School of Mathematical and Computer Sciences

Some recent preprints:

• Derivations and relation modules for inverse semigroups. In Algebra and Discrete Mathematics 12 (2011) 1-19. The full-text published version is available at the journal's site.

  Abstract: We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one.

• (with Rebecca Noonan Heale) The idempotent problem for an inverse monoid. In Internat. J. Algebra Comput 7 (2011) 1179-1194. For access to the published version see here. A preliminary version may be found here PDF.

  Abstract: We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem , the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid aer related to the formal language properties of its idempotent problem. In particular, we show that if an inverse monoid is either E -unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck-Reilly extensions of finite groups.

• (with Elizabeth Miller) The graph expansion of an ordered groupoid. In Algebra Colloquium 18 (Spec 1) (2011) 827-842. A preliminary version may be found here. PDF

  Abstract: We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister's P--theorem for incompressible ordered groupoids.

• (with Mohammad Samman) Endomorphism seminear-rings of Brandt semigroups. In Communications in Algebra 38 (2010) 4028--4041.

  Abstract: We consider the endomorphisms of a Brandt semigroup B_n, and the semigroup of mappings E(B_n) that they generate under pointwise composition. We describe all the elements of this semigroup, determine Green's relations, consider certain special types of mapping which we can enumerate for each n, and give a complete calculations for the size of E(B_n) for small n.

• (with Mohammad Samman) Clifford semigroups and seminear-rings of endomorphisms. In the Int. Electron. J. Algebra 7 (2010) 110-119.

  Abstract: We consider the structure of the semigroup of self-mappings of a semigroup S under pointwise composition, generated by the endomorphisms of S. We show that if S is a Clifford semigroup, with underlying semilattice &Lambda, then the endomorphisms of S generate a Clifford semigroup E(S) whose underlying semilattice is the set of endomorphisms of Λ. These results contribute to the wider theory of seminear-rings of endomorphisms, since E(S) has a natural structure as a distributively generated seminear-ring

• (with Erzsi Dombi) F*-inverse covers for tiling semigroups. In Periodica Math. Hungarica 59 (2009) 185-202.

  Abstract: We introduce the notion of path extensions of tiling semigroups and investigate their properties. We show that the path extension of a tiling semigroup yields a strongly F*-inverse cover of the tiling semigroup and that it is isomorphic to an HNN* extension of its semilattice of idempotents.

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