Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect tothe 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 themultiplicity of a numerical semigroup.Numerical semigroups with maximum and mininmm number ofthis kind of gaps are described.     

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.     

Systems of Inequalities and Numerical Semigroups

A one-to-one correspondence is described between the set S(m)of numerical semigroups with multiplicity m and the set of non-negativeinteger solutions of a system of linear Diophantine inequalities.This correspondence infers in S(m) a semigroup structure andthe resulting semigroup is isomorphic to a subsemigroup of N^m–1.Finally, this result is particularized to the symmetric case.     

Transformational antiatomic characteristic of ordered sets

To any ordered set with a universally maximal element, a semigroup of its transformations with some natural properties that defines the ordered set up to an isomorphism is assigned. The system of such transformation semigroups is proved to be the minimal element in the set of all defining systems of transformation semigroups with respect to the following ordering: one system precedes another if for each ordered set from the class in question, the semigroup of its transformation belonging to the first system is contained in the semigroup of its transformation from the second system. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 112–119, July, 1999.     

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.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Proportionally Modular Diophantine Inequalities and Full Semigroups

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems.     

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.     

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.     

Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in Blanco and Puerto (SIAM J. Discrete Math., 26(3):1210–1237, 2012). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups such as those that are generated by a generalized arithmetic progression.     

Irreducible Numerical Semigroups Having Toms Decomposition

In this article we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.     

The Semigroup of Marcinkiewicz Modulars with Involution

The set M of all concave Marcinkiewicz modulars on [0, 1] is a semigroup with respect to the usual composition of functions. We show that some properties of modulars (which are of importance in interpolation and in general Banach theory) distinguish subsets of M that form ideals of the semigroup. These ideals turn out to be in a natural duality relation, which is also studied. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 121–131.     

On semigroups admitting ring structure

A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results proved in this paper, to show that no nontrivial multiplicative bounded interval semigroup on the real line ℝ admits a ring structure, obtaining the main results of Kemprasit et al. (ScienceAsia 36: 85–88, 2010).     

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.     

The chain of right simple semigroups in ordered semigroups

The relation ≤ is defined on the set of right ideals of an ordered semigroup. The main result of this paper is as follows: an ordered semigroup S is a chain of right simple ordered semigroups if and only if ≤ is an order relation. Bibliography: 3 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 83–88.     

Numerical semigroups with many gaps of the same parity

Let g _e (S) (respectively, g _o (S)) be the number of even (respectively, odd) gaps of a numerical semigroup S. In this work we study and characterize the numerical semigroups S that verify 2|g _e (S)−g _o (S)|+1∈S. As a consequence we will see that every numerical semigroup can be represented by means of a numerical semigroup with maximal embedding dimension with all its minimal generators odd. The first author is supported by the project MTM2007-62346 and FEDER funds. The authors want to thank P.A. García-Sánchez and the referee for their comments and suggestions.     

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 52–60, January, 2009.     

Numerical semigroups with embedding dimension three

Every numerical semigroup generated by three elements is determined by six positive integers that are the solution to a system of three polynomial equations. We give formulas of the Frobenius number and the cardinality of the set of gaps in terms of these six parameters.Received: 6 November 2003     

Submodules of minimal Buchsbaum–Rim multiplicity and applications

We give a characterization, in terms of an equality of integral closures, of a class of submodules having minimal Buchsbaum–Rim multiplicity with respect to a family of ideals. This is a notion motivated by an inequality of multiplicities. We apply our study to the computation of a known invariant in singularity theory. Work supported by DGICYT Grant MTM2006–06027.