David Easdown

Biordered sets come from semigroups

Information about a semigroup can often be gleaned from its partial algebra of idempotents. For example, the idempotents of an inverse semigroup form a semilattice. All isomorphisms between principal ideals of a semilattice E form the Munn inverse semigroup T,, which contains the fundamental image of every inverse semigroup whose idempotents form the semilattice E [ll, 123. Thus semilattices give rise to all fundamental inverse semigroups. Successful efforts have been made to generalize the Munn construction to the wider class of regular semigroups [ 1,6-9, 13, 141. Nambooripad achieved this using the concept of a regular biordered set. The biordered set of a semigroup S means simply the partial algebra consisting of the set E = E(S) of idempotents of S with multiplication restricted to

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.

BIORDERED SETS ARE BIORDERED SUBSETS OF IDEMPOTENTS OF SEMIGROUPS

A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.

TRACE FUNCTIONS ON INVERSE SEMIGROUP ALGEBRAS

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and 5 is completely semisimple of a type that includes countable free inverse semigroups. The concept of a trace function on a real or complex algebra had its origin in matrix theory and is of central importance in many algebraic and analytical contexts. In the case of a group algebra, the trace of an element is defined simply to be the coefficient of the identity and is easily seen to possess all the standard properties. With the growth of interest in inverse semigroups (a class of involution semigroups with many group-like features), it is natural to ask whether the corresponding semigroup algebras also admit trace functions. In this paper we consider the semigroup algebra F[S] of an inverse semigroup 5 over a subring F of C that contains 1 and is closed under complex conjugation. In Section 1, where the basic definitions appear, two simple necessary conditions are obtained for the existence of a trace function on F[S] and attention is drawn to those trace functions (called strong) with the property that their values on the idempotents of S are positive integers. The main result of Section 2 provides a necessary and sufficient condition for F[S] to admit a strong trace function - namely that each principal ideal of the semilattice of S be finite. Section 3 comprises two examples. The notion of a pseudotrace function relative to a submodule is introduced in Section 4 and it is shown that, for any nonempty finite subset T of 5, F[S] admits a

PERIODIC ELEMENTS OF THE FREE IDEMPOTENT GENERATED SEMIGROUP ON A BIORDERED SET

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.

BRAIDS AND FACTORIZABLE INVERSE MONOIDS

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 nmonoid, some aspects of a general theory of which are discussed in this npaper. The coset monoid of a group arises, and turns out to have a universal nproperty within a certain class of factorizable inverse monoids. This theory nis dual to the classical construction of fundamental inverse semigroups from nsemilattices. In our braid examples, we will focus mainly on the ``merge and npart alternative, and introduce a monoid which is a natural preimage of nthe largest factorizable inverse submonoid of the dual symmetric inverse nmonoid on a finite set, and prove that it embeds in the coset monoid of the nbraid group.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

On semigroups with involution

A semigroup 5 with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S, (3t G T)(Vu, v 6 T) tt* = uv => u = v. It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive. 1. DEFINITIONS

The minimal faithful degree of a fundamental inverse semigroup

This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.

Complementation in the group of units of a ring

By adding 1 to elements of the nilradical and Jacobson radical of a ring with identity, normal subgroups of the group of units are obtained. In this paper we record observations about complementation of these subgroups in the group of units of a ring, identifying large classes where complementation takes place and some examples where it fails.