Finitary M-adhesive categories. (26th June 2014) - GABRIEL, KARSTEN; BRAATZ, BENJAMIN; EHRIG, HARTMUT; GOLAS, ULRIKE
<x content-type="archive" xml:space="preserve">Abstract</x> Finitary <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories are <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories with finite objects only, where <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-subobjects. In this paper, we show that in finitary <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories have a unique <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3p6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[E]]> </alternatives>-<alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives> factorisation and initial pushouts, and the existence of an <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-initial object implies we also have finite coproducts and a unique <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3p6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[E]]> </alternatives>′-<alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives> pair factorisation. Moreover, we can show that the finitary restriction of each <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive category is a finitary <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive category, and finitarity is preserved under functor and comma category constructions based on <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories. This means that all the classical results are also valid for corresponding finitary <alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-<alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[M]]> </alternatives>-adhesive categories.