Algebra and coalgebra in computer science

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This work explores various advanced topics in algebraic effects, coalgebra, and their applications in semantics and software development. It begins with a discussion on adequacy for infinitary algebraic effects and moves into parameterized monads and Kleene monads, addressing iteration within a framework of generic effects. The text delves into semantics of higher-order recursion schemes and the theory of coalgebra, highlighting coalgebraic components and the complementation of coalgebra automata. It characterizes behavioral equivalence through coinduction and bisimulation, examining the relationship between different notions of bisimulation and their coalgebraic semantics. Further, the work touches on traces, executions, and schedulers, exploring non-strongly stable orders and their implications for simulation relations. Stone duality is discussed in the context of recognizable languages over algebras, alongside revisiting free Heyting algebras and distributive lattice-structured ontologies. Game theory is examined through Conway games and equilibria, while graph transformation is addressed with a focus on van Kampen colimits and unfolding grammars. Additionally, the text covers software development techniques, including behavior protocols, model-to-model transformation, and various tools such as DBtk for directed bigraphs and a behavioral verification tool based on circular coinduction, showcasing the intersection of the