The Semigroup Gazette

We introduce and investigate the irreducibility of endomorphisms of finitely generated free semigroups, i.e., we investigate when an endomorphism $φ: Σ^+ -> Σ^+$, where $Σ$ is any alphabet, can be nontrivially expressed as a composition $φ= ψ_2 \circ ψ_1$ of endomorphisms $ψ_1, ψ_2: Σ^+ -> Σ^+$. We, hence, study a notion of primality in the endomorphism monoid of the free semigroup---a natural and fundamental concept in this algebraic structure. We establish that irreducibility is a nontrivial property for the class of so-called rank-preserving endomorphisms, and we provide a characteristic condition separating the reducible and irreducible endomorphisms. We also characterise when an endomorphism is a factor of another endomorphism, analyse the non-uniqueness of factorisations of a rank-preserving endomorphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of rank-preserving endomorphisms.

We study the reflexivity property of ideals in numerical semigroups. In particular, we characterize irreducible reflexive ideals and determine the Apéry set of a reflexive ideal. Moreover, we study how reflexivity allows us to characterize some classes of numerical semigroups and the interaction of reflexivity with the property of being a trace ideal.

We show how to solve the conjugacy (search) problem and the problem of determining whether a given element is periodic in symmetric diagram groups based on finite semigroup presentations.

Let $(R,\m)$ be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal $R\subseteq K\subseteq\overline R$, and let $v$ denote its embedding dimension. Put $σ(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$. By definition, $R$ is nearly Gorenstein if and only if $σ(R)>0$. We prove that $R$ is almost Gorenstein if and only if $σ(R)\ge v$, equivalently, $σ(R)=\infty$. Thus every finite value of $σ(R)$ is at most $v-1$. We then determine the structure of rings for which $σ(R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\cong\m$ and $K^v\not\cong\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e \equiv 3 \pmod{v-1}$. As an application, we characterize the numerical semigroup rings $R$ with $σ(R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.

We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an $R$-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an $R$-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.

Let $$ S_n=\left\langle \genfrac{[}{]}{0pt}{}{n}{1},\ldots,\genfrac{[}{]}{0pt}{}{n}{n-1}\right\rangle $$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\genfrac{[}{]}{0pt}{}{n}{n-1}=\binom n2$ and $b_j=\genfrac{[}{]}{0pt}{}{n}{n-2j}$, we determine the complete gcd filtration of the reduced generating system. More precisely, if \[ M_j=\frac12\operatorname{lcm}\{m\ge1:\varphi(m)\le2j\}, \] then $$ \gcd(a,b_1,\ldots,b_j)=\frac{a}{\gcd(a,M_j)}. $$ The proof uses prime-power block polynomials and exact $p$-adic support at threshold degrees. We then establish a divisor-support property for canonical mixed-radix reductions and combine it with an elementary-symmetric growth estimate to prove that, after inactive generators are removed, the resulting generating sequence is telescopic. Consequently, the Apéry set of $S_n$ is rectangular, yielding an explicit Frobenius formula. The same structure shows that $S_n$ is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.

We introduce SemiBase, a project that computes and formally certifies finite identity bases for small semigroups. Deciding finite basability is undecidable for finite algebras and remains open for finite semigroups. The task requires a proof that a candidate basis is complete, or a proof that none exists, rather than a single first-order validity query. LLM-guided agents search for these proofs; a referee agent rebuilds them from source, and the Lean kernel checks the resulting corpus in a final audit. Humans choose targets and approve final outcomes. We certify every semigroup of order at most 6: all 1309 semigroups of order at most 5 and all 15973 of order 6, including proofs that the four known nonfinitely based semigroups have no finite basis. The bases for order 6 define 505 distinct varieties, whose inclusion order Vampire determines except for four pairs. The resulting catalogue is a machine-checked account of results scattered across the literature and a tested foundation for order 7.

