Finitary $\mathcal{M}$ -Adhesive Categories

Abstract

AbstractFinitary \(\mathcal{M}\)-adhesive categories are \(\mathcal{M}\)-adhesive categories with finite objects only, where the notion \(\mathcal{M}\)-adhesive category is short for weak adhesive high-level replacement (HLR) category. We call an object finite if it has a finite number of \(\mathcal{M}\)-subobjects. In this paper, we show that in finitary \(\mathcal{M}\)-adhesive categories we do not only have all the well-known properties of \(\mathcal{M}\)-adhesive categories, but also all the additional HLR-requirements which are needed to prove the classical results for \(\mathcal{M}\)-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 \(\mathcal{M}\)-adhesive categories have a unique \(\mathcal{E}\)-\(\mathcal{M}\) factorization and initial pushouts, and the existence of an \(\mathcal{M}\)-initial object implies in addition finite coproducts and a unique \(\mathcal{E'}\)-\(\mathcal{M'}\) pair factorization. Moreover, we can show that the finitary restriction of each \(\mathcal{M}\)-adhesive category is a finitary \(\mathcal{M}\)-adhesive category and finitariness is preserved under functor and comma category constructions based on \(\mathcal{M}\)-adhesive categories. This means that all the classical results are also valid for corresponding finitary \(\mathcal{M}\)-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-\(\mathcal{M}\)-adhesive categories.KeywordsResource Description FrameworkGraph TransformationCritical PairUnique MorphismPair FactorizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.