Structure Of Semigroups Research Articles

Representations of negative definite functions on polynomial hypergroups

We investigate the relationship between polynomial hypergroups and the usual semigroup structure on the non-negative integers.

FINITE BANDS AND AMALGAMATION BASES FOR FINITE SEMIGROUPS

ABSTRACT Hall and Putcha proved that if a finite semigroup S is an amalgamation base for all finite semigroups, then the -classes of S are linearly ordered. Oknin´ski and Putcha proved that any finite semigroup S is an amalgamation base for all finite semigroups if the -classes of are linearly ordered and the semigroup algebra over the complex field has a zero Jacobson radical. In this paper, we study the structure of semigroups which are amalgamation bases for all finite semigroups. In particular, the structure of finite bands which are amalgamation bases for all finite semigroups is determined.

Computing Transformation Semigroups

This paper describes algorithms for computing the structure of finite transformation semigroups. The algorithms depend crucially on a new data structure for an R -class in terms of a group and an action. They provide for local computations, concerning a single R -class, without computing the whole semigroup, as well as for computing the global structure of the semigroup. The algorithms have been implemented in the share package MONOID within theGAP system for computational algebra.

A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S * of S and a unary operation * in S satisfying: (1) x * ∈ S * ⋂ V_ S * (x) for all x∈ S; (2) (x * ) * =x for all x∈ S * ; (3) (x * y) * =y * x ** and (xy * ) * =y ** x * for all x,y∈ S, then S * is called a regular *- transversal of S ; if (3) is replaced with (xy) * =y * x * for all x,y∈ S, then S * is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.

Semigroups, groups and grammar inference problem

In the paper we analyse a problem o f inferring a grammarfrom a given sample o f a langu- age. We try topresent an algebraicformalism capable ofdescribing the issue. We consider two cases: a case o f inferring canonical finite-state grammars, and a case o f inferring general grammars. In both cases we define a semigroup structure. Finally we look at the possibility o f getting a structure o f a group.

POWER GROUPS OF FREE ABELIAN GROUPS OF FINITE RANK

ABSTRACT The power set of a group G has an induced semigroup structure, some subsets of which will form groups in their own right. We are especially interested in such subsets that are maximal. We demonstrate that even when G is a free abelian group of finite rank, the groups which arise in this way can be diverse profinite abelian groups.

On 2-groups with submultiplicative spectrum

A group G of complex square invertible matrices is said to have submultiplicative spectrum if for any G, H in G we have that σ( GH)⊂ σ( G) σ( H), where σ( G) denotes the spectrum of G. The question whether such a group can be irreducible arises in the study of the structure of matrix semigroups and is answered in the positive for groups with odd exponent (cf. Radjavi and Rosenthal, Simultaneous Triangularization, Springer, New York, 1999). Here we consider the most important remaining case, 2-groups with submultiplicative spectrum. If there is an element in the group of the kind whose determinant has order equal to either the exponent or half of the exponent of the group, then the group is reducible. We give an example of an irreducible group of the kind such that the order of determinant of every element equals one quarter of the exponent of the group.

Some open problems concerning relatively free and free profinite semigroups

Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain.

Some open problems in the structure theory of regular semigroups

The aim of this note is to summarise the most important results which generalise those describing the structure of inverse semigroups via semidirect products of semilattices by groups, and to formulate several open problems whose investigation might contribute to the development of this topic. This is the written version of the author’s talk given at the problem session of the Colloquium on Semigroups, Szeged, 2000. The paper also contains some problems raised there by D. B. McAlister and P. A. Grillet.

Free profinite locally idempotent and locally commutative semigroups

This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety L Sl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.

The pseudoidentity problem and reducibility for completely regular semigroups

Dedicated to George Szekeres on the occasion of his 90th birthdayNecessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

Polytopal linear retractions

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable ‘polytopal’ generalization of linear algebra (and, subsequently, that of algebraic K K -theory).

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed. *The research of the first author is supported by the Natural Science Foundation and the Education Committee Foundation of Yunnan Province, China and also by the Foundation of Yunnan Province, China. #The research of the second author is partially supported by UGC grant (Hong Kong) #2260126. †The research of the third author is supported by a grant of NSF, China and a grant from the Basic Science Research Council of Yunnan Province, China.

