Automorphism invariants for semigroups

Invariant subgroups are associated with each element of a semigroup. These invariants are used to show certain semigroups of continuous functions have only inner automorphisms. In special cases bijections preserving these invariants are necessarily automorphisms and outer automorphisms can be constructed.     

Balanced numerical semigroups

We investigate the Frobenius number, genus, type, and minimal presentation of a class of numerical semigroups of embedding dimension 4 of the form \(S = \langle a_1, a_2, a_3, a_4 \rangle \) such that \(a_1 + a_4 = a_2 + a_3\). The investigation focuses on determining the Apery set of S with respect to the multiplicity.     

Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen–Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen–Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen–Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.     

Dense numerical semigroups

Semigroup Forum - A numerical semigroup S is dense if for all $$sin Sbackslash {0}$$ we have $$left{ s-1,s+1right} cap Sne emptyset $$ . We give algorithms to compute the whole set of...     

Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.     

Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.     

Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n ₁,…,n _p 〉 to computing the genus of a “smaller” numerical semigroup 〈n ₁?an ₁modm,…,n _p ?an _p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.     

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.     

On comparing two chains of numerical semigroups and detecting arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T):=(N-N) and L(T)= \cup { (hN-hN) \colon h \geq 1 } . If S is a numerical semigroup, define strictly increasing finite sequences { B _i (S) \colon 0 ≤ i ≤β (S) } and { L _i (S) \colon 0 ≤ i ≤λ (S) } of semigroups by B ₀ (S):=S=:L ₀ (S) , B _{β (S)} (S):= \Bbb N =: L _{λ (S)} (S) , B _i+1 (S):=B(B _i (S)) for 0<i< β (S) , L _i+1 (S):=L(L _i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B ₂ (S) need not be a subset of L ₂ (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B ₂ (S) \subseteq L ₂ (S) in each of the following cases: S is symmetric;S has maximal embedding dimension;S has embedding dimension e(S) ≤ 3 . Moreover, if either e(S)=2 or S is pseudo-symmetric of maximal embedding dimension, then B _i (S) \subseteq L _i (S) for each i , 0 ≤ i ≤λ (S) . For each integer n \geq 2 , an example is given of a (necessarily non-Arf) semigroup S such that β (S) = λ (S)=n , B _i (S) = L _i (S) for all 0 ≤ i ≤ n-2 , and B _n-1 (S) \subsetneqq L _n-1 (S) . April 4, 2000     

On numerical experiments with symmetric semigroups generated by three elements and their generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r ₁,r ₂,r ₃∈ℤ⁺, r ₁≥2 and gcd(r ₁,r ₂)=gcd(r ₁,r ₃)=1, and calculate their universal Frobenius number Φ(r ₁,r ₂,r ₃) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r₁²,r₃-r₁²k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation.     

Frobenius pseudo-varieties in numerical semigroups

Annali di Matematica Pura ed Applicata (1923 -) - The common behavior of several families of numerical semigroups led up to defining the Frobenius varieties. However, some interesting families were...     

Classes of complete intersection numerical semigroups

We consider several classes of complete intersection numerical semigroups, arising from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory. In particular, we determine all the logical implications among these classes and provide examples. Most of these classes are shown to be well-behaved with respect to the operation of gluing.