Epimorphisms, dominions and H-commutative semigroups

Noor Alam; Noor Mohammad Khan; Peter M. Higgins

Epimorphisms, dominions and H-commutative semigroups

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Pith's one-line read Dominions of H-commutative semigroups are H-commutative, and the minimum condition on principal right ideals makes them saturated.

desk verdict Clean generalization of Isbell and Howie-Isbell from commutative to H-commutative semigroups; the main zigzag proof holds up. read the letter →

arxiv 1908.01813 v2 pith:NA3UR6UB submitted 2019-08-05 math.GR

Peter M. Higgins , Noor Alam , Noor Mohammad Khan This is my paper · ORCID

classification math.GR MSC 20M07

keywords H-commutativesemigroupsdominionepimorphismsaturatedsemigroupGreen'srelationszigzagminimumconditiononprincipalrightidealsarchimedean

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that H-commutativity is hereditary through dominions: if $U$ is an H-commutative subsemigroup of any semigroup $S$, then the set of elements of $S$ that cannot be separated from $U$ by a homomorphism—the dominion $\operatorname{Dom}(U,S)$—is itself H-commutative. This extends a classical closure result from commutative semigroups to the wider H-commutative class. The authors use this dominion closure to prove that every H-commutative semigroup satisfying the minimum condition on principal right ideals is saturated, meaning it cannot be epimorphically embedded as a proper subsemigroup of any semigroup. Along the way they show that all five Green's relations coincide on an H-commutative semigroup, that the semigroup decomposes as a semilattice of archimedean subsemigroups, and that the regular H-commutative semigroups are exactly semilattices of groups.

What carries the argument

The central mechanism is the zigzag characterization of dominion membership: an element $d\in S$ belongs to $\operatorname{Dom}(U,S)$ exactly when it admits a finite alternating chain of factorizations $d = a_0t_1 = y_1a_1t_1 = \cdots = y_m a_{2m}$ with every $a_i\in U$ and the intermediate factors in $S$. The proof of Theorem 4.1 pushes the equation $dh = hwd$ through this chain, moving each $a_i$ past $d$ using the H-commutativity of $U$ and collecting the witnesses into a single element $w\in U$. The same zigzag form, together with the minimum condition on principal right ideals, is what forces the contradiction in the saturation proof.

What would settle it

Find an H-commutative subsemigroup $U$ inside a semigroup $S$ and elements $d,h\in\operatorname{Dom}(U,S)$ such that $dh \neq hwd$ for every $w\in\operatorname{Dom}(U,S)$. No such pair should exist if Theorem 4.1 is correct, so a concrete example would settle it; equivalently, one can compute a finite example and check the zigzag condition directly.

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Extended reading notes

Core claim

The load-bearing claim is Theorem 4.1: for an H-commutative subsemigroup $U$ of an arbitrary semigroup $S$, the dominion $\operatorname{Dom}(U,S)$ is H-commutative. The proof takes two elements $d,h$ of the dominion and, using a finite zigzag factorization of $h$ over $U$, repeatedly swaps the $U$-elements past $d$ by applying $ab=bxa$ inside $U$. Each swap produces a new element of $U$, and the final product of these elements is a single $w\in U$ such that $dh = hwd$. This directly implies that epimorphic images of H-commutative semigroups are H-commutative. The same machinery, combined with a minimality argument on descending chains of principal right ideals, yields the saturation theorem: any H-commutative semigroup with the minimum condition on principal right ideals is saturated.

Load-bearing premise

The argument assumes the zigzag characterization of dominions is complete: every element of $\operatorname{Dom}(U,S)$ can be written as a finite alternating chain of factorizations in $S$ over $U$. If that description missed some dominion elements, the swapping argument would not cover them.

