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Abstract: We define an algebraic structure similar to that of a semiring, but without some of the requirements. As it is somehow also similar to the structure of left brace, we call it an \({\mathscr {M}}\) -brace. We present a connection between Garside monoids and more generally lcm-monoids with this algebraic structure. An lcm-monoid M is a left-cancellative monoid such that 1 is the unique invertible element in M, and every pair of elements in M admit an lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids. PubDate: 2022-05-09

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Abstract: The aim of this paper is to study DRC-semigroups by “categorial approach” and give an Ehresmann–Schein–Nambooripad type theorem (ESN type theorem for short) for this class of semigroups. After exploring some basic properties of these semigroups, we introduce the notion of DRC-generalized categories by which DRC-semigroups are described. More specifically, we show that the category of DRC-semigroups together with (2,1,1)-homomorphisms is isomorphic to the category of DRC-generalized categories together with DRC-mappings. As a special case, an ESN type theorem is obtained for P-Ehresmann semigroups introduced by Jones. Our work may be regarded as extending the ESN type theorems for Ehresmann semigroups and some special classes of P-Ehresmann semigroups in the literature, and as a specific answer of a problem proposed by Stokes (Semigroup Forum 94:442–462, 2017). PubDate: 2022-05-09

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Abstract: As an abstraction of the conjugation on the multiplicative semigroup of quaternions, Garrão, Martins-Ferreira, Raposo, and Sobral (Semigroup Forum 100:806–836, 2020) introduced the notion of a conjugation semigroup, and studied the category of cancellative conjugation semigroups. In this note the conjugations of a group are shown to be in one-to-one correspondence with the endomorphisms of the group whose ranges are in the center. Moreover, cancellative conjugation semigroups are proved to be, up to isomorphism, the conjugation subsemigroups of conjugation groups. PubDate: 2022-05-06

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Abstract: Abstract We present some comments on Theorem 2.4 of ‘Semigroups of transformations whose restrictions belong to a given semigroup’, by Konieczny (Semigroup Forum 104:109–124, 2022). We identify that the statement of Theorem 2.4 is flawed by giving a counterexample and provide the correct statement. PubDate: 2022-04-28

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Abstract: Abstract In this paper we study mid-reversibility of subsemigroups acting on homogeneous spaces. The mid-reversor set of a subsemigroup is defined and it is described in terms of the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group and L a subgroup of G. Assume that S is a subsemigroup of G with nonempty interior. We characterize the mid-reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G. We show that the mid-reversibility of S in G/L is related to the reversibility of S in G/L. We also present sufficient conditions for S to generate G if S is mid-reversible in G/L. PubDate: 2022-04-19

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Abstract: Abstract An associative trioid is an algebra defined on a set with three binary associative operations. Associative trialgebras are linear analogs of associative trioids; they were introduced by Loday and Ronco in the context of algebraic topology. We prove that axioms of an associative trioid (trialgebra) are independent. PubDate: 2022-04-11

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Abstract: In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order \(\alpha \in (0,1]\) , and discuss the continuity of the respective solutions regarding the parameter \(\alpha \) . We also present a study about the continuity of the Mittag-Leffler families of operators (for \(\alpha \in (0,1]\) ), when they are induced by sectorial operators. PubDate: 2022-04-05

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Abstract: Abstract We investigate the sets of countable discrete semigroups that force recurrence, that is, the recurrent properties of a point along a subset of a countable semigroup action. We show that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken IP-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite IP-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence. PubDate: 2022-04-05

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Abstract: Abstract We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures. PubDate: 2022-04-01

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Abstract: Abstract In this paper, we study gapsets and we focus on obtaining information on how the maximum distance between two consecutive elements influences the behaviour of the set. In particular, we prove that the cardinality of the set of gapsets with genus g such that the maximum distance between two consecutive elements is \(\kappa \) is equal to the cardinality of the set of gapsets with genus \(g+1\) such that the maximum distance between two consecutive elements is \(\kappa +1\) , when \(2g \le 3\kappa \) . PubDate: 2022-04-01

