∗-congruences on abundant semigroups with SQ-adequate transversals

We give ?-congruences on an abundant semigroup with an SQ-adequate transversal S ^° by the ?-congruence triple abstractly which consists of congruences on the structure component parts L, T and R. We prove that the set of all ?-congruences on this kind of semigroups is a complete lattice.     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

Proper restriction semigroups – semidirect products and W-products

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product ${T\ltimes {\mathcal {Y}}}$ of a semilattice ${\mathcal {Y}}$ by a monoid T, where the action of T on  ${\mathcal {Y}}$ is injective with images of the action being order ideals of  ${\mathcal {Y}}$ . Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form ${{\mathcal {Y}}\rtimes T}$ .     

Two classes of finite semigroups and monoids involving Lucas numbers

The class of finitely presented groups is an extension of the class of triangle groups studied recently. These groups are finite and their orders depend on the Lucas numbers. In this paper, by considering the three presentations

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

Directed graphs and combinatorial properties of completely regular semigroups

In this paper, we investigate the divisibility graphs and power graphs of completely regular semigroups. We give the structures of these two kinds of graphs and describe a combinatorial property of completely regular semigroups defined in terms of divisibility graphs and power graphs, respectively.     

Continuity and equicontinuity of semigroups on norming dual pairs

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C _b (E) which is continuous with respect to the strict topology β ₀ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β ₀-continuous semigroups on C _b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β ₀-continuous.     

Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics

Cauchy problems for a second order linear differential operator equation

Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups

A suitable notion of hypercontractivity for a nonlinear semigroup {T _t } is shown to imply Nash-type inequalities for its generator H, provided a subhomogeneity property holds for the energy functional (u,Hu). We use this fact to prove that, for semigroups generated by operators of p-Laplacian-type, hypercontractivity implies ultracontractivity. Then we introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is f(x)=x ^α , and write an explicit formula for the associated generator. It is shown that hypercontractivity still holds for the subordinated semigroup and, hence, that Nash-type inequalities hold as well for the subordinated generator.     

Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups

We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” ∂(a)=m satisfies

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

D-saturated property of the Cayley graphs of semigroups

Let D be a finite graph. A semigroup S is said to be Cayley D-saturated with respect to a subset T of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S,T) isomorphic to D with all vertices in V. The purpose of this paper is to characterize the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S. In particular, the Cayley D-saturated property of a semigroup S with respect to any subsemigroup T is characterized.     

Numerical semigroups: Apéry sets and Hilbert series

Let a ₁,…,a _n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a ₁,…,a _n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. Dedicated to the memory of Ernst S. Selmer (1920–2006), whose calculations revealed the “Selmer group”.     

Bruck–Reilly extensions of direct products of monoids and completely (0-)simple semigroups

We say that a class of monoids satisfies the property ℘ if every monoid in that class that admits a finitely presented Bruck–Reilly extension is finitely generated. We show that completely (0-)simple semigroups satisfy ℘, and that the direct product of two monoids in a class that satisfy ℘ also satisfies ℘ subject to a certain condition on the endomorphisms of the direct product. As a consequence of this result we obtain a new class of bands and a new class of completely regular semigroups that satisfy property ℘.     

On some invariants in numerical semigroups and estimations of the order bound

Let S={s _i } _i∈??? be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i ?s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family $\{\mathcal{C}_{i}\}_{i\in\mathbb{N}}Let S={s _i } _i∈ℕ⊆ℕ be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i −s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family {C_i}_{i ? \mathbbN}\{\mathcal{C}_{i}\}_{i\in\mathbb{N}} of one-point algebraic-geometric codes, a good bound for the minimum distance of the code C_i\mathcal{C}_{i} is the Feng and Rao order bound d _ORD (C _i ). It is well-known that there exists an integer m such that d _ORD (C _i )=ν(s _i+1) for each i≥m. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.     

Congruences on abundant semigroups associated with Green’s *-relations

Good congruences are defined as congruences preserving \(\mathcal L^*\)-classes and \(\mathcal R^*\)-classes. In this paper the good congruences on abundant semigroups generated by Green’s \(*\)-relations are investigated. The related results for regular semigroups are enriched and extended.     

Identities determining varieties of semigroups with completely regular power

We provide a transparent syntactic algorithm to decide whether an identity defines a variety of semigroups with completely regular power.