There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid ^x . that is.a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual J$ 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 J x and J^, 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.1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18; secondary 2OM3O.

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Let E be the set of idempotents of a regular semigroup; we prove that V(E") = E n+l (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.Let N denote the set of natural numbers; let n e N and S be any semigroup. We denote by E the set of idempotents of S and by E" the set of all products of n idempotents of S. Further, if £"is not empty, let <£> denote the subsemigroup of 5 generated by E; then <£> =

The monoid 3n of uniform block permutations is the factorisable inverse monoid which arises from the natural action of the symmetric group on the join semilattice of equivalences on an n-set; it has been described in the literature as the factorisable part of the dual symmetric inverse monoid. The present paper gives and proves correct a monoid presentation for ff n . The methods involved make use of a general criterion for a monoid generated by a group and an idempotent to be inverse, the structure of factorisable inverse monoids, and presentations of the symmetric group and the join semilattice of equivalences on an n-set.

What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse monoid, some aspects of a general theory of which are discussed in this paper. The coset monoid of a group arises, and turns out to have a universal property within a certain class of factorizable inverse monoids. This theory is dual to the classical construction of fundamental inverse semigroups from semilattices. In our braid examples, we will focus mainly on the "merge and part" alternative, and introduce a monoid which is a natural preimage of the largest factorizable inverse submonoid of the dual symmetric inverse monoid on a finite set, and prove that it embeds in the coset monoid of the braid group.