Strongly taut finitely generated monoids

A strongly taut monoid is a monoid in which all the powers of any element of the monoid have the same elasticity, that is, the ratio between the maximum and the minimum length of the factorizations of an element remains unchanged under powers. We give a procedure to determine if a finitely generated commutative monoid is strongly taut. The authors are supported by the project MTM2004-01446 and FEDER funds. The authors would like to thank the referee and P. Baginski for their heplful comments and suggestions. Author’s addresses: P. A. García-Sánchez, Departamento de álgebra, Universidad de Granada, E-18071 Granada, Espa?a; D. Llena, Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, 04120 Almería, Espa?a; J. C. Rosales, Departamento de álgebra, Universidad de Granada, E-18071 Granada, Espa?a     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Ideals of finitely generated commutative monoids

We extend the concept of presentation of finitely generated commutative monoids to ideals of finitely generated commutative monoids and give algorithms to obtain information about an ideal from a given presentation.     

Principal ideals of finitely generated commutative monoids

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.     

Higher cohomologies of commutative monoids

Extending the Eilenberg–Mac Lane approach, we introduce and explore higher-level cohomology theories for commutative monoids and compare them with pre-existing theories (Leech, Grillet, etc.). We offer a cohomological classification of symmetric monoidal groupoid structures and work out some explicit computations for cyclic monoids.     

Presentations for subsemigroups of finitely generated commutative semigroups

We introduce the concept of presentation for subsemigroups of finitely generated commutative semigroups, which extends the concept of presentation for finitely generated commutative semigroups. We show that for every subsemigroup of a finitely generated commutative semigroup there are special presentations which solve the word problem in the given subsemigroup. Some properties like being cancellative, reduced and/or torsion free are studied under this new point of view. This paper was supported by the project DGES PB96-1424.     

Maximal ideals of regulous functions are not finitely generated

We prove that every maximal ideal in the ring of k-regulous functions, $k\in \mathbb{N}$, on a smooth real affine algebraic variety of dimension $d\ge 2$ is not finitely generated.     

Presentations for rook partition monoids and algebras and their singular ideals

We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.     

A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X and any relation ρ on X, let T(X,ρ) be the semigroup of all maps a:X→X that preserve ρ. Let S(X) be the symmetric group on X. If ρ is reflexive, the group of automorphisms of T(X,ρ) is isomorphic to N_S(X)(T(X,ρ)), the normalizer of T(X,ρ) in S(X), that is, the group of permutations on X that preserve T(X,ρ) under conjugation. The elements of N_S(X)(T(X,ρ)) have been described for the class of so-called dense relations ρ. The paper is dedicated to applications of this result.     

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus. This is related to classical beautiful results of Castelnuovo and Enriques that we revise here in terms of adjunction theory.     

A combinatorial property of ideals in free profinite monoids

A combinatorial property of ideals in free profinite monoids is proved.     

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of $\mathrm{Aut}(X)$ contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.     

A finitely presented monoid with a non-finitely generated group of units

We exhibit an example of a monoid defined by finitely many generators and defining relations, the group of units of which is not finitely generated. Received: 30 August 2006 Revised: 23 October 2006     

Polynomial cycles in finitely generated domains

LetR be a finitely generated domain of zero characteristics, and letN be a given integer. We show that the lengths of all cycles of polynomial mappingsR ^NR ^N defined overR are bounded by a number depending only onR andN.The work of the second author has been supported by the KBN-grant 2-1037-91-01.     

Two-dimensional representations of finitely generated free group over an arbitrary field

Working over an arbitrary field, we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible, respectively; also, we classify all two-dimensional representations of a finitely generated free group.     

On uniqueness of JSJ decompositions of finitely generated groups

We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in [RS].On the other hand we observe that any two JSJ decompositions of a group are related by an elementary deformation, and that strongly slide-free JSJ decompositions are genuinely unique. These results hold for the decompositions of Rips and Sela, Dunwoody and Sageev, and Fujiwara and Papasoglu, and also for accessible decompositions.     

Chebyshev and L1 solutions of overdetermined systems of linear equations with bounded variables

Two algorithms are here presented. The first one is for obtaining a Chebyshev solution of an overdetermined system of linear equations subject to bounds on the elements of the solution vector. The second algorithm is for obtaining an L₁ solution of an overdetermined system of linear equations subject to the same constraints. Efficient solutions are obtained using linear programming techniques. Numerical results and comments are given.