Products, joins, meets, and ranges in restriction categories

Abstract

Restriction categories provide a convenient abstract formulation of partial function. However, restriction categories can have a variety of structures such as finite partial products (cartesiane s), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, and algebraic geometry. This thesis studies these structures. For finite partial products (cartesiane ) , a construction to add finite partial product to an arbitrary restriction category freely is provided. For joins, we introduce the notion of join restriction categories, describe a construction for the join completion of a restriction category, and show the completeness of join restriction categories in partial map categories using M-adhesive categories and Mgaps. As the join completion for inverse semigroups is well-known in semigroup theory we how the relationships between the join completion for restriction categories and the join completion for inverse semigroups by providing adjunctions among restriction categories, join restriction categories, inverse categories, and join inverse categories. For meet , we introduce the notion of meet restriction categories, show the completeness of meet restriction categories in partial map categories whose M-maps include the regular monies, and provide a meet completion for restriction categories and discuss its connections with the meet completion for inverse semigroups. Finally, for ranges, Schein's representation theorem for a certain class of semigroups ( called type 3 function system) is generalized to range categories and when a partial map category satisfies Schein's condition ([RR.6]) that guarantees each map is an epimorphism onto its range is studied.