The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

The maximum proportionally modular numerical semigroup with given multiplicity and ratio

We prove that the set of all proportionally modular numerical semigroups with fixed multiplicity and ratio has a maximum (with respect to set inclusion). We show that this maximum is a maximal embedding dimension numerical semigroup, for which we explicitly calculate its minimal system of generators, Frobenius number and genus.     

Proportionally modular diophantine inequalities and their multiplicity

Let I be an interval of positive rational numbers. Then the set S （I） = T ∩ N, where T is the submonoid of （Q0＋, ＋） generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S （I） has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.     

Commutative Group Algebras of Summable p-Groups

We study numerical semigroups generated by generalized arithmetic sequences. We present a membership criterion for such a numerical semigroup, and by this we are able to answer fundamental questions concerning a numerical semigroup such as computing the Frobenius number and the type of the numerical semigroup, and decide whether the numercial semigroup is symmetric. Also for this kind of numerical semigroups, we compute the cardinality of a minimal presentation and determine whether they are complete intersections.     

Affine convex body semigroups

In this paper we present a new class of semigroups called convex body semigroups which are generated by convex bodies of ? ^k . They generalize to arbitrary dimension the concept of proportionally modular numerical semigroups of Rosales et al. (J. Number Theory 103, 281–294, 2003). Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of ?² are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.     

The rank of the endomorphism monoid of a uniform partition

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.     

Unital affine semigroups

Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.     

The Lattice of Full Subsemigroups of an Inverse Semigroup

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain.     

The Oversemigroups of a Numerical Semigroup

Abstract. We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.     

Modular translations of numerical semigroups

In this paper we introduce the concept of modular translation. With this tool, if we consider a certain numerical semigroup S, we build another one S′ whose principal invariants are given explicitly in terms of the invariants of S. Some results about irreducible numerical semigroups are also studied.     

The oversemigroups of a numerical semigroup

We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.     

Symmetries on almost symmetric numerical semigroups

The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Fröberg in J. Algebra, 188, 418–442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H ^? (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H ₁ or H ₂ is not symmetric, then the gluing of H ₁ and H ₂ is not almost symmetric.     

A note on automatic semigroups

It is well known that for monoids and completely simple semigroups the property of being automatic does not depend on the choice of the semigroup generating set. In this paper we extend these results to semigroups satisfying the equality S=SS. As an immediate corollary of this result we obtain that for regular semigroups being automatic is invariant under the change of generators.     

The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms.This procedure allows the construction of a tree for each given genus containing all semigroups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some light to a former conjecture saying that the number of semigroups of a given genus increases with the genus.We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.     

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

Group ideals in a semigroup of measures

In this paper we introduce the notion of a group ideal in a semigroup. We shall prove that all group ideals of a compact affine semigroup are convex and dense. This generalizes many results in the literature concerning ideals in semigroups.     

Semigroups with Boolean congruence lattices

A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, respectively, with Boolean (...) congruence lattice are completely determined.     

Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups. This paper was supported by the project BFM2000-1469. The fourth author wishes to acknowledge support from the Universidade de Evora and the CIMA-UE.     

Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect to the multiplicity of the semigroup. The semigroup is fully determined by its multiplicity and these gaps.We study the case when a set of non-negative integers is the set of fundamental gaps with respect to the multiplicity of a numerical semigroup, Numerical semigroups with maximum and minimum number of this kind of gaps are described.