We establish asymptotics for the degree and height growth rate functions of partially commutative semigroups of endomorphisms of varieties, i.e., semigroups where the only relations between generators, if any, are commuting ones. To do this, we prove a general result connecting the growth rates of two abstract functions, one multiplicative and one simply nonnegative, subject to a Tate telescoping hypothesis and a sufficiently nice normal form on the underlying semigroup. Finally, we construct several new families of endomorphisms on which our asymptotics apply. To do this, we generalize the standard ping-pong lemma for elements to a ping-pong lemma for subsemigroups.

We give a necessary and sufficient condition for a combinatorial D-class of a finite semigroup to have a permutation matching: that its structure matrix admit a balanced weighting, that is, non-negative weights with constant row sums and constant column sums. This weakens the regularity hypothesis, the role of Hall's lemma being taken up by the Birkhoff--von Neumann theorem. We then apply it to show that the monoid of all order-preserving self-maps of an n-element chain, has a permutation matching.

We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index $d$, the criterion reduces to the existence of an element with nonzero $(d-1)$st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.

In this paper, we study the presence of perfect powers in orbits of polynomials, or of semigroups generated by a finite family of polynomials, defined over a number field. In particular, we correct and generalize a result of Ahmadi et al. (2012), showing that multiplicative shifts of perfect powers in orbits can be detected by a finite algorithm. We also give a finiteness result on elements in orbits satisfying certain multiplicative relations.

We study the finite basis problem for flat semirings of the forms \(S_c(W)\), \(M_c(W)\), and \(M(W)\), where \(W\) is a nonempty set of words in a free commutative semigroup, a free commutative monoid, and a free monoid, respectively. We completely classify such flat semirings with respect to the finite basis property, allowing \(W\) to be infinite. We prove that \(S_c(W)\) is finitely based if and only if every word in \(W\) is either a cube of a letter or has length at most two, whereas \(M_c(W)\) and \(M(W)\) are finitely based if and only if \(W\) consists solely of the empty word. As applications, we recover the nonfinite basability of \(\flat(\mathbb{Z})\) and the max-plus semiring \((\mathbb{Z},\max,+)\).

We consider the 653 equational laws of order up to 4 for an associative binary operation, and all of their conjunctions. We determine that there are only 456 equivalence classes of such conjunctions (associative equational theories), and find all implications between them. The conjunction operation makes this set of theories into a semi-lattice. These results are formalized in Lean. This is a semigroup analogue of the Equational Theories Project, but extended to conjunctions of equations.

The finite basis problem for small semirings differs from its semigroup counterpart even in order three. Recent work classifies several additive types of four-element additively idempotent semirings, including all 386 semirings whose additive reduct is a chain. We consider all four-element semirings with commutative addition, in the binary signature without named constants. Combining the existing classifications with structural reductions and polynomial normal forms, we obtain 2284 finitely based and 57 nonfinitely based isomorphism types among the 2341 types. The nonfinitely based cases consist of 45 additively idempotent semirings and twelve with nonidempotent addition. The positive arguments give explicit bases or finite constructions with specified bounds. The negative arguments use cited results, term retractions and a hypergraph obstruction. A complete catalogue records the applicable result for each representative; separate correspondence tables identify precisely the cases supplied by the earlier classifications.

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal $I_H$ has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of $H$ has form $\{h+α,h+2α,\ldots,h+(n-1)α\}$ for $h\ge 0, α>0$. We prove this conjecture under the condition that $h$ has a unique factorization in $H$. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by $m=cA_{n-1}+1$, $1\le c\le a$. We prove that the relevant element $h$ has a unique factorization and the defining ideal is generated by the 2-minors of a $2\times n$-matrix. In particular, the Eagon--Northcott complex of the matrix mentioned above gives the graded minimal free resolution of $k[H]$.

