Semigroups uniquely determined by one-sided identity and zero sets
Pith reviewed 2026-05-23 19:20 UTC · model grok-4.3
The pith
Every right group with maximal subgroup size 2 and every commutative-rectangular band is uniquely determined by the one-sided identity and zero sets of its elements.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A groupoid or semigroup is stabilized with respect to the one-sided identity and zero sets if any other structure on the same set with identical such sets must have the same binary operation. Every right group with maximal subgroup size 2 is a stabilized semigroup with respect to these sets. A commutative-rectangular band is defined as a band where every pair of elements either commutes or are generalized inverses, and every such band is likewise stabilized.
What carries the argument
The stabilized property: the binary operation is the unique one compatible with given left and right identity sets and left and right zero sets for each element.
If this is right
- The maximal subgroups and maximal left and right zero subsemigroups can be read directly from the sets.
- The rectangular band subsemigroups are likewise recoverable without the full table.
- Any two semigroups in these classes that share the same collection of sets must be identical as algebraic structures.
- Classification or enumeration of these semigroups can be carried out by specifying consistent assignments of the identity and zero sets.
Where Pith is reading between the lines
- The same uniqueness property may hold for additional classes of semigroups whose axioms constrain products via identity and zero relations.
- An algorithm could reconstruct the multiplication table from the sets alone for any semigroup known to belong to these classes.
- The approach of fixing an operation via relational sets might extend to other semigroup varieties or to structures defined by Green's relations.
Load-bearing premise
The semigroup must obey the axioms of a right group with subgroups no larger than size 2 or of a commutative-rectangular band, which are used to show that matching sets force identical products.
What would settle it
Two distinct multiplications on the same finite set that produce identical left-identity, right-identity, left-zero, and right-zero sets for every element, while forming a right group whose largest subgroup has size 2.
read the original abstract
For a groupoid $S$ with elements $a$ and $b$, if $ba = a$, then $b$ is a left identity of $a$ and $a$ is a right zero of $b$. We define the left identity set of $a$ to be the set of all left identities of $a$ in $S$, and similarly for the right identity set of $a$ in $S$. We defined the left zero set of $a$ to be the set of all left zeroes of $a$ in $S$, and similarly for the right zero set of $a$. The one-sided identity and zero sets of a semigroup can be utilized in the determination of its maximal subgroups, maximal left and right zero subsemigroups, maximal left and right subgroups, and rectangular band subsemigroups. A band is an idempotent semigroup. Every commutative band is a semilattice and uniquely determined by the left and right identity sets of its elements or equivalently by the left and right zero sets of its elements. We generalize this notion by defining a groupoid or semigroup to be stabilized with respect to binary relations, in particular the binary relations defined by the one-sided identity and zero sets of its elements, if and only if for any groupoid or semigroup on the same set with the same binary relations, their binary operations are identical. We prove every right group with maximal subgroup size $2$ is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements. We define a commutative-rectangular band to be a band in which every pair of elements either commutes or are generalized inverses of each other, and we prove a commutative-rectangular band is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines left/right identity sets and left/right zero sets for elements of a groupoid or semigroup. It introduces the notion of a semigroup being stabilized w.r.t. these sets (i.e., the sets uniquely determine the multiplication table). It recalls that every commutative band is stabilized and proves two new results: every right group whose maximal subgroups have size 2 is stabilized w.r.t. the one-sided identity or zero sets, and every commutative-rectangular band (a band in which every pair of elements either commutes or forms a pair of generalized inverses) is stabilized w.r.t. the same sets. The sets are also claimed to determine maximal subgroups and certain subsemigroups.
Significance. If the two stated theorems hold, the work extends the known uniqueness result for commutative bands to two additional classes, providing concrete families of semigroups whose multiplication is recoverable from the one-sided identity/zero sets. This may be useful for the classification tasks mentioned in the abstract. The explicit definition of the new commutative-rectangular band class and the stabilization notion constitute a modest but concrete contribution to semigroup theory.
minor comments (3)
- [Abstract] Abstract, first paragraph: the sentences beginning 'We defined the left identity set...' and 'We defined the left zero set...' employ past tense; standard mathematical style uses present tense ('We define') when introducing definitions.
- [Abstract] Abstract, second paragraph: the bracketed notation 'one-sided identity [zero] sets' is ambiguous and should be replaced by an explicit phrase such as 'one-sided identity sets or zero sets'.
- [Abstract] Abstract, definition of stabilized: the long sentence beginning 'We generalize this notion by defining a groupoid or semigroup...' is run-on and would benefit from being split or rephrased for readability.
Simulated Author's Rebuttal
We thank the referee for their careful reading, accurate summary of the paper, and recommendation of minor revision. The referee's assessment correctly identifies the main results: the stabilization of right groups with maximal subgroups of size 2 and of commutative-rectangular bands with respect to one-sided identity and zero sets. No specific major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper defines the one-sided identity and zero sets directly from the groupoid operation, introduces the stabilized property as the sets determining the multiplication uniquely, recalls the standard fact that commutative bands are semilattices uniquely determined by these sets, and then proves two new results: right groups with maximal subgroup size 2 are stabilized, and commutative-rectangular bands (newly defined) are stabilized. Both proofs proceed from the defining axioms of right groups or bands together with the set definitions; no parameters are fitted, no predictions reduce to inputs by construction, and no self-citations or imported uniqueness theorems appear in the provided text. The derivation is therefore self-contained against the structural definitions.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The binary operation is associative (semigroup axiom)
- domain assumption Left and right identity/zero sets are well-defined subsets of the underlying set
discussion (0)
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