N-Transitivity and the complementation property

We investigate the structure of (minimal) n-transitive semigroups of matrices of rankat most n in M k( F) , by considering an equivalent problem regarding certain families of (n−k)-dimensional subspaces of F k .

LIKE ME?- MEASURES OF CORRESPONDENCE AND IMITATION

Imitation is a powerful mechanism for efficient learning of novel behaviors that both supports and takes advantage of sociality. A fundamental problem for imitation is to create an appropriate (partial) mapping between the body of the system being imitated and the imitator. By considering for each of these two systems an associated automaton (respectively, transformation semigroup) structure, attempts at such mapping can be considered (partial) relational homomorphisms. This article shows how mathematical techniques can be applied to characterize how far a behavior is from a successful imitation and how to evaluate attempts at imitation arising from a particular correspondence between the imitator and model. For the imitator and the imitated, affordances in the agent-environment structural coupling are likely to be different, all the more so in the case of dissimilar embodiment. We argue that the use of what is afforded to the imitator to attain corresponding effects or, as in dance, sequences of effects, is necessary and sufficient for successful imitation. However, the judged degree of success or failure of an attempted behavioral match depends on some externally imposed or in the case ofautonomous agents internally determined criteria on effects of the attempted imitative behavior (including effects attained successively as well as final effects). These criteria correspond to metrics measures of difference which can guide the evaluation of a correspondence, the learning of a correspondence, or learning how to apply one. Metrics on states and sequences of action events in the system-environment coupling allow judgment of similarity for observer-dependent' purposes. This allows one to formally define successful imitation with respect to such criteria. The resulting measures can be used to compare various candidate mappings (e.g., body plan or perception-action correspondences). Additionally, this may be applied in the automated construction and learning of mappings to be used in imitation for artificial, hardware, and software systems.

Semigroups of zero entropy

Let S be the multiplicative semigroup generated by a finite set {s 1,…,s k} of n by n matrices over a field. For every positive integer m, let l(m) denote the number of sequences i 1,…,i m such that the corresponding product s i 1 ⋯s i m is nonzero. We say that the entropy of S is zero if 0 is the limit of the sequence logl(m)/m . The structure of semigroups of entropy zero is described.

An Algebraic Model for Generating and Adapting Neural Networks by Means of Optimization Methods

This paper describes a new scheme of binary codification of artificial neural networks designed to generate automatically neural networks using any optimization method. Instead of using direct mapping of strings of bits in network connectivities, this particular codification abstracts binary encoding so that it does not reference the artificial indexing of network nodes; this codification employs shorter string length and avoids illegal points in the search space, but does not exclude any legal neural network. With these goals in mind, an Abelian semi-group structure with neutral element is obtained in the set of artificial neural networks with a particular internal operation called superimposition that allows building complex neural nets from minimum useful structures. This scheme preserves the significant feature that similar neural networks only differ in one bit, which is desirable when using search algorithms. Experimental results using this codification with genetic algorithms are reported and compared to other codification methods in terms of speed of convergence and the size of the networks obtained as a solution.

On the structure of simplicial affine semigroups

We give a structure theorem for simplicial affine semigroups. From this result we deduce characterizations of some properties of semigroup rings of simplicial affine semigroups. We also compute an upper bound for the cardinality of a minimal presentation of a simplicial affine semigroup.

Semilattice Decompositions of Semigroups Revisited

Two relations introduced by M. S. Putcha and T. Tamura, denoted by A! and , play a crucial role in semilattice decompositions of semigroups. General properties of the graphs that correspond to these relations were studied by M. S. Putcha in [14], 1974, and the structure of semigroups in which the minimal paths in the graph corresponding to A! are bounded was described by the Þrst two authors in [7], 1996. In this paper this will be done for the relation . We consider the same problem for the left-hand analogue of this relation and for two radicals of the Green’s J -relation. The obtained results generalize various results given by M. S. Putcha, T. Tamura, L. N. Shevrin and the Þrst two authors of this paper.

Finite semigroups with slowly growing p n -sequences

The p n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p n (S) ≤cn r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p n -sequence and give some examples.