Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite

Eilenberg Theorems for Free

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words

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

A systematic study of models of abstract data types

AbstractThe term-generated models of an abstract data type can be represented by congruence relations on the term algebra. Total and partial heterogeneous algebras are considered as models of hierarchical abstract data types.Particular classes of models are studied and it is investigated under which conditions they form a complete lattice. This theory allows also to describe programming languages (and their semantic models) by abstract types. As example we present a simple deterministic stream processing language