Dg FitzGerald

On inverses of products of idempotents in regular semigroups

Let E be the set of idempotents of a regular semigroup; we prove that V(E n ) = E n+1 (see below for the meaning of this notation). This generalizes a result of Miller and Clifford ([3], theorem 4, quoted as exercise 3(b), p. 61, of Clifford and Preston [1]) and the converse, proved by Howie and Lallement ([2], lemma 1.1), which together establish the case n = 1. As a corollary, we deduce that the subsemigroup generated by the idempotents of a regular semigroup is itself regular.

DUAL SYMMETRIC INVERSE MONOIDS AND REPRESENTATION THEORY

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps of a set X. The present paper describes the structure of a categorical dual I*_X to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in I_X and I*_X, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

Ideal Structure of the Kauffman and Related Monoids

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljević, Došen, and Petrić gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.

A presentation of the dual symmetric inverse monoid

The dual symmetric inverse monoid is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

Enumeration of idempotents in diagram semigroups and algebras

We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various R -, L - and D -classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.

Computing the maximum generalized inverse of a Boolean matrix

Abstract This paper gives an efficient, direct method for testing whether an arbitrary Boolean matrix is regular and, if it is regular, for computing its maximum generalized inverse.

The impact of residential socio-economic profile on medical service utilisation and nursing home attendance claims by older Australians

In other developed countries, older persons in deprived personal circumstances make increased use of primary care and nursing homes compared with those less disadvantaged. The influence of living in a more deprived area on the use of these services by older residents has not been so well studied. This study appraises if older Australians living in more disadvantaged locations have similar increased service usage. Retrospective analyses of fee-for-service data from 1 October 2001 to 30 September 2002 for all Medicare and Department of Veterans? Affairs claimants aged 77 years and over, not admitted to residential aged care, were undertaken. Data were analysed by age, gender and quartile of disadvantage of postcode of residence as categorised by the Australian Bureau of Statistics 1996 Census Index of Relative Socio-economic Disadvantage. The main outcome measures were: mean number of claims for general practitioner primary medical care (GP) services; longer GP consultations; specialist consultations; rate of GP health assessments; and, first claims for GP nursing home attendances. Mean number of claims for GP primary medical care services generally decreased as the disadvantage of postcode increased. Men aged 77-79 years were an exception, with no difference in the mean across quartiles of disadvantage. The mean number of claims for longer primary care consultations and all specialist consultations also decreased as the disadvantage of postcode increased. Claims for GP health assessments were more frequent as the disadvantage of the residential postcode increased. Claim rates for first nursing home attendance items by men did not differ across the range of disadvantage. This lack of difference was also seen for women aged 77-79 years. Claims for first nursing home attendance items by women aged 80 or more years decreased as the disadvantage of postcode increased. This study concluded that there are fewer claims for medical and nursing home services by older Australians living in more disadvantaged postcodes. Whether this represents under-provision of medical care leading to sub-optimal health outcomes needs further examination.

Organisational characteristics, cultural qualities and excellence in leading Australian-owned information technology firms

This paper reports empirical findings of an initial series of field case studies aimed at identifying and modelling the organisational characteristics and cultural qualities of excellence in top Australian-owned information technology firms. A common set of critical success factors, missions, and core business activities was identified by executive managers and owners of 10% of the top Australian IT firms. Management described other organisational characteristics including size, revenue, growth projections, the role of software, information systems functions, project teams, customer relations, organisational resources and technology uptake. Staff were asked to identify and describe cultural aspects of the organisations such as the importance of key individuals, orthodoxy, watershed events, key functional values and beliefs, folklore, rites and rituals. They provided metaphors for the organisation culture, the organization itself, their manager an employee, the ideal work environment, and the ideal manager.

Inverse Semigroups of Bicongruences on Algebras, Particularly Semilattices

Isomorphisms between quotient algebras of an algebra make up an inverse semigroup which carries information about the algebra. The extent of this information is explored in the case of semilattices; the main result is that any (meet) semilattice admitting no joins of incomparable elements, and satisfying chain conditions, is determined up to isomorphism by the inverse semigroup of isomorphisms between its quotient semilattices.