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Abstract: Abstract We study the separately continuous actions of semitopological monoids on pseudocompact spaces. The main aim of this paper is to generalize Lawson’s results to some class of pseudocompact spaces. Also, we introduce the concept of a weak \(q_D\) -space and prove that a pseudocompact space and a weak \(q_D\) -space form a Grothendieck pair. As an application of the main result, we investigate the continuity of multiplication and taking inverses in subgroups of semitopological semigroups. In particular, we get that if \((S, \bullet )\) is a Tychonoff pseudocompact semitopological monoid with a quasicontinuous multiplication \(\bullet \) and G is a subgroup of S, then G is a topological group. Also, we study the continuity of operations in semitopological semilattices. PubDate: 2023-12-04

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Abstract: Abstract Jackson and Lee proved that certain six-element monoid generates a hereditarily finitely based variety \({{\mathbb {E}}}^1\) whose lattice of subvarieties contains an infinite ascending chain. We identify syntactic monoids which generate finitely generated subvarieties of \({{\mathbb {E}}}^1\) and show that one of these finite monoids together with certain seven-element monoid generates a new limit variety. PubDate: 2023-12-04

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Abstract: Abstract We study the structure of strongly 2-chained semigroups, which can be defined alternatively as semigroups whose regular elements are completely regular. The main result is a semilattice decomposition of these semigroups in terms of ideal extensions of completely simple semigroups by poor semigroups and idempotent-free semigroups. PubDate: 2023-11-08

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Abstract: Abstract Taking formations of groups and of inverse semigroups as the starting point, formations of orthodox semigroups are defined, as well as the wider class of i-formations (i standing for idempotent-separating). The relation between the nature of a class of inverse semigroups \(\mathscr {F}\) [of groups \(\mathscr {G}\) ] and that of certain classes of orthodox semigroups with associated inverse semigroups in \(\mathscr {F}\) [groups in \(\mathscr {G}\) ] is discussed. The product of formations of orthodox semigroups, in particular of R-unipotent semigroups, is considered, and a product like the Gaschütz product known for groups is presented for i-formations. The paper concludes with a list of questions. PubDate: 2023-11-08

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Abstract: Abstract Given a continuous binary operation \(\star \) on the positive real numbers \({\textbf{R}}_+\) with ordinary topology, we consider the homeomorphism group \(Homeo_{\star }({\textbf{R}}_+)\) consisting of all homeomorphisms on \({\textbf{R}}_+\) which preserve \(\star \) . We show that if \(\star \) is any continuous cancellative semigroup operation on \({\textbf{R}}_+\) , then \(Homeo_{\star }({\textbf{R}}_+)\) is isomorphic to either one of the three well-known groups. Furthermore, we reveal homeomorphism groups defined by some continuous noncancellative semigroup operations on \({\textbf{R}}_+\) . PubDate: 2023-11-08

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Abstract: Abstract We consider the inverse submonoids \(\mathcal {OPDI}_n\) , \(\mathcal {MDI}_n\) and \(\mathcal {ODI}_n\) of the dihedral inverse monoid \(\mathcal{D}\mathcal{I}_n\) of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids. PubDate: 2023-10-31

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Abstract: Abstract A recent paper studied an inverse submonoid \(M_{n}\) of the rook monoid, by representing the nonzero elements of \(M_{n}\) via certain triplets belonging to \({\mathbb {Z}}^3\) . In this note, we allow the triplets to belong to \({\mathbb {R}}^3\) . We thus study a new inverse monoid \(\overline{M}_{n}\) , which is a supermonoid of \(M_{n}\) . We point out similarities and find essential differences. We show that \(\overline{M}_{n}\) is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly \({E}^{*}\) -unitary inverse monoid. PubDate: 2023-10-19

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Abstract: Abstract In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid G being effective. One of these conditions is that G satisfies the C*-algebraic local bisection hypothesis; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras. PubDate: 2023-10-17

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Abstract: Abstract In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup \(F_{1}\) weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of \(F_{1}\) . We define \(F_{1}\) using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of \(F_{1}\) . In particular, we show that the ‘free regular semigroup \({\textrm{FI}}_2\) weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of \(F_{1}\) weakly generated by \(\{xx',x'x\}\) . PubDate: 2023-10-11

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Abstract: By proving existence, regularity and uniqueness of solutions to Cauchy problems governed by abstract unbounded operators with finite pseudo-spectral bounds as an alternative and a serious enhancement of results by Melnikova and Filinkov, we establish a new generation criterion theorem for \(C_0\) -semigroups in general Banach spaces. A generalization of Kaiser–Weis–Batty’s perturbation generation theorem in reflexive Banach spaces is therefore derived. We apply our last theoretical result to a singular transport model in \(L^p\) -spaces, \(p\in ]1,+\infty [\) , where the streaming (unperturbed) semigroup cannot be explicit. PubDate: 2023-10-10

