Semigroup Forum最新文献

Abstract

We study some nil-ai-semiring varieties. We establish a model for the free object in the variety (textbf{FC}) generated by all commutative flat semirings. Also, we provide two sufficient conditions under which a finite ai-semiring is nonfinitely based. As a consequence, we show that the power semiring (P_{scriptstyle {dot{S}}_{c}(W)}) of the finite nil-semigroup ({dot{S}}_{c}(W)) is nonfinitely based, where W is a finite set of words in the free commutative semigroup (X_{c}^{+}) over an alphabet X, whenever the maximum of lengths of words in W is (kge 3) and W does not contain the kth power of a letter. This partially answers a problem raised by Jackson et al. (J Algebr 611: 211–245, 2022).

We study the atomic generalized numerical semigroups (GNSs), which naturally extend the concept of atomic numerical semigroups. We introduce the notion of corner special gap and we characterize the class of atomic GNS in terms of the cardinality of the set of corner special gaps and also in terms of a maximal property. Using this maximal property we present some properties concerning irreducibility of Frobenius GNSs. In particular, we provide sufficient conditions for certain Frobenius GNSs to be atom non-irreducible. Furthermore, we given necessary and sufficient conditions so that the maximal elements of a set of Frobenius GNSs with two fixed gaps to be all irreducible or not.

We revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood–Paley–Stein theory for symmetric diffusion semigroups.

We generalize the result of Reid (J Lond Math Soc 13:454–458, 1976), namely, we prove that a curve of genus (geqq g^2+4g+6) having a double cover of a hyperelliptic curve of genus (ggeqq 2) does not lie as a non-singular curve on any K3 surface. Applying this result we construct non-K3 Weierstrass numerical semigroups. A numerical semigroup H is said to be Weierstrass if there exists a pointed non-singular curve (C, P) such that H consists of non-negative integers which are the pole orders at P of a rational function on C having a pole only at P. We call the numerical semigroup K3 if we can take the curve C as a curve on some K3 surface. A non-K3 numerical semigroup means that it cannot be attained by a pointed non-singular curve on any K3 surface. We also give infinite sequences of non-K3 Weierstrass numerical semigroups.

A class of Möbius monoids leads us to Möbius categories of (Q_{3})-type via a particular breaking process, where (Q_{3}) is a quiver with three arrows (atoms). In this paper we show that a quasi-commutativity regarding composable atoms uniquely determines (via a certain local congruence) a half-factorial broken Möbius category of (Q_{3})-type as a quotient category of the path category of (Q_{3}). Some examples shed light on the development of the topic under discussion. On the other hand, the Möbius breaking process can be extended for inverse semigroups as well. The Leech inverse semigroups of the two broken Möbius categories (the path category of (Q_{3}) and its quotient category) are split inverse semigroups in the sense that both are unions of two proper inverse subsemigroups, that is, both are with covering numbers 2. The connections on the two planes (broken Möbius categories and split inverse semigroups) are made on one hand by a local congruence (varrho ^{+}) of the path category of (Q_{3}), and on the other hand by a normal inverse subsemigroup (G^{+}) namely a gauge inverse subsemigroup. This gauge inverse semigroup is a full inverse subsemigroup of the almost Cartesian product (Btimes _{0}B_{{mathbb {N}}}) of the bicyciclic semigroup B and the Brandt semigroup (B_{{mathbb {N}}}). Special properties of this almost Cartesian product are examined by comparison with the well known polycyclic monoid.

We answer two orderability questions about the prefix expansion semigroup Pr(G) of an inverse semigroup G. We show that if G is a left ordered inverse semigroup, then Pr(G) is a left ordered inverse semigroup if and only if it is an ordered inverse semigroup, if and only if G is a semilattice. We also prove that when G and Pr(G) are left ordered, Pr(G) is proper if and only if G is proper. Positivity of the canonical map from G into Pr(G) is also proved. At the end we correct an existing result in the literature by showing that for two arbitrary inverse semigroups G and H the map Pr((pi )): Pr(G) (longrightarrow ) Pr(H) induced by the partial homomorphism (pi ): G (longrightarrow ) H is not necessarily a homomorphism, but is a partial homomorphism.

Given two operations (*) and (circ ) on a set S, an operation (star ) on S is said to be a collaboration between (*) and (circ ) if for all (a,b in S), (a star b) (in {a *b, acirc b }). Another term for collaborations is two-option operations. We are interested in learning what associative collaborations of two given operations (*) and (circ ) there may be. We do not require that (*) and (circ ) themselves be associative. For this project, as an initial experiment, we consider Plus-Minus operations (i.e. collaborations between addition and subtraction on an abelian group) and Plus-Times operations (i.e. collaborations between the addition and multiplication operations on a semiring.) Our study of Plus-Minus operations focuses on the additive integers but extends to ordered groups. For Plus Times operations, we make some headway in the case of the semiring of natural numbers. We produce an exhaustive list of associative collaborations between the usual addition and multiplication on the natural numbers ({mathbb {N}}). The Plus-Times operations we found are all examples of a type of construction which we define here and that we call augmentations by multidentities. An augmentation by multidentities combines two separate magmas A and B to create another, A(B), having (A sqcup B) as underlying set, and in such a way that the elements of B act as identities over those of A. Hence, B consists of a sort of multiple identities (explaining the moniker multidentities.) When A and B are both semigroups then so is A(B). Understanding the connection between certain collaborations and augmentation by multidenties removes, in several cases, the need for cumbersome computations to verify associativity. A final section discusses connections between group collaborations and skew braces.

Given a number field K with at least one real embedding, we generalize the notion of the classical Frobenius problem to the ring of integers ({mathfrak {O}}_K) of K by describing certain Frobenius semigroups, (textrm{Frob}(alpha _1,ldots ,alpha _n)), for appropriate elements (alpha _1,ldots ,alpha _nin {mathfrak {O}}_K). We construct a partial ordering on (textrm{Frob}(alpha _1,ldots ,alpha _n)), and show that this set is completely described by the maximal elements with respect to this ordering. We also show that (textrm{Frob}(alpha _1,ldots ,alpha _n)) will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as n is fixed and (alpha _1,ldots ,alpha _nin {mathfrak {O}}_K) vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.

If the Krull dimension of the semigroup ring is greater than one, then affine semigroups of maximal projective dimension ((textrm{MPD})) are not Cohen–Macaulay, but they may be Buchsbaum. We give a necessary and sufficient condition for simplicial (textrm{MPD})-semigroups to be Buchsbaum in terms of pseudo-Frobenius elements. We give certain characterizations of (prec )-almost symmetric ({mathcal {C}})-semigroups. When the cone is full, we prove the irreducible ({mathcal {C}})-semigroups, and (prec )-almost symmetric ({mathcal {C}})-semigroups with Betti-type three satisfy the extended Wilf conjecture. For (e ge 4), we give a class of MPD-semigroups in ({mathbb {N}}^2) such that there is no upper bound on the Betti-type in terms of embedding dimension e. Thus, the Betti-type may not be a bounded function of the embedding dimension. We further explore the submonoids of ({mathbb {N}}^d), which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are (textrm{PI})-monoids.

This work is devoted to describing the completion of a posemigroup by cuts. We introduce cut-stable morphisms between posemigroups and obtain that the category ({mathsf {RQuant_{wedge }}}) of quantales with meet and residuals preserving morphisms is a full reflective subcategory of the category ({textsf{CSPoSgr}}) of posemigroups with cut-stable morphisms. As an application, we also characterize other kinds of completions for posemigroups by means of lower cuts.