Editorial extensions

If this is right

Epimorphic images of H-commutative semigroups are H-commutative (Corollary 4.2), so H-commutativity is preserved by the broadest notion of surjective-like morphism.
Every H-commutative semigroup satisfying the minimum condition on principal right ideals is saturated (Theorem 4.9), generalizing the commutative case.
Every H-commutative archimedean semigroup containing an idempotent is saturated (Theorem 4.10).
Inside any H-commutative semigroup, all five Green's relations coincide and the semigroup is a semilattice of archimedean semigroups (Theorems 2.3 and 2.10).
The regular members of the class are exactly semilattices of groups (Theorem 2.6).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because dominion closure is the key step, any semigroup that appears as an epimorphic image of an H-commutative semigroup must itself be H-commutative; this makes H-commutativity an 'epi-invariant' property and suggests that saturation questions for the class reduce to one-sided ideal conditions.
The direct-product counterexample in Example 3.3 shows the first-order definition does not behave like a variety; one testable extension is whether the dominion closure proof can be rerun for left or right semicommutative semigroups, where the same zigzag commutation might fail.
Since the archimedean components of an H-commutative semigroup need not be H-commutative (Example 3.6), the saturation theorem cannot be proved component-wise; it would be natural to test whether the minimum-condition hypothesis can be replaced by a condition on each archimedean component.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this one. First, Theorem 4.1 is the real item: the dominion of an H-commutative subsemigroup is H-commutative, and the proof is a genuine zigzag chase, not a repackaging of the commutative case. Second, Theorem 4.9 follows from it in a straightforward way and gives the advertised Howie-Isbell analogue. I read the proof of 4.1 in detail, including the delicate case (iv) where both elements lie outside U. The manipulations are legitimate: each swap uses H-commutativity of U on a product that indeed lies in U, and the intermediate products lie in Dom(U,S) because dominions are subsemigroups. I did not find a hidden step.

What is new: the paper extends Isbell's theorem and the saturatedness theorem to a class that includes all groups, and it does so with a completely explicit argument. Section 2 is mostly expository, restating Nagy's structural facts with proofs, but that is fine because the later sections depend on those facts. The examples are useful and not thrown in; Example 3.3 showing HC is not closed under direct products is a nice counterexample to a naive expectation.

The soft spots are minor. The abstract says Howie and Isbell proved the result for H-commutative semigroups, but the original was for commutative semigroups; that is a wording slip, not a mathematical one. In the proof of Theorem 4.8, the sentence "Since T (and not just S) is H-commutative" silently relies on Corollary 4.2 — true, but it should be cited there. The typesetting has several typos ("explor e", "i.e." spacing), but those look like copyediting issues, not substantive errors.

I checked for circularity: the arguments use Isbell's zigzag theorem as an external characterization, plus standard facts about H-commutative semigroups. There is no parameter fitting and no reuse of the target result. The citation pattern is honest, mostly to Nagy's book and the original Isbell/Howie papers.

Who is this for? People working on epimorphisms and dominions in semigroup theory. It is a solid advance within that niche, not a field-shifting result, but it is a complete and correct one. My recommendation: send it to peer review. A competent referee will not struggle with soundness; the main work is checking whether the generalizations are as broadly stated, and they are.

Referee Report

0 major / 6 minor

Summary. The paper studies H-commutative semigroups, that is, semigroups satisfying ab = bxa for some x in S^1. It establishes structural results (coincidence of Green's relations, semilattice decompositions, regular cases), gives examples and closure properties, and then proves the two main generalizations: Theorem 4.1 states that the dominion of an H-commutative subsemigroup in any containing semigroup is H-commutative, and Theorem 4.9 states that every H-commutative semigroup satisfying the minimum condition on principal right ideals is saturated. The proof of Theorem 4.1 is a detailed zigzag chase using Isbell's zigzag theorem, and the saturation theorem follows by adapting Howie and Isbell's arguments with the help of the dominion result. The paper is written as a conventional semigroup-theory contribution and relies on standard external results such as Isbell's zigzag theorem and Nagy's structure theory.

Significance. The main contribution is Theorem 4.1, which generalizes Isbell's theorem on dominions of commutative semigroups to H-commutative semigroups. The zigzag proof is carefully structured and the later saturation theorem (Theorem 4.9) is a natural and nontrivial consequence. The structural results in Section 2 and the examples in Section 3 (especially the failure of HC to be closed under direct products and the non-H-commutativity of archimedean components) are useful additions. The argument is based on standard tools—Isbell's zigzag theorem, Green's relations, Nagy's semilattice decomposition—and I found no circularity or unsupported parameter fitting. If the minor presentation issues are corrected, the paper would be a solid contribution to the dominion theory of semigroups.

minor comments (6)

