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.     

Idempotent reducts of abelian groups

In this paper we construct two maps between the polynomials of abelian groups and the polynomials of idempotent reducts of abelian groups and show that these can be used to “lift” the finite basis property from abelian groups to their idenpotent reducts. It follows that this equational class of all idempotent reducts of abelian groups has the finite basis property. The research of this author was supported by a grant from the National Research Council of Canada. Presented by J. Mycielski     

On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis-Meakin expansion M(H) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin-Zelmanov varieties K_n of all locally finite groups of exponent dividing n.     

\mathcal{P}\mathcal{A}{\text{-isomorphisms}} of inverse semigroups

A partial automorphism of a semigroup S is any isomorphism between its subsemigroups, and the set all partial automorphisms of S with respect to composition is an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be if their partial automorphism monoids are isomorphic. A class of semigroups is called if it contains every semigroup to some semigroup from Although the class of all inverse semigroups is not we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is A semigroup is called if it is isomorphic or antiisomorphic to any semigroup that is to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are To Ralph McKenzieReceived April 15, 2004; accepted in final form October 7, 2004.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Right projective semigroups with 0

We describe the semigroups S with 0 that are projective in the category of right S-objects, i.e. of centered right S-operands. Among them are the free semigroups with an adjoined zero, and several characterizations of this subclass are given. Two other classes, one of them close to free semigroups with 0, the other rather far apart, are discussed in detail. The first class consists of semigroups of finite and infinite sequences, with multiplication based on concatenation. The second is formed by the projectives in the category of (S,S)-biobjects.     

Strong Morita equivalence of semigroups with local units

In this paper we study Morita contexts for semigroups. We prove a Rees matrix cover connection between strongly Morita equivalent semigroups and investigate how the existence of a unitary Morita semigroup over a given semigroup is related to the existence of a ‘good’ Rees matrix cover of this semigroup.     

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ (S denotes a monoid, and is a right congruence onS), and determined conditions onS under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which is generated by a single pair of elements ofS), and further results on such collapsing of flatness properties are obtained. An observation which is used extensively in this study is the fact that forw andt inS withwtt,S/(wt,t) is flat if and only ift is a regular element ofS.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A4494.Research supported by Estonian Research Foundation Grant No. 930.     

Translation numbers in a Garside group are rational with uniformly bounded denominators

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.     

Identities without the star for *-bands

Adair determined the lattice of varieties of *-bands and provided a single identity for each of them. We construct a new set of identities for the same purpose. Both of these systems involve at least one variable which is starred. Using the second system of identities, we show that the system of identities constructed by Gerhard and the author for varieties of bands also represents a set of identities determining the varieties of *-bands. The same type of result is proved for varieties of band monoids and varieties of *-band monoids using the results of two papers by Wismath.Presented by R. W. Quackenbush.     

On generating countable sets of endomorphisms

Sierpiski proved that every countable set of mappings on an infinite set X is contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this paper we prove that every countable set of endomorphisms of an algebra which has an infinite basis (independent generating set) is contained in a 2-generated subsemigroup of the semigroup of all endomorphisms of .     

The structure of solid binary algebras

A solid binary algebra in an abstract characterisation of the idempotents of a completely regular semigroup. We present here a structure theorem for solid binary algebras in terms of semilattices and rectangular bands. We also show that a free solid binary algebra can be embedded in a free completely regular semigroup; thus the word problem for the free solid binary algebra can be solved by using a solution of the word problem for the free completely regular semigroup.Presented by B. M. Schein.The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award.     

Stability of collocation-based Runge-Kutta-Nyström methods

We analyse the attainable order and the stability of Runge-Kutta-Nyström (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.These investigations were supported by the University of Amsterdam who provided the third author with a research grant for spending a total of two years in Amsterdam.     

On same-realization prediction in an infinite-order autoregressive process

Let observations come from an infinite-order autoregressive (AR) process. For predicting the future of the observed time series (referred to as the same-realization prediction), we use the least-squares predictor obtained by fitting a finite-order AR model. We also allow the order to become infinite as the number of observations does in order to obtain a better approximation. Moment bounds for the inverse sample covariance matrix with an increasing dimension are established under various conditions. We then apply these results to obtain an asymptotic expression for the mean-squared prediction error of the least-squares predictor in same-realization and increasing-order settings. The second-order term of this expression is the sum of two terms which measure both the goodness of fit and model complexity. It forms the foundation for a companion paper by Ing and Wei (Order selection for same-realization predictions in autoregressive processes, Technical report C-00-09, Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC, 2000) which provides the first theoretical verification that AIC is asymptotically efficient for same-realization predictions. Finally, some comparisons between the least-squares predictor and the ridge regression predictor are also given.     

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.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

On the automorphism groups of algebraic bounded domains

This paper is a part of author's doctoral thesis. It was partially supported by Sonderforschungsbereich 237 Unordnung und große Fluktuatuionen of the Deutsche Forschungsgemeinschaft     

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety of ordered semigroups, let be the class of sofic subshifts whose syntactic semigroup lies in . It is proved that if contains then is closed under taking shift equivalent subshifts, and conversely, if is closed under taking conjugate subshifts then contains and . Almost finite type subshifts are characterized as the irreducible elements of , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

Uniqueness of the bounded solution to a strongly degenerate parabolic problem

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value problem is unique under some essentially necessary condition on the growth of the convection.