Every positive integer is the Frobenius number of an irreducible numerical semigroup with at most four generators

Letg be a positive integer. We prove that there are positive integersn ₁,n ₂,n ₃ andn ₄ such that the semigroupS=(n ₁,n ₂,n ₃,n ₄) is an irreducible (symmetric or pseudosymmetric) numerical semigroup with g(S)=g. This work has been supported by the project BFM2000-1469.     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

A mathematical programming approach to the computation of the omega invariant of a numerical semigroup

In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.     

A note on a special case of the Frobenius problem

For a set of positive and relative prime integers A = {a ₁…,a _k }, let Γ(A) denote the set of integers of the form a ₁ x ₁+…+a _k x _k with each x _j ≥ 0. Let g(A) (respectively, n(A) and s(A)) denote the largest integer (respectively, the number of integers and sum of integers) not in Γ(A). Let S*(A) denote the set of all positive integers n not in Γ(A) such that n + Γ(A) \ {0} ? Γ((A)\{0}. We determine g(A), n(A), s(A), and S*(A) when A = {a, b, c} with a | (b + c).     

Atoms of the set of numerical semigroups with fixed Frobenius number

Given a positive integer g, we denote by F(g) the set of all numerical semigroups with Frobenius number g. The set (F(g),∩) is a semigroup. In this paper we study the generators of this semigroup.     

A note on the gradient of heat semigroup

A new representation for the gradient of heat semigroup on Riemannian manifold is given by using the integration by parts formula on Wiener space, which reflects in some sense the asymptotic property of Brownian motion.     

A note on the units of semigroup rings

In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to extend the results in [2] and 3] to the semigroup ring RG, G an u.p, semigroup.

In the last section we give a necessary and sufficient condi-tion for an element to be a divisor of zero in RG where G is an u.p. semigroup.     

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 .

     

A note on the product of random elements of a semigroup

LetX=(X ₀,X ₁, ...) be a Markov chain on the discrete semigroupS. X is assumed to have one essential classC such thatCK, whereK is the kernel ofS. We study the processY=(Y ₀,Y ₁,...) whereY _n =X ₀ X ₁ ...X _n using the auxiliary process which is a Markov chain onS×S. The essential classes and the limiting distribution of theZ-chain are determined. (These results were obtained earlier byH. Muthsam, Mh. Math.76, 43–54 (1972). However, his proofs contained an error restricting the validity of his results.Supported in part by the Danish Ministry of Education and the Toroch Ellida Ljungbergs fond.     

A note on sensitivity of semigroup actions

It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey (Am. Math. Mon. 99, 332–334, 1992), Silverman (Rocky Mt. J. Math. 22, 353–375, 1992) and Glasner and Weiss (Nonlinearity 6, 1067–1075, 1993). In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner (Ergodic Theory via Joinings, 2003). Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.     

A short note on the Frobenius norm of the commutator

This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ? ^n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY ? XY‖ _F ² ≤ 2‖X‖ _F ² ‖Y‖ _F ² is finally improved into $$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$ . In addition, a further improvement is made.     

The numerical duplication of a numerical semigroup

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup S and a semigroup ideal E?S, produces a new numerical semigroup, denoted by S? ^b E (where b is any odd integer belonging to S), such that S=(S? ^b E)/2. In particular, we characterize the ideals E such that S? ^b E is almost symmetric and we determine its type.     

A note on model structures on arbitrary Frobenius categories

We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category F such that the homotopy category of this model structure is equivalent to the stable category F as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When F is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed) model structure in the sense of Gillespie (2011).     

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 note on the computation of the CP-rank

The purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix (over the reals) is completely positive and finding its CP-rank when it is. We show that these questions can be resolved by finite algorithms and we provide (non-polynomial) complexity bounds on the number of arithmetic/Boolean operations that these algorithms require. We state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over {0, 1} matrices.