2021/01/04 by Adamek, Jiri, Chen, Liang-Ting, Milius, Stefan +1 · 1 citation
#Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL)
paper · doi:10.48550/arxiv.2101.00942
Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman's theorem states that they precisely specify pseudovarieties, i.e.~classes of finite algebras closed under finite products, subalgebras and quotients. In this paper, Reiterman's theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T-algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad associated to the monad T, which gives a categorical view of the construction of the space of profinite terms.