This paper investigates the exponential stability of contraction semigroups on Banach spaces under Miyadera–Voigt perturbations. Necessary and sufficient conditions for exponential stability are established in terms of integral estimates involving the unperturbed semigroup and the perturbation operator. Several examples are provided to illustrate the applicability of the theoretical results.
We prove that, for each fixed genus, the proportion of semigroups of that genus belonging to infinite chains in the semigroup tree approaches 0 as the genus grows to infinity. This means that most numerical semigroups have a finite number of descendants in the semigroup tree. This problem has been open since 2009.
This article develops a theory of formations of completely simple semigroups, extending earlier work on inverse semigroups and orthodox semigroups. Firstly, we discuss the relation between the algebraic properties of a class of groups 𝒢 and that of certain classes of completely simple semigroups with associated groups in 𝒢 . Secondly, we prove that i-varieties of completely simple semigroups are closed under the product, while f-formations of completely simple semigroups are closed under the Gaschütz product, thereby extending corresponding results from group to semigroup theory.
Let _n [respectively, _n ] be the plactic-like monoid obtained by factoring the free monoid over a finite ordered alphabet 𝒜_n by the meet of the taiga [respectively, stalactic] congruence and its dual. In this paper, we prove that _n can only be equipped with two different involutions and give a faithful representation for _n under each involution. Further, we prove that _n and _n are equationally equivalent under the certain involution.
This paper will use the kernel-trace approach of congruences to investigate one-sided congruences on a class of inverse semigroups. The gauge inverse submonoids, as kernels of congruences, play a leading role in this investigation. A congruence whose kernel is the gauge inverse submonoid and whose trace is the universal relation is a group congruence. The quotient group is isomorphic to the additive group of integers. This group congruence generates a Möbius category such that the corresponding breaking process preserves the Möbius function.
We investigate the Frobenius problem for a specific family of numerical semigroups with embedding dimension four. Specifically, the members of this family are balanced and have common quotient 2. We show that numerical semigroups in this family come in two distinct varieties and we provide formulas for the Frobenius number and genus for both. This research is related to previous work on balanced numerical semigroups with common quotient 1 and provides insights into finding the Frobenius number of a balanced numerical semigroup for any value of the common quotient.
We investigate higher derivations (Hasse–Schmidt derivations) on semirings, establishing a comprehensive framework for these operators in the context of matrix semirings over commutative additively idempotent semirings. While derivations on rings and algebras have undergone extensive development, their higher-order analogues in semiring theory remain largely unexplored, particularly regarding inheritance properties from base semirings to matrix extensions. Building upon Vladeva’s foundational work on usual derivations, we prove that every higher derivation on the matrix semiring M_n(S) is hereditary whenever S is commutative and additively idempotent; specifically, each higher derivation on M_n(S) acts entrywise via a higher derivation on the base semiring S. This result extends the classical hereditary theorem to the Hasse–Schmidt setting and reveals that the idempotent structure fundamentally simplifies the iterativity condition to d_i d_j = d_i+j . We further characterize multiplicative higher derivations through generating sequences satisfying idempotent convolution identities, establish that these sequences constitute a commutative monoid under Cauchy convolution, and provide the Hasse–Schmidt encoding into truncated polynomial semirings. Infinite-rank analogues are discussed subject to appropriate topological hypotheses. Our analysis encompasses both finite-rank and infinite-rank derivations, with explicit constructions over tropical and Boolean semirings illustrating the general theory.
The monoid of all binary operations was first introduced by H. S. Kim and J. Neggers in 2008. Since then, different aspects and applications of this monoid were studied, while several questions about its semigroup-theoretic properties remain unanswered. We employ a transformation semigroup perspective to fully characterize principal left and right ideals, idempotent and regular elements of this monoid, as well as provide precise combinatorial enumerations of them. This approach gives a general framework for most of the existing results on ideals in the magma monoid. We also answer several open questions posed in the 2023 PhD dissertation of A. Rafieipour. Finally, we correct an error regarding the description of the center of the magma monoid from the 2011 paper of H. F. Fayomi.