We study g-dimonoids, which are algebraic structures equipped with two associative binary operations satisfying a prescribed system of axioms. We develop a construction of g-dimonoids based on inflations of semigroups, construct several classes of 2-dinilpotent commutative g-dimonoids, and investigate the isomorphism problem for these classes. As an application of these results, we obtain a structural description of all commutative g-dimonoids of order 3, up to isomorphism.

The Wagner-Preston theorem states that every inverse semigroup can be embedded into an inverse semigroup of partial bijections on some set. However, this embedding does not respect lattice operations in the natural ordering. In this paper, we show that an abstract pseudogroup can be faithfully represented by partial bijections on a set if and only if its frame of idempotents is spatial. More generally, we show that any abstract pseudogroup can be faithfully represented by partial homeomorphisms on a locale.

Motivated by recent work on conjugacy languages in groups, we introduce a general framework for studying \emph{languages of representatives} associated to equivalence relations on finitely generated semigroups. After developing this notion in full generality, we particularize it to Green's relations and obtain the corresponding Green languages, which form the central objects of this article. For free inverse monoids, we describe these languages via Munn trees and establish several structural relations among them, including characterizations of the $\mathcal{D}$-, $\mathcal{R}$-, $\mathcal{L}$-, and $\mathcal{H}$-languages. We then investigate the formal language-theoretic complexity of these Green languages. For free inverse monoids of rank at least two, we show that $\mathcal{H}$-Geo is context-free and co-context-free but not regular, while $\mathcal{R}$-Geo, $\mathcal{L}$-Geo, and $\mathcal{D}$-Geo are neither context-free nor co-context-free. In contrast, in the monogenic case, $\mathcal{D}$-Geo is regular and the remaining Green languages are deterministic context-free. We further relate these results to the ShortLex language and the growth of free inverse monoids, proving in particular that $(\text{FIM}_X,X)$ is growth tight. The article concludes with a collection of open problems and directions for future research.

We solve (modulo groups) the word problem for free semigroup satisfying $xy = (xy)^n$. This enables us to show that the variety ${\cal {SCR}}_n$ given by this identity is not equal to the join of the varieties ${\cal{CR}}_n$ and ${\cal{SI}}$ defined by the identities $x = x^n$ and $xy = (xy)^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\cal {SCR}}_n)$ is modular does not follow from the earlier results concerning $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ consisting of all semigroups from ${\cal {SCR}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\cal O}_n$ of all orthogroups and ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.

We study embeddings of Cohen--Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summand. Motivated by almost Gorenstein rings, we prove an Ext-vanishing criterion for finite projective dimension, which implies G-regularity and the generalized Auslander--Reiten condition. We characterize this summand condition for one-dimensional fiber products and numerical semigroup rings. For Stanley--Reisner rings of graphs, we characterize the corresponding condition for graded embeddings in terms of the graph and show that it is equivalent to a strict multiplicity inequality.

Using the duality between the categories of inverse semigroups and ample groupoids, we prove the triviality (global and local) of the central term of the extension in terms of twisted crossed products.

An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup $\mathbb{R}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigroup homomorphism to $\mathbb{R}_{>0}$. We show that every uniquely divisible $\mathfrak{N}$-semigroup $S$ admits enough semigroup homomorphisms to $\mathbb{R}_{>0}$ to detect divisibility in $S$. %They were studied by Tamura and Kobayashi. The main ingredient is a version of the Eidelheit and Kakutani hyperplane separation theorem for vector spaces over $\mathbb{Q}$. A counterexample due to Ravsky explains why this result does not generalise to the larger class of $\mathfrak{N}$-semigroups.

