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Abstract: We observe that for each \(n\ge 2\) , the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of \(\mathscr {J}\) -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids. PubDate: 2022-08-01

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Abstract: A C-monoid is a suitably defined submonoid of a factorial monoid with finite reduced class semigroup. This monoid plays a key role in an arithmetical investigation of a large class of Mori domains. It is well understood that a C-monoid is Krull if and only if the reduced class semigroup coincides with the (v-)class group of a Krull monoid, and the arithmetic of a Krull monoid can be determined by the structure of its (v-)class group. The finiteness of the reduced class semigroup allows us to prove the similar arithmetical finiteness for a general C-monoid to results known in the Krull case. Recently, the algebraic structure of the reduced class semigroup has begun to be studied for a non-Krull C-monoid. Every Krull monoid is a root-closed weakly Krull Mori monoid, and under the mild conditions, a root-closed weakly Krull Mori monoid is a C-monoid. In this paper, we study the structure of a root-closed weakly Krull Mori monoid and of its class semigroup. PubDate: 2022-07-25

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Abstract: Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as \(a^{-1}b\) , where \(a, b \in S\) and \(a^{-1}\) is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be \(\mathscr {R}\) -related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of \({\mathscr {R}}\) and \({\mathscr {L}}\) from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the \(\mathscr {L}\) -classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight. PubDate: 2022-07-22

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Abstract: Let S be a semigroup. We shall consider the centres of the semigroup \((\beta \,S, \,\Box \,)\) and of the algebra \((M(\beta \,S), \,\Box \,)\) , where \(M(\beta \,S)\) is the bidual of the semigroup algebra \((\ell ^{\,1}(S),\,\star \,)\) , and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of \(S^*\) and of \(M(S^*)\) that are ‘determining for the left topological centre’ (DLTC sets) of \(\beta \,S\) and \(M(\beta \,S)\) . It is known that, when the semigroup S is cancellative, \(\ell ^{\,1}(S)\) is strongly Arens irregular and that there is a DLTC set consisting of two points of \(S^*\) . In contrast, there is little that has been published about the Arens regularity of \(\ell ^{\,1}(S)\) when S is not cancellative. Totally ordered, abelian semigroups, with the map \((s,t)\rightarrow s \wedge t\) as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of \(\beta \,S\) and of \(M(\beta \,S)\) for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for \(\beta S\) or for \(M(\beta S)\) . There was no previously-known example of an abelian semigroup S for which \(\beta \,S\) or \(M(\beta S)\) did not have a finite DTC set. PubDate: 2022-07-18

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Abstract: The value semigroup \(\Gamma \) and the value set \(\Lambda \) of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Given a plane branch \({\mathcal {C}}\) with value semigroup \(\Gamma \) there are a finitely number of distinct possible sets \(\Lambda \) according to the analytic class of \({\mathcal {C}}\) . In this work we show that the value set of 1-forms \(\Lambda \) determines the value semigroup \(\Gamma \) and we present an effective method to recover \(\Gamma \) by \(\Lambda \) . In particular, this allows us to decide if a subset of \({\mathbb {N}}\) is a value set of 1-forms for a plane branch. PubDate: 2022-07-11

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Abstract: This article explores a generalisation of the theory of formations of groups. Taking formations of groups as the starting point, formations of inverse semigroups are defined, as well as the wider classes of i-formations (i standing for idempotent-separating) and some classes of the kind named f-formations (f standing for fundamental). The relation between the nature of a class of groups and that of certain classes of inverse semigroups with associated groups in the first is discussed. The product of formations is considered, and a product like the Gaschütz’s product known for groups is presented for f-formations. PubDate: 2022-07-07

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Abstract: We define a subsemigroup \(S_n\) of the rook monoid \(R_n\) and investigate its properties. To do this, we represent the nonzero elements of \(S_n\) (which are \(n\times n\) matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that \(S_n\) consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, compute nilpotency indexes, determine Green’s relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth root of a nonzero element to exist in \(S_n\) , show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of \(S_n\) (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results. PubDate: 2022-07-07

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Abstract: In this article we fully describe the domain of the infinitesimal generator of the optimal state semigroup which arises in the theory of the linear-quadratic problem for a specific class of boundary control systems. This represents an improvement over earlier work of the authors, joint with I. Lasiecka, where a set inclusion was established, but not an equality. The novel part of the proof of this result develops through appropriate asymptotic estimates that take advantage of the regularity analysis carried out in the study of the optimization problem, while the powers of positive operators and interpolation are still key tools. We also attest to the validity of an assumed relation between two significant parameters in the case of distinct systems of coupled hyperbolic–parabolic partial differential equations which are pertinent to the underlying framework. PubDate: 2022-06-28