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Abstract: Abstract For a positive integer n, consider the submonoid \(S_n\) of \({\mathbb {N}}\) generated by the sequence of positive integers \({\textbf{s}}_j=ca^{n+j}-d\) , \(j\in {\mathbb {N}}\) , where a, c, and d are integers, \(a\ge 2\) , and c is positive. We unify ideas and results from previous works on specific cases for a, c, and d, and we prove some conjectures that arise in the general case. Under fairly general conditions on a, c and d, we characterize the embedding dimension of \(S_n\) in terms of a, c, and d, and provide a general characterization of the Apéry set \(\textrm{Ap}(S_n, {\textbf{s}}_0)\) . We then derive formulas for the Frobenius number and genus of the numerical semigroup \((1/\lambda )S_n\) , where \(\lambda =\gcd (S_n)\) . We also estimate the set of pseudo-Frobenius numbers of \((1/\lambda )S_n\) and prove an inequality involving the type of \((1/\lambda )S_n\) , which implies that Wilf’s conjecture is true for these numerical semigroups. PubDate: 2023-10-02

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Abstract: Abstract The concept of a weak factorization system for modules has found an application in one of the proofs of the celebrated flat cover conjecture for modules over a ring. We examine this notion in the context of S-acts over a monoid S. Bailey and Renshaw constructed a weak factorization system related to the existence of precovers of S-acts and showed that the class of flat right S-acts satisfies a related property they call saturated. In this paper we provide a factorization system, namely a weak factorization system with the unique mapping property, which is related to the existence of covers of S-acts with the unique mapping property. Moreover, we show that in this case, a class of monomorphisms associated with principally weakly flat right S-acts is saturated. PubDate: 2023-10-01

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Abstract: Abstract The semilattice congruence \(\mathscr {N}\) , which identifies two elements if they generate the same principal filter, plays a significant role in studying the decomposition of semigroups. We investigate the remarkable properties of the semilattice congruence \(\mathscr {N}\) on n-ary semigroups, where \(n\ge 3\) , and use these properties to describe the structure of n-ary semigroups which are decomposable into i-simple and regular components for all \(1<i<n\) . In particular, we show that each n-ary semigroup which is both regular and intra-regular is decomposable into a semilattice of i-simple and regular n-ary semigroups, and the reverse assertion also holds. Moreover, we prove that an n-ary semigroup is intra-regular if and only if it is a semilattice of i-simple n-ary semigroups. Finally, we discuss the connection between semilattices of i-simple (and regular) n-ary semigroups and semilattices of simple (and regular) n-ary semigroups. PubDate: 2023-10-01

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Abstract: Abstract Simple quantales were introduced by Paseka and are closely related to \(C^{*}\) -algebras. The sublattice \({\mathcal {S}}(L)\) of \(L^{L}\) is a prototypical example of a simple quantale under the multiplication defined by composition \(\circ \) , where \({\mathcal {S}}(L)\) denotes the set of sup-preserving endomaps of a nontrivial complete lattice L and \(L^{L}\) denotes the set of order-preserving endomaps of L. However, \((L^{L},\circ )\) is not a quantale. We define a new composition operation \(\cdot \) on the set of order-preserving maps so that \((L^{L},\cdot )\) forms a quantale, where L is a completely distributive lattice. Moreover, \(({\mathcal {S}}(L),\circ )\) is quantale isomorphic to a quotient of \((L^{L},\cdot )\) . In addition, if there exists a nontrivial surjective quantale homomorphism from \(L^{L}\) to a unital quantale M, then M is a simple quantale. PubDate: 2023-09-25

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Abstract: Abstract We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green’s \({\mathscr {R}}\) -, \({\mathscr {L}}\) - and \({\mathscr {H}}\) -classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green’s \({\mathscr {R}}\) - and \({\mathscr {L}}\) -classes also represent extensions of Drazin’s (b, c)-inverses and Mary’s inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper. PubDate: 2023-09-20

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Abstract: Abstract Under suitable conditions on the unknown functions, solutions of Levi-Civita functional equations on commutative monoids with no prime ideals are exponential polynomials. This is not generally the case on commutative monoids with prime ideals. Here we describe the solutions of Levi-Civita equations on commutative monoids in which every prime ideal is tractable. Monoids with this property include those which are regular or generated by their squares, as well as many others. Our results also give the continuous solutions on topological commutative monoids with tractable prime ideals. PubDate: 2023-09-13 DOI: 10.1007/s00233-023-10381-y

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Abstract: Abstract We consider a topological dynamical system under the action of a finitely generated free semigroup. Using the Carathéodory structure, we define BS-dimension on an arbitrary subset and obtain a Bowen’s equation which illustrates the relation of BS-dimension to Pesin–Pitskel topological pressure defined in Zhong and Chen (Acta Math Sinica Engl Ser 37(9):1401–1414, 2021). Then, an analogue of Billingsley’s theorem is obtained, by which we get a variational principle between BS-dimension and measure-theoretical lower BS-dimension. PubDate: 2023-08-30 DOI: 10.1007/s00233-023-10379-6