We study filter-Laver forcing on omegaxomega with successor sets containing final quadrants. The observation retaining the minimum coordinate and orientation is a complete projection. Its finite nontrivial hidden-label quotients are Cohen, whereas the full offset quotient has Boolean density equal to the ground-model dominating number dV . For the associated grid-meager ideal I_box, we prove cov(I_box) = add(M) for the usual meager ideal M, and obtain further cardinal bounds. We also analyze threshold games, oracle degrees and the distinction between positive and filter-large certificate sets: recurrent filter-large acceptance admits refinements whose every branch succeeds. As an arithmetic application, we prove Wilfs inequality c \leq 13e for numerical semigroups with thirteen elements below the conductor c, where eis the embedding dimension. A complete arithmetic proof, including a bounded exact computation, is given in the appendix. Its terminating certificate procedures supply a computable quadrant tree and uniform refinements for every forcing condition. Finite support kernels describe persistence of the registered proofs; arithmetic absoluteness explains precisely why forcing organizes these certificates without replacing their arithmetic justification.

Given power series $f_1,\dots,f_r\in z^2K[[z]]$ over a field $K$, we show that the semigroup $\langle f_1,\dots,f_r\rangle$ generated by the $f$'s under composition is free of rank $r$ whenever the multiplicative semigroup $\langle\mathrm{lc}(f_1),\dots,\mathrm{lc}(f_r)\rangle$ in $K^\times$ is free commutative of rank $r$; here $\mathrm{lc}(f)$ denotes the first nonzero coefficient of $f$. To do this, we associate an affine linear transformation to every $F\in\langle f_1,\dots,f_r\rangle$ and then apply a ping-pong lemma. In particular, combining this with work of Pappalardi, Sha, Shparlinski, and Stewart, it follows that most semigroups of polynomials over a number field $K$ are free, and we make this statement precise through a counting argument.

In this paper we explore the concept of smoothness in arithmetic semigroups. In particular, we give explicit asymptotics for the counting function $Ψ_G(x,y)$ when the arithmetic semigroup $G$ satisfies Axiom A (in the spirit of Knopfmacher). The asymptotics generalise the standard results in the literature for the integer case, and are expressed in terms of the Dickman $ρ$ function. Finally, we give applications of our results to counting smooth families of finite abelian groups and finite semisimple rings.

We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel--Şahin--Singh--Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family \[ Γ_m(n)=\langle m,m+1,\ldots,\widehat{m+n},\ldots,2m-1\rangle, \qquad 2\leq n<m, \] where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity $m$ and $3\leq n<m$, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.

We introduce semiconstellations and restriction semiconstellations, together with their actions on sets, providing a common setting for partial actions of categories, inverse semigroupoids, restriction semigroups, and semigroupoids. We then investigate the \emph{globalization problem}, which asks whether every partial action can, up to isomorphism, be obtained by restricting a global action. A distinctive feature of our approach is the use of reflectors not only to construct globalizations, but also to study when partial actions are globalizable. We show that every partial action of a restriction semiconstellation admits a reflector in the full subcategory of $S$-algebras consisting of what we call \emph{almost global actions}, and investigate conditions under which this reflector provides a globalization. As an application, we obtain a positive answer to the globalization problem for partial actions of restriction semigroupoids on sets, and consequently for restriction semigroups and restriction categories. In contrast, the answer is negative for restriction semiconstellations in general. Finally, given a semiconstellation $S$ and a suitable equivalence relation $R$ on $S$, we introduce the notion of an \emph{$R$-compatible action of} $S$ and construct a restriction semiconstellation $S^R$. We then show that partial and $R$-compatible actions of $S$ correspond, respectively, to partial and global actions of $S^R$. In this way, the globalization problem for semiconstellations can be reduced to the corresponding problem for restriction semiconstellations.

