Categorically Closed Unipotent Semigroups

Round 1

Reviewer 1 Report

Let C be a class of T1 topological semigroups which contains all Hausdor zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in anytopological semigroup Y 2 C that contains X as a discrete subsemigroup; X is injective-ly C-closed if for any injective homomorphism h : X ! Y to a topological semigroupY 2 C, the image h(X) is closed in Y . A semigroup X is unipotent if it contains a uniqueidempotent. The authors prove that a unipotent commutative semigroup X is (injective-ly) C-closed if and only if X is bounded and nonsingular (and group-nite). The results

of this manuscript are complete and the paper is written enough well. So I recommend publication of the paper but after some revision. My suggestions are as follows.

1. In the third line of Abstract, replace \(injective)" with \injective".

2. In the ninth line of Page 3, replace \For a semigorup X let" with \For a semigroup X, let".

3. In the twelfth line of Page 3, replace \proved" with \proven".

4. In the fourteenth line of Page 4, replace \ n p A = fx 2 S : xn 2 Ag and N p

S A =n2N n p

A = fx 2 S : A \ xN 6= ?g" with \ n p

A = fx 2 X : xn 2 Ag and N

p

A =

S

n2N

n p

A =

fx 2 X : A \ xN 6= ?g".

5. In the eighteenth line of Page 4, replace \is proved" with \is proven".

6. In the proof of Lemma 6, for the number of claims, why start with Claim 5?

7. the twenty-fth line of Page 6, replace \applying Lemma 1, conclude that" with \applying Lemma 1, we conclude that".

8. In the rst sentence of section 4, replace \Lemma 5 and 6" with \Lemmas 5 and 6".

9. For the proof of Theorem 3, it follows from Lemmas 5 and 6. It is not clear. So I suggest that the authors explain it in detail.

 

Comments for author File: Comments.pdf

Author Response

Thenk you very much for the careful reading the manuscript and many helpful suggestions, all of which were taken into account in the revised version. Concerning your last remark (how to deduce Theorem 3 from Lemmas 5 and 6), we added a sentence that Lemmas 5 and 6 imply the ``only if'' and ``if'' parts of  Theorem 3, respectively.  

Reviewer 2 Report

Report on the manuscript

 Title: Categorically closed unipotent semigroups

Taras Banakh and Myroslava Vovk

 The paper prove that a unipotent commutative 5 semigroup X is (injectively) C-closed if and only if X is bounded and nonsingular (and group-finite).

I think the readers of this journal will appreciate the results presented in the manuscript.  Generally speaking, the manuscript is well written, the material is judiciously divided and organized and correct from scientific point of view. Some changes are, however, necessary. For these reasons I can recommend the acceptance of this paper after some corrections presented in the attached file.

Comments for author File: Comments.pdf

Author Response

Thank you very much for careful reading our manuscript and many valuable comments. All of them were taken into account in the revised version of the paper. In particular, the following changes have been made:

1) The names Banakh--Bardyla are removed from Theorem 1;

2) Many sentences are reworded in oder to avoid using the informal ``we''.

3) We have removed four self-citations and added some more references in order to make the self-citation rate equal to 20%. Unfortunatley, this forced us to remove also Remark after Problem 1, saying that this Problem is partially answered in some our preprints (which were removed from the list of references in odered to keep the number of self-citation at 20%).

4) At the end of the paper we have added Conclusion explaining the new results obtained in the paper and their importance.

With best regards,

The authors

Round 2

Reviewer 2 Report

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