[Abstract] The abstract states that the paper generalises 'Howie and Isbell's result that any H-commutative semigroup satisfying the minimum condition on principal ideals is saturated'. This misattributes the original theorem: Howie and Isbell proved this for commutative semigroups, and the introduction correctly states so. Please rephrase the abstract, and also change 'a H-commutative' to 'an H-commutative'.
[§4, Theorem 4.1 proof] In Case (iv), the first step dh = da0y1 = a0w1dy1 is an application of Case (iii), not Case (ii), since d ∈ Dom(U,S)\U and a0 ∈ U. Later citations of Case (ii) for products such as w1d are also technically outside the stated case when the product falls in U; the conclusion still holds by H-commutativity of U, but a clarifying remark would remove the ambiguity.
[§4, Proposition 4.3] From a^r K^1 = a^{2r}K^1 the inference 'a^r = a^{2r}c for some c ∈ K' is not immediate, because the element provided by K^1 could be the identity. If it is the identity, then a^r is idempotent and one may take c = a^r ∈ K (since a^r ∈ K for a right ideal K by induction); this one-line justification should be added.
[§2, Theorem 2.5(a)] The displayed equation 'ab = bxa = xyba' is terse: the second equality follows by applying H-commutativity to b and x, yielding bxa = x y b a. As written, 'xyba' is ambiguous and should be written with appropriate spacing or a short explanation.
[§4, Proposition 4.4] The symbol B is used both for the set of left divisors of b in K and for the set of principal right ideals generated by those divisors. Rename the second collection to avoid confusion.
[References and abstract] There are several typographical errors: '[6, Corollay 2.5]' should be 'Corollary 2.5', the reference list contains 'confrence', and the abstract contains 'explor e'. Please proofread these.

Circularity Check

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No significant circularity: the central derivation is self-contained from Isbell's external zigzag theorem.

full rationale

I walked the derivation chain. Theorem 4.1 is proved by a zigzag chase: every rearrangement is an application of H-commutativity of U, and membership of intermediate products in Dom(U,S) follows from Dom(U,S) being a subsemigroup. The proof cites Case(ii) in Case(iv) in places where the needed orientation is Case(iii), but Case(iii) is explicitly stated as the left-right dual and is available, so this is a citation typo rather than a circular reliance. Isbell's zigzag theorem (Result 1.4) is quoted from standard external references [6,4] and is a full characterization; no target result is assumed in its proof. Corollary 4.2 legitimately transfers H-commutativity along epis using Theorem 4.1. Theorem 4.8's use of 'T is H-commutative' is justified by Corollary 4.2 from Dom(S,T)=T. The saturation theorems rely on standard facts about H-commutative semigroups and absolute closedness of groups [5, Theorem 2.3]. Self-citations to Khan [7] and Khan-Shah [8] are contextual and not load-bearing in the main proof. I find no fitted parameters called predictions, no self-definitional reductions, and no uniqueness assertions imported from the authors.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on a small number of external theorems from the semigroup literature, listed above. The proofs of the new results are derived from these and from the definition of H-commutativity.

assumptions (4)

standard math Isbell's zigzag theorem characterizes dominion membership (Result 1.4, from [6] and [4]).
Used as the foundation of Theorem 4.1 and Propositions 4.4-4.7 to represent elements of Dom(U,S) as zigzags. The paper does not prove this theorem.
standard math Nagy's theorem: a semigroup is H-commutative if and only if Green's relation H is a commutative congruence (Result 1.2, [11, Theorem 5.2]).
Invoked in Remark 2.4, Theorem 2.5, Lemma 2.9 and the proof of Theorem 4.1 to use divisibility and congruences. The paper cites this as a known characterization.
standard math Nagy's theorem: every H-commutative semigroup is a semilattice of archimedean semigroups (Result 1.3, [11, Theorem 5.3]).
Used as motivation in the introduction and effectively re-proved as Theorem 2.10. The paper relies on the archimedean decomposition concept.
standard math A group is absolutely closed (Howie-Isbell [5, Theorem 2.3]).
Used in Theorem 4.8 to obtain a contradiction from a zigzag over the group He.

