Finitary M-Adhesive Categories
Karsten Gabriel, Benjamin Braatz, Hartmut Ehrig, Ulrike Golas · 2010
Finitary M-adhesive categories are M-adhesive categories with finite objects only, where the notion M-adhesive category is short for weak adhesive high-level replacement (HLR) category. We call an object finite if it has a finite number of M-subobjects. In this paper, we show that in finitary M-adhesive categories we do not only have all the well-known properties of M-adhesive categories, but also all the additional HLR-requirements which are needed to prove the classical results for M-adhesive systems. These results are the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension, and Local Confluence Theorems, where the latter is based on critical pairs. More precisely, we are able to show that finitary M-adhesive categories have a unique E-M factorization and initial pushouts, and the existence of an M-initial object implies in addition finite coproducts and a unique E'-M' pair factorization. Moreover, we can show that the finitary restriction of each M-adhesive category is a finitary M-adhesive category and finitariness is preserved under functor and comma category constructions based on M-adhesive categories. This means that all the classical results are also valid for corresponding finitary M-adhesive systems like several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-M-adhesive categories.