We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon $\mathcal{M}$-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.

翻译：我们把最近在规则代数框架文献中研究的组合组合重写概念扩大到以条件重写规则,我们的方法具有分类理论性质,规则组合作业的定义将非决定性的先后并存规则同时适用于双普什图(DPO)和塞斯基普什图(SqPO),根据$\mathcal{M}$-adheive类别改写适用条件。我们发现了规则和形态学条件的分类理论概念、某些转变和运输结构的构成性和兼容性以及规则构成的关联性特性三者之间错综复杂的相互作用。