Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

We construct symmetric numerical semigroups for every minimal number of generators and multiplicity , . Furthermore we show that the set of their defining congruence is minimally generated by elements.

     

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.     

Every numerical semigroup is one half of a symmetric numerical semigroup

Let be a numerical semigroup. Then there exists a symmetric numerical semigroup such that .

     

On additive semigroups starting parts

We present geometrical arguments suggesting that the part of the segment {0,1,…,N−1} covered by the additive semigroup generated by (a,b,c) between 0 and the Frobenius number N(a,b,c) should exceed λ V for some constant λ (which might be 1/3 or even more).      

Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.     

A classification of maximal subsemigroups of finite order-preserving transformation semigroups

We describe the maximal subsemigroups of the semigroup of all order-preserving transformations of a finite chain and completely obtain their classification. We also count the number of its maximal subsemigroups.     

Delta sets for symmetric numerical semigroups with embedding dimension three

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.     

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.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

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     

Embedding the bicyclic semigroup into countably compact topological semigroups

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that a topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonoff semigroup containing a copy of C(p,q).     

The maximal Abelian dimension of linear algebras formed by strictly upper triangular matrices

We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra of n×n strictly upper triangular matrices, where n ∈ ℕ \ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 419–429, September, 2007.     

Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups

In this paper we study locally maximal idempotent-generated subsemigroups of finite singular orientation-preserving transformation semigroups. This work is supported by N.S.F. and E.S.F. of Guizhou.     

Maximal properties of some subsemigroups in finite order-preserving transformation semigroups

Let [n] = {1,2,…,n} be a finite set, ordered in the usual way. The order-preserving transformation semigroup O_n is the set of all order-preserving transformations of [n] (excluding the identity mapping) under composition. In this paper we first describe maximal idempotent-generated subsemigroups of O _n, and show that O_n has 2n - 2 such subsemi-groups. Secondly, we investigate maximal regular subsemigroups of O_n , and obtain the number of such subsemigroups as 2n - 3. Thirdly, we describe maximal idempotent-generated regular subsemigroups of O_n , and also obtain their classification and number.     

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.