Semigroups of I-quotients

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On Ehresmann semigroups

We describe an alternative approach to describing Ehresmann semigroups by categories in which a class of \'etale actions plays an important r\^ole. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As an application, we prove that every birestriction semigroup can be suitably embedded into a birestriction semigroup constructed from a category. As a corollary, we determine when a birestriction semigroup can be suitably embedded into an inverse semigroup

Some Undecidable Problems on Representability as Binary Relations

We establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for J onsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions

Partial maps with domain and range: extending Schein's representation

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised

Faithful functors from cancellative categories to cancellative monoids with an application to abundant semigroups

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup

Recognisable languages over free algebras

This thesis considers notions of recognisability for languages over (universal) algebras. The main motivation here is the body of work on recognisable languages over the free monoid, which in particular connects several, equivalent, approaches. The free monoid X^* on a set X consists of all finite strings of elements of X; these are thought of as words, and hence a subset of X^* is known as a language (i.e. a collection of words). The term is then used for a subset of any (free) algebra. Our first approach to recognisability is via finite index of syntactic congruences; the latter may be defined for any kind of algebra. We consider how to define syntactic congruences in the most efficient way: absolutely, or with regard to a particular class of algebras or languages. We give examples where only finitely many terms are needed to determine syntactic congruences. For a particular class of free algebras we find an infinite list of terms, each built from the previous, and give an example of a language such that we need to check terms of every kind. Using syntactic congruences we consider closure properties of recognisable languages. We give many examples, including critical examples of languages that are themselves free algebras (in some sense) but are contained in the free inverse monoid. Our second approach is in the context of unary monoids. We introduce a new kind of formal machine we call a +-automaton. Our main result in this regard is to show that a language over a free unary monoid has syntactic congruence of finite index if and only if it is recognised by a +-automaton. This result exactly parallels the well known result for languages over free monoids