Abstract We construct a free 2-generated band $$\{e,p,e+p,p+e,e+p+e,p+e+p\}$$ { e , p , e + p , p + e , e + p + e , p + e + p } in $$\beta \mathbb {N}$$ β N . As a consequence we obtain a new Ramsey theoretic result.
We introduce Algebraic Magnetism, a theory of attractors associated to magnets within the framework of actions of diagonalizable monoid schemes on algebraic spaces. For a diagonalizable monoid scheme A(M)_S acting on an algebraic space X, we introduce for any submonoid N of M an attractor space X^N . We then investigate and study various aspects of attractors associated to monoids. This leads to the notion of pure magnets, which encode the combinatorial structure underlying attractor phenomena in terms of semigroups. In the affine case, we give an explicit description of attractors in terms of graded algebra, yielding representability as closed subspaces in this case. Using descent methods, technical preliminary results, and fixed-point-reflecting atlases, we prove representability for general algebraic spaces. We establish compatibility of attractors with fiber products, base change, faces of monoids, intersections of submonoids, subgroups, dilatations, and many other operations. We study pure magnets and prove, and conjecture, finiteness results under suitable hypotheses. In this way, Algebraic Magnetism provides invariants to study diagonalizable actions in algebraic geometry.
Let E(X, f) be the Ellis semigroup of a discrete discrete dynamical system (X, f) where X is a compact countable metric space. Set P(X,f): = { |𝒪_f(x)|: x is a periodic point of X } , for an eventually periodic point x∈ X let l_x ∈ℕ be its waiting time, and set L(X,f):= { l ∈ℕ: ∃ x ∈ X(x is eventually periodic and l = l_x) } . We give two necessary and sufficient statements equivalent to the assertion “E(X, f) is a compactification of ℕ with the discrete topology”. And we also prove the following assertions: If each x∈ X has a finite orbit and L(X, f) and P(X, f) are finite, then |E(X,f)| ≤∏ _s∈ P(X,f){0,… , s-1} + max L(X,f). Besides, if all the periods are relative prime numbers, then |E(X,f)|= ∏ _s∈ P(X,f){0,… , s-1}+max L(X,f). Two necessary conditions on a discrete dynamical system are given in order that its Ellis semigroup be countable. We also include several examples that exemplify the diversity of dynamical properties in relation to the cardinality of the Ellis semigroup.
We study a one-dimensional wave equation posed on a two-edge network with constant wave speeds on each edge, coupled at a transmission point and equipped with boundary feedback. A set-valued boundary damping law is imposed at one endpoint and is described by a subset Σ⊂ℝ^2 . We aim at determining conditions on Σ ensuring existence and uniqueness as well as strong and exponential stability of solutions.
Wilf’s conjecture gives the relationship between the embedding dimension, the Frobenius number, and the genus of a numerical semigroup. Consider a numerical semigroup S = ⟨ a_1,a_2,… ,a_d ⟩ minimally generated by d coprime positive integers, where a_1 denotes the multiplicity of S. In 2015, Moscariello and Sammartano [9] established the validity of Wilf’s inequality for all those numerical semigroups where a_1 is sufficiently large and all the prime divisors of a_1 are greater than or equal to ρ , for every fixed value of ρ = ⌈a_1/d⌉ . In this article, we relax the arithmetic constraint on the prime divisors of a_1 by replacing it with the more natural condition (a_1,a_2) = 1, thereby significantly enlarging the family of numerical semigroups known to satisfy Wilf’s Conjecture. Moreover, under this hypothesis, we sharpen the bound on a_1 appearing in the work of Moscariello and Sammartano. As a consequence, our results not only extend their theorem to a wider class of numerical semigroups but also demonstrate that Wilf’s inequality holds under strictly improved numerical bounds.
Given a unital topological semigroup S and a unital semitopological semigroup T, whenever there is a morphism ρ : T→𝒮urj(S) , the set of morphisms of S onto itself, we consider the semitopological semigroup S⋊ _ρ T (the so called semidirect product of S and T with respect to ρ ) and give sufficient conditions that ensure the existence of a left invariant mean on the associated function spaces: AP(S⋊ _ρ T), WAP(S⋊ _ρ T) , LUC(S⋊ _ρ T) , and WLUC(S⋊ _ρ T ) of the semidirect product S⋊ _ρ T .