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Cite this review

Pith. "Pith review of Epimorphisms, dominions and H-commutative semigroups." pith.science (2026). https://pith.science/paper/NA3UR6UB

@misc{pith2026190801813,
  author       = {Pith},
  title        = {Pith review of: Epimorphisms, dominions and H-commutative semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NA3UR6UB}},
  note         = {Machine review of arXiv:1908.01813}
}

read the original abstract

In the present paper, a series of results and examples that explore the structural features of H-commutative semigroups are provided. We also generalise a result of Isbell from commutative semigroups to H-commutative semigroups by showing that the dominion of an H-commutative semigroup is H-commutative. We then use this to generalise Howie and Isbell's result that any H-commutative semigroup satisfying the minimum condition on principal ideals is saturated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Bogdanovic, Ciric, S.M. D. and Z.L. Popovic, Semilattice decompositions of semi- groups, Faculty of Economics, University of Niˇs (2011)

work page 2011
[2]

Cliﬀord, A.H. and G.B. Preston, The Algebraic Theory of Semigroups , Math. Surveys No.7 (Amer. Math. Soc., Providence, R.I., Vol. I, 1961; Vol II, 1967)

work page 1961
[3]

Higgins, P.M., Techniques of semigroup theory , OUP, (1992). 19

work page 1992
[4]

Howie, J.M., Fundamentals of Semigroup Theory , London Math. Soc. Monograph (1995), Clarendon Press, Oxford

work page 1995
[5]

Howie, J.M. and J.R. Isbell, Epimorphisms and dominions II , J. Algebra, 6 (1967), 7-21

work page 1967
[6]

of the confrence on Categorical Algebra, La Jolla, (1966), 232-246, Lange and Springer, Berlin

Isbell, J.R., Epimorphisms and dominions , Proc. of the confrence on Categorical Algebra, La Jolla, (1966), 232-246, Lange and Springer, Berlin

work page 1966
[7]

Rajasthan Acad

Khan, N.M., On permutative saturated semigroups , J. Rajasthan Acad. Phy. Sci., Vol. 5, No. 2, (2006), 167-174

work page 2006
[8]

Khan, N.M. and A.H. Shah, Epimorphisms, dominions and permutative semi- groups, Semigroup Forum, Vol. 80 (2010), 181-190

work page 2010

Show all 13 references

[9]

Mary, X., Classes of semigroups modulo Green ’s relation H, Semigroup Forum 88, (2014), 647-669

2014
[10]

Mary, X., Reverse order law for the group inverse semigroups and rings , Commu- nications in Algebra, 43(6), (2015), 2492-2508

2015
[11]

Nagy, A., Special Classes of Semigroups , Springer (2001)

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[12]

Strecker, R., H-commutative ∆ - semigroup , Rostock. Math. Kolloq. 49, 98-104 (1995)

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Tully, E.J.jr., H-commutative semigroups in which each homomorphism is uniq uely determined by its kernel , Paciﬁc J. Math. 45, 669-681 (1973). 20

1973

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bogdanovic, Ciric, S.M. D. and Z.L. Popovic, Semilattice decompositions of semi- groups, Faculty of Economics, University of Niˇs (2011)

work page 2011

[2] [2]

Cliﬀord, A.H. and G.B. Preston, The Algebraic Theory of Semigroups , Math. Surveys No.7 (Amer. Math. Soc., Providence, R.I., Vol. I, 1961; Vol II, 1967)

work page 1961

[3] [3]

Higgins, P.M., Techniques of semigroup theory , OUP, (1992). 19

work page 1992

[4] [4]

Howie, J.M., Fundamentals of Semigroup Theory , London Math. Soc. Monograph (1995), Clarendon Press, Oxford

work page 1995

[5] [5]

Howie, J.M. and J.R. Isbell, Epimorphisms and dominions II , J. Algebra, 6 (1967), 7-21

work page 1967

[6] [6]

of the confrence on Categorical Algebra, La Jolla, (1966), 232-246, Lange and Springer, Berlin

Isbell, J.R., Epimorphisms and dominions , Proc. of the confrence on Categorical Algebra, La Jolla, (1966), 232-246, Lange and Springer, Berlin

work page 1966

[7] [7]

Rajasthan Acad

Khan, N.M., On permutative saturated semigroups , J. Rajasthan Acad. Phy. Sci., Vol. 5, No. 2, (2006), 167-174

work page 2006

[8] [8]

Khan, N.M. and A.H. Shah, Epimorphisms, dominions and permutative semi- groups, Semigroup Forum, Vol. 80 (2010), 181-190

work page 2010