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Abstract: Abstract We describe structural and quantitative properties of type-dependent sets in monoids with suitable analytic structure, including simple analytic monoids, introduced by Kaczorowski (Semigroup Forum 94:532–555, 2017. https://doi.org/10.1007/s00233-016-9778-9), and formations, as defined by Geroldinger and Halter-Koch (Non-unique factorizations, Chapman and Hall, Boca Raton, 2006. https://doi.org/10.1201/9781420003208). We propose the notions of rank and degree to measure the size of a type-dependent set in structural terms. We also consider various notions of regularity of type-dependent sets, related to the analytic properties of their zeta functions, and obtain results on the counting functions of these sets. PubDate: 2022-04-01

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Abstract: Abstract We provide a close analysis of the connections between pseudogroups, groupoids, and toposes. This analysis provides a topos perspective on both the localic germ groupoid of a pseudogroup defined by Resende, and the topological groupoid of a pseudogroup defined by Lawson and Lenz. In particular, we show how to analyze the topos of a pseudogroup using sheaf theory, leading to an examination of pseudogroup torsors. Consequently we obtain a concrete description of the category of points of an étendue. PubDate: 2022-03-10

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Abstract: Abstract A partial transformation \(\alpha \) on an n-element chain \(\mathbf{n} =\{1,\dots , n\}\) is called order-preserving if \(x \le y\) implies \(x\alpha \le y\alpha \) for all \(x,y\in \mathop {\mathrm {dom}}\nolimits (\alpha )\) and it is called extensive if \(x\le x\alpha \) for all \(x\in \mathop {\mathrm {dom}}\nolimits (\alpha )\) . The set of all partial order-preserving extensive transformations on \(\mathbf{n} \) forms a semiband \(\mathcal {POE}_n\) . In this paper, we compute the rank and the idempotent rank of the ideals of the monoid \(\mathcal {POE}_n\) . Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid \(\mathcal {POE}_n\) . Our work extends previous results found in the literature. PubDate: 2022-03-10

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Abstract: Abstract We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Apéry set, as well as bounds on the elements of the Apéry set. We also find bounds on the Frobenius number and genus, and the asymptotic behavior of the Frobenius number and genus. Finally, we find the embedding dimension of all such numerical semigroups. PubDate: 2022-03-07

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Abstract: Abstract A topologized semilattice X is called complete if each non-empty chain \(C\subset X\) has \(\inf {C}\) and \(\sup {C}\) that belong to the closure \(\overline{C}\) of the chain C in X. In this paper, we introduce various concepts of completeness of topologized semilattices in the context of operators that generalize the closure operator, and study their basic properties. In addition, examples of specific topologized semilattices are given, showing that these classes do not coincide with each other. Also in this paper, we prove theorems that allow us to generalize the available results on complete semilattices endowed with a topology. PubDate: 2022-03-07

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Abstract: Abstract For \(n\in {\mathbb {N}}\) , let \(\mathcal {OPE}_{n,k}\) be the monoid of all orientation-preserving k-extensive full transformations on \(\{1, \ldots , n\}\) ordered in the standard way. In this paper, it is shown that \(\mathcal {OPE}_{n,k}\) is an abundant semi-band, and a sufficient and necessary condition of an element in \(\mathcal {OPE}_{n,k}\) to be regular is obtained. Moreover, a minimal generating set and the minimum idempotent generating set of the monoid \(\mathcal {OPE}_{n,k}\) are determined, and so the rank and the idempotent rank of \(\mathcal {OPE}_{n,k}\) are established. PubDate: 2022-03-04

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Abstract: Abstract We characterise the respective semigroups of mappings that preserve, or that preserve or reverse orientation of a finite cycle, in terms of their actions on oriented triples and oriented quadruples. This leads to a proof that the latter semigroup coincides with the semigroup of all mappings that preserve intersections of chords on the corresponding circle. PubDate: 2022-02-19 DOI: 10.1007/s00233-022-10256-8

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