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.     

On the Number of Nilpotents in the Partial Symmetric Semigroup

Abstract

In this note, we obtain and discuss formulae for the total number of nilpotent partial and nilpotent partial one–one transformations of a finite set.     

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 .

     

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {a^p, a^p?1b, a^p?2b²,…, b^p} to allow the p copies of a (resp. b) to all be different. We call the sequence {a₁a₂a₃… a_p, b₁a₂a₃…a_p, b₁b₂a₃…a_p,…, b₁b₂b₃…b_p} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.     

Apery Sets of Numerical Semigroups #

Let S be a numerical semigroup. We examine a particular subset of the Apery set of S and establish a correspondence between this subset and the holes of S . This correspondence allows us to establish conditions for S to be almost symmetric.     

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.     

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.

     

Lévy–Khinchin Formula and Existence of Densities for Convolution Semigroups on Symmetric Spaces

We prove the existence of a smooth density for a convolution semigroup on a symmetric space and obtain its spherical representation.      

Unitary Numerical Semigroups and Perfect Bricks

This article completes a previous investigation of balanced and unitary numerical semigroups. The main result establishes the equivalence of unitary numerical semigroups and perfect 2 × 2 bricks.     

The Commutativity Relation in the Symmetric Semigroup

We compute the cardinality of the centralizer of an injection or surjection provided that the basic set is countable. We prove a formula for the cardinality of the set of conjugacy classes in the infinite symmetric group.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

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.     

Perfect Pairs of Ideals and Duals in Numerical Semigroups

This article considers numerical semigroups S that have a nonprincipal relative ideal I such that μ _S (I)μ _S (S ? I) = μ _S (I + (S ? I)). We show the existence of an infinite family of such pairs (S, I) in which I + (S ? I) = S\{0}. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.     

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.     

Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this article, we give new characterizations of the Buchsbaum and Cohen–Macaulay properties of the tangent cone gr _𝔪 (R), where (R, 𝔪) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr _𝔪 (R).     

Minimal Generating Sets for Relative Ideals in Numerical Semigroups of Multiplicity Eight

Abstract

Let S be a numerical semigroup and let I be a relative ideal of S. Let S ? I denote the dual of I and let μ _S (?) represent the size of a minimal generating set. We investigate the inequality μ _S (I)μ _S (S ? I) ≥ μ _S (I + (S ? I)) under the assumption that S has multiplicity 8. We will show that if I is non-principal, then the strict inequality μ _S (I)μ _S (S ? I) > μ _S (I + (S ? I)) always holds.     

Combinatorics of Nilpotents in Symmetric Inverse Semigroups

We show how several famous combinatorial sequences appear in the context of nilpotent elements of the full symmetric inverse semigroup . These sequences appear either as cardinalities of certain nilpotent subsemigroups or as the numbers of special nilpotent elements and include the Lah numbers, the Bell numbers, the Stirling numbers of the second kind, the binomial coefficients and the Catalan numbers.AMS Subject Classification: 05A15, 20M18, 20M20, 05A19.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Star Operations on Numerical Semigroups

It is proved that the number of numerical semigroups with a fixed number n of star operations is finite if n > 1. The result is then extended to the class of analytically irreducible residually rational one-dimensional Noetherian rings with finite residue field and integral closure equal to a fixed discrete valuation domain.