Free-Algebra Functors from a Coalgebraic Perspective
Abstract
We continue our study of free-algebra functors from a coalgebraic perspective as begun in [8]. Given a set Σ
of equations and a set X of variables, let FΣ(X)
be the free Σ−
algebra over X and V(Σ)
the variety of all algebras satisfying Σ.
We consider the question, under which conditions the Set-functor FΣ
weakly preserves pullbacks, kernel pairs, or preimages [9].We first generalize a joint result with our former student Ch. Henkel, asserting that an arbitrary Set−
endofunctor F weakly preserves kernel pairs if and only if it weakly preserves pullbacks of epis.By slightly extending the notion of derivative Σ′
of a set of equations Σ
as defined by Dent, Kearnes and Szendrei in [3], we show that a functor FΣ
(weakly) preserves preimages if and only if Σ
implies its own derivative, i.e. Σ⊢Σ′
, which amounts to saying that weak independence implies independence for each variable occurrence in a term of V(Σ)
. As a corollary, we obtain that the free-algebra functor will never preserve preimages when V(Σ)
is congruence modular.Regarding preservation of kernel pairs, we show that for n-permutable varieties V(Σ),
the functor FΣ
weakly preserves kernel pairs if and only if V(Σ)
is a Mal’cev variety, i.e. 2-permutable.
Domains
Origin | Files produced by the author(s) |
---|