Introduction

Abstract

The motivation for the work in this monograph was to explicate the connection between graphs and monoids dictated by the catalan construction which the author introduced in [2]. The resulting work comprises two papers.Paper I is a conceptual framework in which to study monoids of endomorphisms (concrete monoids). Paper II introduces the category of automata and refinements. We formulate the catalan construction as two adjunctions, the first between the categories of automata and concrete monoids, the second between the categories of automata and graphs. The adjunctions each produce a monad on the category of automata. These we interpret as closure operators with respect to notions familiar from computer science, namely, macros and overloading.A very widely studied monoid is the monoid On of endomorphisms of a linear order of length n. In [1] Aizenstat gave a presentation for On using the generating set consisting of maps [n] → [n] which fix all points but one. Put another way, one may consider the chain as a directed graph with n vertices and an edge from i to i +1 and another from i +1 to i, for each 0<= i < n. Then the generators are those maps which fix all vertices but one and move that vertex along an edge. In [2] such a map is called an elementary transition of the directed graph, and the monoid generated by the elementary transitions of a directed graph is referred to as the catalan monoid of the graph. A number of results relating the structure of graphs to the algebraic properties of their catalan monoids have been found in [2] and [3] but no general pattern has been established.Therefore we adopt the technique of putting our constructions in a categorical framework in the hope that the constraints so imposed would illuminate the nature of the catalan relationship between monoids and graphs. Paper II is the result of this work.By analysing the intuition behind the catalan construction, we came to realize that it is a process in two steps. Given a graph G, the first step is to form a canonical automaton which has state graph G, the second step is to take the action monoid of this automaton.In the process of these investigations into automata, technical difficulties forced us to give careful consideration to the relationship of division between concrete monoids as distinct from division of abstract monoids. A number of examples were found which illustrate the fact that this relationship is common in mathematics and we indicate how it may be of use in studying concrete monoids, and also in making constructions of concrete monoids functorial. This is the subject of Paper I.This monograph presupposes no mathematial knowledge beyond the definitions and elementary notions in the theories of semigroups and categories. As such it is accessible to any beginning graduate student, and will be of particular interest to workers in the fields of automata and semigroups and to those interested in the applications of category theory. NoteIn [2] the catalan monoid of a graph is defined to be the monoid of transformations of the vertex set generated by elementary transitions which fix all but a single point and move that point along an edge. In Paper II, the catalan automaton has letters (generators) which move any number of points, but only along a single edge. It has become clear that the latter definition of the generating set is more meaningful and has a universal property.Significantly, the two definitions give rise to the same transformation monoid when the graph is finite and has no directed cycles. In particular, the catalan monoid of a finite directed tree is precisely the action monoid of its catalan automaton, and a presentation for this monoid is found in [2].