Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.     

Characterization of Prüfer Multiplication Monoids and Domains by Means of Spectral Module Systems

 We generalize and unify several well-known characterizations of Prüfer domains and PVMDs by properties of their overrings. Received November 19, 2001; in revised form March 18, 2002 Published online August 14, 2002     

On Retracts of the Random Graph and Their Natural Order

 We prove that the natural order on the idempotents of the endomorphism monoid of the countably infinite random graph is -universal; that is, it embeds every countable order. We therefore extend in a strong fashion a result of [2] which showed that the natural order embeds every countable linear order. We consider a refinement of the natural order which embeds every countable quasi-order. Received 26 January 2001; in revised form 3 October 2001     

Embedding Semigroups into Idempotent Generated Ones

We present a concrete model of the embedding due to Pastijn and Yan of a semigroup S into an idempotent generated semigroup now in terms of a Rees matrix semigroup over S¹. The paper starts with a comparison of the two embeddings. Studying the properties of this embedding, we prove that it is functorial. We show that a number of usual semigroup properties is preserved by this embedding, such as periodicity, finiteness, the cryptic property, regularity, complete semisimplicity and various local properties, but complete regularity is not one of them.     

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     

The Anosov Relation for Nielsen Numbers of Maps of Infra-Nilmanifolds

The relative ranks of the monoid of endomorphisms of a strong independence algebra of infinite rank modulo two distinguished subsets are calculated. These subsets are the group of automorphisms and the endomorphisms α satisfying α² = α. The extra generators are characterised in each case.     

Partial Orders on Transformation Semigroups

In 1986, Kowol and Mitsch studied properties of the so-called natural partial order on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), ). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare with another natural partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.This author gratefully acknowledges the generous support of Centro de Matematica, Universidade do Minho, Portugal during his visit in May–June 2001.Received May 27, 2002; in revised form November 27, 2002 Published online May 16, 2003     

Orbits of Inner Mapping Groups

 Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way. (Received 5 March 2001; in revised form 11 June 2001)     

Hyperbolic Structures and Measured Foliations on 2-Dimensional Complexes

 In this paper, we introduce hyperbolic structures and measured foliations on 2-dimensional complexes and we describe moduli spaces for these structures. Received November 19, 2001; in revised form August 5, 2002 Published online February 28, 2003     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣. Supported in part by NSF DMS-0074531. Received February 14, 2002; in revised form July 18, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Corrigendum: “On the Exceptional Set for Goldbach’s Problem in Short Intervals”

We correct the proof of the log-free hybrid zero-density estimate given in On the Exceptional Set for Goldbachs Problem in Short Intervals, Monatsh Math 132: 49–65.     

Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${\cal C}$ Grou ps

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FP_m. This generalises work of J. King for ¹(G) in the case when G is metabelian [9].Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on ¹(G) and on the automorphism group of G.Both authors are partially supported by CNPq, Brazil.     

Partial Actions of Groups on Cell Complexes

 In this paper, we study partial group actions on 2-complexes. Our results include a characterization, in terms of generating sets, of when a partial group action on a connected 2-complex has a connected globalization. Using this result, we give a short combinatorial proof that a group acting without fixed points on a connected 2-complex, with finite quotient, is finitely generated. This result is then generalized to characterize finitely generated groups as precisely those groups having a partial action, without fixed points, on a finite tree, with a connected globalization. Finally, using Bass-Serre theory, we determine when a partial group action on a graph has a globalization which is a tree. The author was supported in part by NSF-NATO postdoctoral fellowship DGE-9972697, by Praxis XXI scholarship BPD 16306 98 and by FCT through Centro de Matemática da Universidade do Porto. Received September 20, 2001; in revised form June 25, 2002     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Commutators in Residually Finite Groups

 The following result is proved. Let G be a residually finite group satisfying the identity ([x ₁, x ₂][x ₃, x ₄]) ⁿ  ≡ 1 for a positive integer n that is not divisible by p ² q ² for any distinct primes p and q. Then G′ is locally finite. Received 7 May 2001; in revised form 3 December 2001     

The <Emphasis Type="Italic">p</Emphasis>-Cockcroft Property of Central Extensions of Groups II

In [6], ?evik defined necessary and sufficient conditions for the presentation of a central extension of a cyclic group by an ordinary group to be p-Cockcroft, where p is a prime or 0. In this paper, as a next step of this above result, we will define the p-Cockcroft property for the presentation of a central extension of an abelian group by any group. Finally, as an application of the main result, we will present an example.     

The First Power Mean of the Inversion of L-Functions and General Kloostermann Sums

 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula. Received 30 March 2001; in revised form 12 November 2001     

Plurisubharmonic Defining Functions,Good Vector Fields,and Exactness of a Certain One Form

 We show that the approaches to global regularity of the -Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation. Received 14 May 2001; in revised form 8 November 2001