On r-Ideals and m-k-Ideals in BN-Algebras
Round 1
Reviewer 1 Report
Dear Editor,
The new referee report you requested is attached containing the questions and answers you want,
Respects.
New Review Report(mdpi-Axioms–A-Manuscript ID Number-axioms-1698907)-mcancan
Article title: On r-Ideals and m-k-Ideals in BN-algebras
Possible questions and answers about the article:
- What is the main question addressed by the research?
In the article, the illustrates, necessary properties, definitions and theorems,formulas in finding variationality features of a special “ideality equation” structure modelling are characterized different type ideal equations(based structure ideals and structure special rules).
- Do you consider the topic original or relevant to the field? Does it address a specific gap in the field?
This topic fills the specific gap in characterizing the original “algebraic,topological, geometric and Application maps , applied mathematics, representation approach, comparative analysis, r-m-k-type-ideality theory, BN-algebra theory” model issue in applied science fields (with BN type algebra theory and subsets of new ideal type sets).
- What does it add to the subject area compared with other published material?
The compared with other published material,there are adding “in determining special ideal approach field, sufficient conditions for a special complex ideality equations and its contents to be the modelling with special ideal set structures on the ideality approach theory” the subject area(under BN1-algebras).
- What specific improvements should the authors consider regarding the methodology? What further controls should be considered?
Authors can improve by developing new aspects of a special r-ideal approach structure on connecting with the structure m-k-ideal -problem equation to the algebraic ideality representation theory equations of common ideal problem transforms regarding the Methodology, The authors can provide a control mechanism by making new different aspects from the phenomenon of developing special ideality approach features(in BN-algebra; BN1-algebra approximation method models).
- Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?
The results appear consistent with the evidence and arguments.
The arguments presented adequately address the main question.
- Are the references appropriate?
In the article, references appear to be relevant to the study subject.
- Please (if any) include any additional comments on the tables and figures.
model mechanisms formed from special Axiomity features on ideal-algebraic approach theory and special ideality counting algebraic-mapping forms are developed and shown in terms of article images (under the structure ideal connection and with the ideals).
report:In the article, it is seen that the special " in algebraic topology,and applied mathematical field method(the theory of applied mathematics and science)( applied theoretic method)" featured definitions and calculations are made (the special mapping forms and enumerating the numbers of algebraic on a ideal net) for some special applied approach form equations (with ideal algebraic representation and strong enumerate the number calculations under r-ideal; m-k-ideal).
findings: In the article, the special “applied math and structural properties of the common complex ideal topology” process approach form equations are seen (the special ideal calculations) with the help of some special forms (under special ideal algebraic map and ideality -theoretic tool and with a binary operation).
strengths: In the article, there are proficiency-enhancing conditions for the special applied ideal algebraic approach forms.
weaknesses: In the article, it may be necessary to develop some extra new ideal algebraic conditions for the special ideal algebraic algorithms and applied ideal algebraic approach forms of this type(with axiomatic ideals and homomorphisms).
any minor issue: Although there are some minor weaknesses in terms of mathematical language, the article can be considered sufficient.
result: The article is suitable for publication for the journal.
Author Response
On this occasion, we would like to thank you very much for the axioms-1698907 reviewer reports. The reviewer’s comments/suggestions have improved the contents of our paper. In the following appendix file, we respond to a number of the reviewer’s suggestions and questions.
Author Response File: 
Author Response.docx
Reviewer 2 Report
Review Report on Paper Axioms-1698907
On r-Ideals and m-k-Ideals in BN-algebras
by
Sri Gemawati, Musraini, Abdul Hadi, La Zakaria, and Elsi Fitria
In this article, the authors define several new ideal types in BN-algebras, namely r-ideal,
k-ideal, and m-k-ideal. Further, they give some of their properties. They also provide
relationship between ideal in BN-algebra with r-ideal, k-ideal, and m-k-ideal. Moreover, the
authors give the concept of r-ideal and m-k-ideal homomorphisms in BN-algebra and BN1-
algebra.
This is not written very well. Also, the problem of study is not novel as well as there is
no very strong evidence to show novelty of the present work in this paper.
This paper has very big flaw from beginning. The authors have used wrong definition of
BN-algebra.
Terminologies used in the paper are standard and well-known. Paper does not contain
any new approaches to proof results.
Further, the following are suggestions and comments:
Define (BN) in abstract is wrong. This condition is same as (B3) for B-algebra. Same are
repeated in introduction at line 27.
Moreover, see Definition 2.1 and Definition 2.2, both have exactly same lines. I do not
find any difference between these two definitions.
Page 2, line 50: Provide reference here, “In 2017, the author discussed .....”
Page 2, line 65: Correct the sentence here since reference [9] contains many authors.
Page 3, line 99: Delete “Proof. Theorem 2.3 has been proved in [3].”
