On embedding of finitely generated profinite semigroups into 2-generated profinite semigroups

Koryakov recently asked the following question: when a pseudovariety V does not satisfy any non-trivial identity, does there exist an embedding from any finitely generated V-free profinite semigroup into the 2-generated V-free profinite semigroup? During the conference “Semigroups, Automata and Languages” in Porto (June 1994 [2]), a positive answer to this question was conjectured. We give here a counterexample to this conjecture. Received December 15, 1994; accepted in final form June 5, 1997.     

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.     

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...     

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.     

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...     

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

Most factorization invariants in the literature extract extremal factorization behavior, such as the maximum and minimum factorization lengths. Invariants of intermediate size, such as the mean, median, and mode factorization lengths are more subtle. We use techniques from analysis and probability to describe the asymptotic behavior of these invariants. Surprisingly, the asymptotic median factorization length is described by a number that is usually irrational.     

Distinct degree factorizations in additive arithmetical semigroups

The probability that an element of degreen has a given factorization pattern is computed within the context of a certain class of additive arithmetical semigroups. Concrete cases of these semigroups include the semigroup, of monic polynomials in one indeterminate over a finite fieldF _q , the multiplicative semigroups of ideals in principal orders within algebraic function fields overF _q and semigroups of integral divisors in algebraic function fields overF _q .     

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     

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 numerical semigroups and the order bound

Let S={s₀=0<s₁<?<s_i…}⊆N be a numerical non-ordinary semigroup; then set, for each . We find a non-negative integer m such that d_ORD(i)=ν_i+1 for i≥m, where d_ORD(i) denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, the value of m is also found.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.