Finitary $\mathcal{M}$-adhesive categories. (26th June 2014)

• Record Type: Journal Article
• Authors: GABRIEL, KARSTEN; BRAATZ, BENJAMIN; EHRIG, HARTMUT; GOLAS, ULRIKE
• Is Part Of: Mathematical structures in computer science. Volume 24:Number 4(2014)
• Publication Date: 2014-06-26
• Subjects: Computer science -- Mathematics -- Periodicals; 004.015105
• DOI: 10.1017/S0960129512000321
• Languages: English
• ISSNs: 0960-1295

Abstract

<abstract abstract-type="normal"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>Finitary <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories are <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories with finite objects only, where <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-subobjects. In this paper, we show that in finitary <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories have a unique <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3p6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{E}$]]></tex-math></alternatives></inline-formula>-<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula> factorisation and initial pushouts, and the existence of an <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-initial object implies we also have finite coproducts and a unique <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3p6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{E}$]]></tex-math></alternatives></inline-formula>′-<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula> pair factorisation. Moreover, we can show that the finitary restriction of each <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive category is a finitary <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive category, and finitarity is preserved under functor and comma category constructions based on <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories. This means that all the classical results are also valid for corresponding finitary <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-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-<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghmrxd3j4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{M}$]]></tex-math></alternatives></inline-formula>-adhesive categories.</p> </abstract>