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We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO + , allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO + can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO + can define all rewrite relations definable by PBPO, AGREE and DPO, as well as additional ones. Additionally, we show that PBPO + is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.

Quasitoposes have proven to be an interesting framework for many graph rewriting formalisms. Since presheaves categories are toposes and fuzzy sets are quasitoposes, we prove that the combination of both, fuzzy presheaves, are also quasitoposes. This question was recently conjectured more specifically for fuzzy graphs and is now therefore proven. This entails that many fuzzy presheaves of interest can be used for graph rewriting.

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Monads and their composition via distributive laws have many applications in program semantics and functional programming. For many interesting monads, distributive laws fail to exist, and this has motivated investigations into weaker notions. In this line of research, Petrişan and Sarkis recently introduced a construction called the semifree monad in order to study semialgebras for a monad and weak distributive laws. In this paper, we prove that an algebraic presentation of the semifree monad M s on a monad M can be obtained uniformly from an algebraic presentation of M . This result was conjectured by Petrişan and Sarkis. We also show that semifree monads are ideal monads, that the semifree construction is not a monad transformer, and that the semifree construction is a comonad on the category of monads.