Proonite Categories, Implicit Operations and Pseudovarieties of Categories

The last decade has seen two methodological advances of particular direct import for the theory of nite monoids and indirect import for that of rational languages. The rst has been the use of categories (considered as \algebras over graphs") as a framework in which to study monoids and their homomorphisms, the second has been the use of implicit operations to study pseudovarieties of monoids. Still more recent work has emphasized the r^ ole of prooniteness in nite monoid theory. This paper fuses these three topics by means of a general study of proonite categories, with applications to C-varieties (pseudovarieties of categories) in general, to those C-varieties arising from M-varieties (pseu-dovarieties of monoids) in particular, to implicit operations on categories and to recognizable languages over graphs. Before discussing the contents of the paper in detail, we provide some brief background on each of the three topics mentioned above. In an expository paper 21], B. Tilson argued persuasively that (\small") categories, considered not as classifying constructs, but as algebras over graphs (see x1), have an important r^ ole to play in monoid (and semigroup) theory. Developing earlier work by I. Simon, R. Knast, D. Th erien and others, he introduced the \derived category" of a monoid morphism and through his \Derived Category Theorem" showed that this category, and categories in general, play a central part in the study of semidirect products of monoids and semidirect products of M-varieties, with applications to varieties of languages. Since his paper, these techniques have been exploited in many directions, including \general" semigroup theory. Further arguments have been made, by D. Th erien in particular, ((18], 19]) for a more direct connection with language theory, by means of languages over graphs, where the inputs are no longer \linear" words but \branching" paths in graphs. (See x10). The theory of implicit operations on monoids (and on semigroups, indeed on universal algebras) was initiated by J. Reiterman 16] and developed extensively by J. Almeida (and A. Azevedo) in a series of papers. The papers of relevance to this one