Browsing by Author "Guo, Xiuzhan"
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- ItemOpen AccessProducts, joins, meets, and ranges in restriction categories(2012) Guo, Xiuzhan; Cockett, James Robin B.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.
- ItemOpen AccessRanges, restrictions, partial maps, and fibrations(2004) Guo, Xiuzhan; Cockett, James Robin B.In this thesis, we study range restriction categories and their properties. Range restriction categories with split restriction idempotents are shown to be equivalent to the partial map categories of ℳ-stable factorization systems. The notions of a restriction fibration, a range restriction fibration, a stable meet semilattice fibration, and a range stable meet semilattice fibration are introduced and it is shown that (range) stable meet semilattice fibrations provide a bridge between the category of (range) restriction categories and the category of categories and that (range) restriction fibrations are the same as (range) restriction categories so that these fibrations provide a useful setting for studying (range) restriction categories. Finally, we construct the free range restriction structures over directed graphs using deterministic trees.