Geometrical illustration of numerical semigroups and of some of their invariants

Let \(p,q\in \mathbb {N}\) be relatively prime integers with \(1 . In Knebl et al. (J Algebra 348:315–335, 2011) the numerical semigroups containing \(p\) and \(q\) have been illustrated by certain lattice paths in the plane. In this paper we derive some further information from this visualization, for example formulas to compute or estimate the number of certain semigroups by counting lattice paths, and formulas for their genus and Frobenius number.     

Weakly B-orthodox semigroups

We introduce the notions of a band category and of a weakly orthodox category over a band. Our focus is to describe a class of weakly $B$ -orthodox semigroups, where $B$ denotes a band of idempotents. In particular, we investigate orthodox semigroups, by using orthodox groupoids. Weakly $B$ -orthodox semigroups are analogues of orthodox semigroups, where the relations $\widetilde{\mathcal {R}}_B$ and $\widetilde{\mathcal {L}}_B$ play the role that ${\mathcal {R}}$ and $\mathcal {L}$ take in the regular case. We show that the category of weakly $B$ -orthodox semigroups and admissible morphisms is equivalent to the category of weakly orthodox categories over bands and orthodox functors. The same class of weakly $B$ -orthodox semigroups was studied in an earlier article by Gould and the author using generalised categories. Our approach here is more akin to that of Nambooripad. The significant difference in strategy is that it is more convenient to consider categories equipped with pre-orders, rather than with partial orders.     

The Lattice of Varieties of Completely Regular Semigroups

Polák’s theorem on the structure of the (lattice of) varieties of completely regular semigroups provides an isomorphic copy of the interval $[{\cal S,CR}]$ of varieties which contain semilattices in terms of certain functions. We give a variant of this theorem for the lattice ${\cal L(CR)}$ of all varieties of completely regular semigroups in terms of pairs with componentwise inclusion. The first entry of these pairs is a band variety and the second consists of a ?₀-tuple of members of ${\cal K}_0$ . Here ${\cal K}_0$ is the set of varieties which satisfy ${\cal V}_K={\cal V}$ where ${\cal V}_K$ is the least element of the K-class containing ${\cal V}$ . We have based the proof of our theorem on Polák’s theorem for the sake of expediency and comparison. It utilizes a set of varieties which we term canonical. Several corollaries treat various special cases.     

Some Conditions for the Embeddability of Semigroups in Groups

We indicate two new classes of group embeddable semigroups, containing some earlier-known classes of semigroups with this property (Theorems 1 and 3), and construct examples of nonembeddable semigroups of class $K_{p,r}^2$ for ${p + r \geqslant 6}$ (Theorem 2), thus disproving an earlier conjecture.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.     

Commutative orders revisited

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order \(S\) , the square-cancellable elements \(\mathcal {S}(S)\) constitute a well-behaved separable subsemigroup. Indeed, \(\mathcal {S}(S)\) is also an order and has a maximum semigroup of quotients \(R\) , which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of \(\mathcal {S}(S)\) and families of ideals of \(S\) . We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of \(S\) are homomorphic images of the tensor product \(R\otimes _{\mathcal {S}(S)} S\) . By introducing the notions of generalised order and semigroup of generalised quotients, we show that if \(S\) has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on \(\mathcal {S}(S)\) that are restrictions of congruences on \(S\) .     

Relatively congruence-free regular semigroups

Yu, Wang, Wu and Ye call a semigroup \(S\) \(\tau \) -congruence-free, where \(\tau \) is an equivalence relation on \(S\) , if any congruence \(\rho \) on \(S\) is either disjoint from \(\tau \) or contains \(\tau \) . A congruence-free semigroup is then just an \(\omega \) -congruence-free semigroup, where \(\omega \) is the universal relation. They determined the completely regular semigroups that are \(\tau \) -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is \(\mathrel {\mathcal {J}}\) -congruence-free if and only if it is either a semilattice or has a single nontrivial \(\mathrel {\mathcal {J}}\) -class, \(J\) , say, and either \(J\) is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for \(\mathrel {\mathcal {L}}\) and \(\mathrel {\mathcal {R}}\) . In the case of \(\mathrel {\mathcal {H}}\) , only the completely semisimple case is fully resolved, again specializing to those of Yu et al.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Fibonacci-like growth of numerical semigroups of a given genus

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n _g is the number of numerical semigroups of genus g, we prove that $$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$ where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n _g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.     

A graph associated to a lattice

In this paper, we associate a simple graph to a lattice $\mathcal L $ , in which the vertex set is being the set of all elements of $\mathcal L $ , and two distinct vertices $x$ and $y$ are adjacent if $x\vee y\in S$ , when $S$ is a multiplicatively closed subset of $\mathcal L $ . We denote this graph by $\Gamma _S(\mathcal L )$ . We study some properties of $\Gamma _S(\mathcal L )$ . Moreover, we investigate the planarity of $\Gamma _S(\mathcal L )$ , whenever $S$ is a saturated multiplicatively closed subset of $\mathcal L $ .     

A note on residually small varieties of semigroups

It is proved for any varietyG of groups that if the subdirectly irreducible groups inG form a set, and if the subdirectly irreducible representation algebras of groups inG form a set, then every finite group inG is Abelian. The result is essential for the characterization of residually finite varieties of semigroups.     

Classes of semigroups modulo Green’s relation \mathcal{H}

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation \(\mathcal{H}\) , or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.     

Good congruences on abundant semigroups with PSQ-adequate transversals

In this paper, we will characterize good congruences on abundant semigroups with \(PSQ\) -adequate transversals by using the so called good congruence triple, which generalizes some results on abundant semigroups with multiplicative adequate transversals.     

Axioms for function semigroups with agreement quasi-order

The agreement quasi-order on pairs of (partial) transformations on a set X is defined as follows: ${(f, g) \preceq (h, k)}$ if whenever f, g are defined and agree, so do h, k. We axiomatize function semigroups and monoids equipped with this quasi-order, thereby providing a generalisation of first projection quasi-ordered ${\cap}$ -semigroups of functions. As an application, axiomatizations are obtained for groups and inverse semigroups of injective functions equipped with the quasi-order of fix-set inclusion. All axiomatizations are finite.     

E-local pseudovarieties

Generalizing a property of the pseudovariety of all aperiodic semigroups observed by Tilson, we call E -local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular $\mathcal{D}$ -classes. In this paper, we present several sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.     

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

On a sublattice of the lattice of congruences on a θ-regular semigroup

Let S be a regular semigroup. The lattice of all idempotent-separating congruences on S and the lattice of all group congruences on S are both modular sublattices of the full lattice of congruences on S. It is evident that the set theoretical union of these two sublattices, (S), is also a sublattice of the full lattice of congruences on S. It is natural to ask: Under what conditions is the sublattice (S) modular? In this paper we obtain a necessary and sufficient condition for the sublattice (S) to be modular when S is what we call a θ-regular semigroup. Bisimple ω-semigroups and simple regular ω-semigroups are θ-regular semigroups and so this paper extends the work of Munn [5] and Baird [1].