Page 4, line 127: Delete “Proof. Theorem 2.8 has been proved in [4].”
About Section 3 and Section 4: The authors should recheck all results in these sections
since the authors have given wrong definition of BN-algebra all the places in the paper.
Moreover, the authors should justify that how they are using correct definition of BN-
algebra.
The authors should give some examples to justify their results.
The authors should provide some applications of this work.
Conclusion: The authors have written about future. It should mention that what are
advantages of future aspects over the present work?
References: Almost all references are written in very rough style. Correct all of them
carefully and write them properly.
Note: (i) The authors have attached certificate for editing proper English language,
grammar, punctuation, spelling, and overall style. Somehow, it is ok but grammar and
language editing are not scientifically (especially mathematically). It should recheck by
mathematicians.
(ii) By writing style of paper, it does not seem that “All authors have read and agreed to
the published version of the manuscript.”
We provide one chance to improve this paper. This paper needs major improvements.
Author Response
On this occasion, we would like to thank you very much for the axioms-1698907 reviewer reports. The reviewer’s comments/suggestions have improved the contents of our paper. In the following appendix file, we respond to a number of the reviewer’s suggestions and questions.
Author Response File: Author Response.docx
Reviewer 3 Report
The paper defines new types of ideas in BN algebras and proves some of their properties. The results are new and the proofs appear to be correct, I recommend publication after some improvements to the exposition.
The paper offers a kaleidoscope of various definitions and properties, which is hard to follow without keeping prototypical examples in mind. There is not a single example of BN algebra in the paper, let alone examples that distinguish between their different flavors. The inclines are discussed in the Introduction without a word as to what that is. The only example of an incline ([0,1] with max and min), which is buried later in Preliminaries, and even then, a relationship between BN algebras and inclines is not explained at all. As a result, it is unclear what the motivation for considering these structures is, or whether various distinctions made among them have non-trivial scopes.
I suggest opening the Introduction with motivating examples of BN algebras (e.g. Coxeter algebras) before giving formal definitions, moving the incline example to the Introduction as well, to explain the concept properly, and spelling out how the former relate to the latter. This is particularly sensible considering that one of the main results of the paper is to transplant certain notions of ideals from inclines to BN algebras. It would also help to give examples of those various types of ideals when they are introduced.
In the Introduction there are phrases “In 2017, the author discussed…” and “In 2020, the author discussed…”. Considering that the paper has five authors, it would help to be more specific. I suspect that it is the first author that is referred to. In the references there are a couple of typos: BN1 should be BN_1 (1 in the index) in [4], and Mathe-matics should be Mathematics in [9].
Author Response
On this occasion, we would like to thank you very much for the axioms-1698907 reviewer reports. The reviewer’s comments/suggestions have improved the contents of our paper. In the following appendix file, we respond to a number of the reviewer’s suggestions and questions.
Author Response File: Author Response.docx
Reviewer 4 Report
The originality is based on the definitions of several ideals
in BN-algebras, namely r-ideal, k-ideal, and m-k-ideal. The situation is that the results are easy to obtain. So the contribution is less. However, these are correct.
Author Response
On this occasion, we would like to thank you very much for the axioms-1698907 reviewer reports. The reviewer’s comments/suggestions have improved the contents of our paper. In the following appendix file, we respond to a number of the reviewer’s suggestions and questions.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
The authors have not read all comments and suggestions carefully. They have ignored many of them. Almost all references are wrongly written. Moreover, the quality of manuscript has not been improved. See the following and include all of them carefully:
About Section 3 and Section 4: The authors should recheck all results in these sections since the authors have given wrong definition of BN-algebra all the places in the paper.
Moreover, the authors should justify that how they are using correct definition of BN-algebra.
The authors should give some examples to justify their results.
The authors should provide some applications of this work.
Conclusion: The authors have written about future. It should mention that what are advantages of future aspects over the present work?
References: Almost all references are written in very rough style. Correct all of them carefully and write them properly.
Author Response
We thank you profusely for the second response/response submitted by the reviewer for our article (Axiom-1698907). We have improved the article based on reviewers' comments/suggestions in the second revised version of the article. We also provide feedback as in the attached file that we include below.
Author Response File: 
Author Response.pdf
Reviewer 4 Report
The fundamental problem remains that the results presented in the paper are very simple, so it is important to highlight why these are relevant for an algebraist and in general for the mathematical community. It is also convenient to present a counterexample for the converse of Theorem 4.5
Author Response
We thank you profusely for the second response/response submitted by the reviewer for our article (Axiom-1698907). We have improved the article based on reviewers' comments/suggestions in the second revised version of the article. We also provide feedback as in the attached file that we include below.
Author Response File: Author Response.pdf
Round 3
Reviewer 2 Report
The authors have included all suggestions and comments in the paper. Now, we recommend manuscript for possible publication in Axioms.
