Liam O'Carroll

Embedding theorems for proper inverse semigroups

Abstract Let S be an inverse semigroup with semilattice of idempotents E, and let ϱ(S), or ϱ if there is no danger of ambiguity, be the minimum group congruence on S. Then S is said to be proper if Eϱ = E, or alternatively, if the equation ex = e for some e ϵ E and x ϵ S implies that x ϵ E. For example, free inverse semigroups and fundamental ω-inverse semigroups are proper. In a recent paper, McAlister has given a remarkable structure theorem for an arbitrary proper inverse semigroup, and using this theorem we show (Theorem 1.3) that any proper inverse semigroup P can be embedded in a semidirect product P of a semilattice and a group. Some consequences of this result are given; for example, if P is bisimple with identity then P is simple (see Theorem 1.6). Reilly has proved that an arbitrary inverse semigroup can be embedded in a bisimple inverse semigroup with identity. Given a proper inverse semigroup P it is shown that a proper inverse semigroup P′, arising from P by blowing up P g9 , can indeed be embedded in a bisimple proper inverse semigroup with identity (Theorem 2.4). We also investigate those homomorphic images of bisimple proper inverse semigroups which are themselves proper. In the final section, Rees quotient images of proper inverse semigroups are characterised (Theorem 3.4), and some other constructions for (simple) semidirect products of semilattices and groups are considered.

Note: On the symmetry of Welch- and Golomb-constructed Costas arrays

We prove that Welch Costas arrays are in general not symmetric and that there exist two special families of symmetric Golomb Costas arrays: one is the well-known Lempel family, while the other, although less well known, leads actually to the construction of dense Golomb rulers.

Degree and Algebraic Properties of Lattice and Matrix Ideals

We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of

On some properties of Costas arrays generated via finite fields

\mathbb{Z}^s

IDEALS OF HERZOG-NORTHCOTT TYPE

and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud--Sturmfels theory of binomial ideals over algebraically closed fields. We then use these results to study certain families of integer matrices (positive critical binomial (PCB), generalized positive critical binomial (GPCB), critical binomial (CB), and generalized critical binomial (GCB) matrices) and the algebra of their corresponding matrix ideals. In particular, the family of GPCB matrices is shown to be closed under transposition, and previous results for PCB ideals are extended to GPCB ideals. Then, more particularly, we give some applications to the theory of

Direct limit systems, generalized fractions and complexes of cousin type

1

On a theorem of Knörrer concerning Cohen–Macaulay modules

-dimensional binomial ideals. If

Free resolutions for deformations of maximal cohen-macaulay modules

G

On a surmise of McAdam concerning quintasymptotic primes

is a connected graph, we show as a further application that the order of...

LOCAL COHOMOLOGY: AN ALGEBRAIC INTRODUCTION WITH GEOMETRIC APPLICATIONS (Cambridge Studies in Advanced Mathematics 60)

We prove that Welch-constructed Costas arrays are in general not symmetric and that the Golomb-constructed ones are symmetric in two cases only, namely the Lem-pel one and a (rare) second one leading to the construction of dense Golomb rulers. Finally, we look into the (hard) problem of the number of fixed points of a Welch-constructed Costas array and formulate a conjecture.