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. Specifically, we show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the associated semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Moreover, both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, we show that this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that it is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several additional key results. 1) A rigidity theorem: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result can be regarded as the ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 3) Herzog--Watanabe question: en route to proving the main theorem, we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

For an additively idempotent semiring (ai-semiring) $S$, let $S^0$ denote the ai-semiring obtained from $S$ by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval $[\mathsf{V}(S), \mathsf{V}(S^0)]$ of ai-semiring varieties between the variety generated by $S$ and that generated by $S^0$. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to $[\mathsf{V}(S_7), \mathsf{V}(S_7^0)]$, where $S_7$ is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra \textbf{611} (2022), 211--245). The same conclusion holds for $[\mathsf{V}(B_2^1), \mathsf{V}((B_2^1)^0)]$, where $B_2^1$ is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. \textbf{17} (2007), no.~8, 1537--1551) that the $7$-element ai-semiring $(B_2^1)^0$ has no finite basis for its identities.

We classify, up to quasi-isometry, the large-scale geometry of Galton--Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the \emph{ray class}, the \emph{full tree class} (which includes the binary tree), one \emph{chain class} $(\mathrm{C}_Λ)$ for every possible branching semigroup $Λ$, and the \emph{bushy} class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely \textit{not} quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on $\{1,2\}$, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving $D$-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.

We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.

Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $γ_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.

Let $(a,b,c,d) = \left(2mn, 2mp, m^2 - n^2 - p^2, m^2 + n^2 + p^2\right)$ be a primitive Pythagorean quadruple and let $S=\langle a, b, c, d\rangle$ be the numerical semigroup generated by $a,b,c,$ and $d.$ For convenience, we also let $Q = n^2 + p^2, δ= \gcd(n,p),$ and $n = δn_0$ for some $n_0 \in \mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\geq 2Q$ for $δ= 1$ and $m \geq \frac{2Q}δ - 1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.

The rank of a semigroup is the cardinality of a smallest generating set. In this paper, we introduce iterated wreath products of transformation semigroups. We determine the rank of iterated wreath products of full transformation monoids and symmetric groups.

For every semigroup $S$, the set $\mathcal{P}(S)$ of all subsets of $S$ and the set $\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the finite basis problem for the full and nonempty power semirings $\mathcal{P}(S_7)$ and $\mathcal{P}^{+}(S_7)$ of the multiplicative reduct of $S_7$, where $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For $\mathcal{P}^{+}(S_7)$, we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval $[\mathsf{V}(\mathcal{P}^{+}(S_7)), \mathsf{V}(\mathcal{P}(S_7))]$ in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.

We study numerical semigroups whose defining ideals admit determinantal presentations by the $2\times2$ minors of a $2\times n$ matrix. A conjecture of Cuong, Kien, Matsuoka, and Truong relates such presentations to arithmetic progressions of pseudo-Frobenius numbers. We prove a numerical criterion for this determinantal presentation, reducing the conjecture to a problem on factorizations of maximal elements of an Apéry set in the numerical semigroup. As an application, we verify the conjecture in embedding dimension four under a unique factorization assumption on an Apéry set.

Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimension three, we obtain such ideals as inverse images $I_h=\varphi_H^{-1}(t^hk[t]\cap k[H])$. If $\ell$ is the least degree of a defining relation of $k[H]$, then $I_\ell$ is monomial, whereas $I_{\ell+1}$ has a binomial generator in the cases considered here. We determine $I_{\ell+1}$ for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal $I$ in these families, we prove that its Rees algebra is a Cohen--Macaulay normal domain. In the non-symmetric multiplicity-four case, $I_{\ell+1}$ is seven-generated; for $H=\langle4,9,15\rangle$, we prove that its Rees algebra is again a Cohen--Macaulay normal domain.

Semigroups in which right divisibility is a partial order, that is, $\mathscr R$-trivial semigroups, have been studied by people in algebraic combinatorics in connection with Markov chains (especially left regular bands). Margolis and the author showed that the quiver of the algebra of an $\mathscr R$-trivial monoid is field independent. We prove here that the integral semigroup ring of an $\mathscr R$-trivial semigroup has a bound quiver presentation provided that it is unital. This then gives a uniform quiver presentation over any field via extension of scalars.

We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length $n$, every first nontrivial deduction between them requires an identity with at least $n+1$ variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.