14 research outputs found
Free-algebra functors from a coalgebraic perspective
Given a set of equations, the free-algebra functor
associates to each set of variables the free algebra over
. Extending the notion of \emph{derivative} for an arbitrary set
of equations, originally defined by Dent, Kearnes, and Szendrei, we
show that preserves preimages if and only if , i.e. derives its derivative . If weakly
preserves kernel pairs, then every equation gives rise to a
term such that and . In
this case n-permutable varieties must already be permutable, i.e. Mal'cev.
Conversely, if defines a Mal'cev variety, then weakly
preserves kernel pairs. As a tool, we prove that arbitrary endofunctors
weakly preserve kernel pairs if and only if they weakly preserve pullbacks
of epis
Duality of Equations and Coequations via Contravariant Adjunctions
International audienceIn this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve the duality between equations and coequations for automata proved by Ballester-Bolinches, Cosme-Llópez and Rutten, and generalize it to dynamical systems