For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R i...

Braided Thompson’s groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups Bn, the finitary braid group B∞ and Thompson’s group F as subgroups. We describe some of the metric properties of braided Thompson’s groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams use...

Games on graphs with ω-regular objectives provide a model for the control and synthesis of reactive systems. Every ω-regular objective can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens “eventually.” Two main strengths of the classical, infinite-limit formulation of liveness are robustness (independence from the granularity of transit...

We consider the equivalence of Lawvere theories and finitary monads on Set from the perspective of Endf (Set)-enriched category theory, where Endf (Set) is the category of finitary endofunctors of Set. We identify finitary monads with one-object Endf (Set)-categories, and ordinary categories admitting finite powers (i.e., n-fold products of each object with itself) with Endf (Set)-categories ad...

The operation V → V ω is a fundamental operation over finitary languages leading to ω-languages. It produces ω-powers, i.e. ω-languages in the form V , where V is a finitary language. This operation appears in the characterization of the class REGω of ω-regular languages (respectively, of the class CFω of context free ω-languages) as the ω-Kleene closure of the family REG of regular finitary la...

We investigate how finitary functors on Set can be extended or lifted to finitary functors on Preord and Poset and discuss applications to coalgebra.