auxiliary Statements are included in the Introduction, but should be a separate paragraph. The main result of this paper in the form of Theorem 1 is included in the Introduction, and this is unacceptable.

The structure of the entire paper is incorrect.

3. Examples should be at the end of the article in the form of an Appendix or Application. And the fact that Example 1 motivates the problem in the article looks very strange.

4. Now about the results. In the presented paper, the results obtained are very weak; they look primitive and too obvious. All Lemmas are direct consequences of the well-known Remsey Theorem from [3], and the main result is a consequence of these Lemmas.

Therefore, we cannot evaluate anything.

Author Response

We agree with some of the remarks of the Referee. We improved introduction and gave better motivation of this result. Also, we agree that proofs of some lemmas are  not so complicated. Our idea was to write this result as simply as possible, to make it more accessible to readers. Thats why we divide the proof of the main theorem into 5 lemmas. But in general, we don't think that our result is "primitive and obvious". 

Reviewer 3 Report

In this work the authors provide a  characterization of finite semigroups in terms of finite chains and antichains. Namely prove that  a semigroup is finite if and only if it is chain-finite and antichain-finite. The proofs are presented adequetly and the work has a clear structure. To the best of my knowledge the result is of not o great importance but yet is original and i support the publication following some minor improvments. 

1. The introduction is too short, please elaborate a bit more into the previous results and mention some potential impact of this result. 

2. I understand that work is more expository in nature nevertheless please a include a few concluding remarks to further enhance the importance.

 

Author Response

We totally agree with the first remark of the Referee. According to it, we improve the introduction by adding information about chain-finite semilattices and posets. Hopefully, this time the motivation will be more clear.

However, the second remark of the Referee is less clear. This manuscript is a short note which contains basically one theorem. So, we think that each reader can draw his own conclusions. If the Referee means further applications, then probably this theorem  can be applied to the combinatorial structure of semigroups. Maybe, somebody can use it in the theory of complete or H-closed topological semigroups. We don't know this in advance and so we think that it will be better not to insert our hypotheses into the text.