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Abstract: We study the fish population model integrating age and weight structures, introduced in E. Sánchez, M. L. Hbid, R. Bravo de la Parra. (J. Evol. Equ. 14:603–616, 2014). We reformulate the model in the nonautonomous past setting, and then as a boundary perturbation problem with unbounded operators in the boundary. Using semigroup theory of linear operators in Banach spaces, and via the theory of time-invariant regular system with feedback, we prove the existence and uniqueness of a classical solution with a form of a variation of parameters formula. We give an explicit criterion of the uniform exponential-stability by determining the characteristic equation of the model. We also prove the asynchronous exponential growth behaviour for the system unconditionally with the system parameters, and we give the associated projection in an explicit form. The results in this paper represent an improvement of the ones given in loc. cit. for the well-posedness and asymptotic behaviour. PubDate: 2022-06-23

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Abstract: We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to \(\mathbb {Z}^2\) . For this purpose, we solve this task for three special types of such monoids in Theorems 5 and 6, and then transfer these results to the general case. It turns out that all atoms can be obtained from the (regular) continued fraction expansion of the slopes of the bounding rays of the cone, which is spanned by the monoid. PubDate: 2022-06-21

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Abstract: In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular). PubDate: 2022-06-20

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Abstract: We prove that for every archimedean positively ordered semigroup \(S\) with a maximal element, if \(S\) satisfies one minor condition, then \(S\) is isomorphic to a mixture of \(P_1\) and \(P_1^*\) in the notation of [2]. We also discuss a variation of metrizable spaces where we use a positively ordered semigroup to describe distances. PubDate: 2022-06-16

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Abstract: Let H be a cancellative commutative monoid, let \(\mathcal {A}(H)\) be the set of atoms of H and let \(\widetilde{H}\) be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counterexamples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if \(\widetilde{H}\) is a DVM, then H is transfer Krull if and only if \(H\subseteq \widetilde{H}\) is inert. Moreover, we prove that if \(\widetilde{H}\) is factorial, then H is transfer Krull if and only if \(\mathcal {A}(\widetilde{H})=\{u\varepsilon \mid u\in \mathcal {A}(H),\varepsilon \in \widetilde{H}^{\times }\}\) . We also show that if \(\widetilde{H}\) is half-factorial, then H is transfer Krull if and only if \(\mathcal {A}(H)\subseteq \mathcal {A}(\widetilde{H})\) . Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids. PubDate: 2022-06-14

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Abstract: Let \(S=\left\langle s_1,\ldots ,s_n\right\rangle \) be a numerical semigroup generated by the relatively prime positive integers \(s_1,\ldots ,s_n\) . Let \(k\geqslant 2\) be an integer. In this paper, we consider the following k-power variant of the Frobenius number of S defined as $$\begin{aligned} {}^{k\!}r\!\left( S\right) := \text { the largest } k \text {-power integer not belonging to } S. \end{aligned}$$ In this paper, we investigate the case \(k=2\) . We give an upper bound for \({}^{2\!}r\!\left( S_A\right) \) for an infinite family of semigroups \(S_A\) generated by arithmetic progressions. The latter turns out to be the exact value of \({}^{2\!}r\!\left( \left\langle s_1,s_2\right\rangle \right) \) under certain conditions. We present an exact formula for \({}^{2\!}r\!\left( \left\langle s_1,s_1+d \right\rangle \right) \) when \(d=3,4\) and 5, study \({}^{2\!}r\!\left( \left\langle s_1,s_1+1 \right\rangle \right) \) and \({}^{2\!}r\!\left( \left\langle s_1,s_1+2 \right\rangle \right) \) and put forward two relevant conjectures. We finally discuss some related questions. PubDate: 2022-06-13

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Abstract: In the literature, parametrizations are given for Arf numerical semigroups with small multiplicity and arbitrary conductor. From those parametrizations, formulas are obtained for the number of such Arf numerical semigroups. These formulas show that the number of Arf numerical semigroups with multiplicity 3, 5 or 7 and arbitrary conductor depends only on the congruence class of the conductor modulo the multiplicity. In a recent work with S. İlhan and M. Süer, observing that the same is true for Arf numerical semigroups with multiplicity 11 and 13, we asked if that was true for Arf numerical semigroups with arbitrary prime multiplicity. In the present work this question is answered affirmatively. PubDate: 2022-06-10

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Abstract: We investigate a semigroup construction generalising the two-sided wreath product. We develop the foundations of this construction and show that for groups it is isomorphic to the usual wreath product. We also show that it gives a slightly finer version of the decomposition in the Krohn–Rhodes Theorem, in which the three-element flip-flop monoid is replaced by the two-element semilattice. PubDate: 2022-06-06

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Abstract: We give an overview of a number of Schreier-type extensions of monoids and discuss the relation between them. We begin by discussing the characterisations of split extensions of groups, extensions of groups with abelian kernel and finally non-abelian group extensions. We see how these characterisations may be immediately lifted to Schreier split extensions, special Schreier extensions and Schreier extensions respectively. Finally, we look at weakenings of these Schreier extensions and provide a unified account of their characterisation in terms of relaxed actions. PubDate: 2022-06-01

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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: 2022-06-01

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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 DOI: 10.1007/s00233-022-10278-2