Skew Pairs of Idempotents in Partial Transformation Semigroups

Round 1

Reviewer 1 Report

I believe that the text of the manuscript should be revised according to my comments, and then it can be considered for publication.

Comments for author File: Comments.pdf

Author Response

Dear Referee

 

We modify the manuscript as the following:

1. We change the statement of Theorem 1 to avoid confusion. In Theorem 1, they presented that there are exactly four distinct types of skew pairs.
2. We add a description in line 34 that all mappings are written on the left.
3. We change Theorem 3 (Theorem 2 in the previous version) and Corollary 1 to avoid confusion.
4. We add “setminus” between ran ef and fix ef.

Other modified points are modified as other referee’s comments.

 

Sincerely yours,

Thawhat Changphas

Author Response File: Author Response.pdf

Reviewer 2 Report

I think this paper excellent and is an important addition to the literature. The authors gave a characterization when four types of skew pairs for partial transformation semigroups .The topic and results are well represented and seem interesting, but the authors must provide more references linked with their work.

Typos:

-Put Keywords in alphabetical order;

 -line 24, page 1: put “and ” between authors, according abstract;

-line 46, page 2: Do we need this “fix” ? Explain better;

-line 68, page 4: Is it true?

-line 109, page 7:  “we can show” ? Why is it not shown?

Reference 2: Put 2002 in bold and add a comma after Semigroup Forum. In reference 1 add a comma after Algebra.

Author Response

Dear Referee

 

We modify the manuscript as the following:

1. We put Keywords in alphabetical order.
2. We put “and” between authors.
3. We explain the definition of “fix” and add “setminus” between ran ef and fix ef.
4. We show mappings ef and fe in Corollary 3.
5. We show mappings ef, fe, fefe, efef, efe, fef in Theorem 8 (Theorem 7 in the old manuscript).

Other modified points are modified as other referee’s comments.

 

Sincerely yours,

Thawhat Changphas

Author Response File: Author Response.pdf

Reviewer 3 Report

In this paper, the authors consider the problem of characterization  of skew pairs of idempotents in partial transformation semigroups. They give such characterization for a regular semigroup consisting of partial transformations on a non-empty set based on cardinality of this set.

The article is well organized and the proofs are precise and concise.

In my opinion, the conclusion of the article can be extended a little bit. Moreover, I think that the word each in Corollary 1 can lead to confusion.

Author Response

Dear Referee

 

We change the text in Theorem 3 (Theorem 2 in the old file) and Corollary 1 to avoid confusion. Other modified points in the manuscript are modified as other referee’s reports.

 

Sincerely yours,

Thawhat Changphas

Author Response File: Author Response.pdf