Tue, 12 Dec 2023, 1:25 pm MT
Universal Coalgebra is a general theory of state based systems, encompassing all sorts of automata, transition systems, Kripke structures, etc., be they deterministic, non-deterministic, probabilistic, second order, or else. This diversity of examples is covered by choosing as signature $F$ of the mentioned coalgebras an appropriate Set-Functor, rather than just a sequence of arities, like in Universal Algebra.
Preservation properties of that signature functor $F$ determine much of the structure of the corresponding classes of coalgebras. Of particular interest, it turns out, is the question, whether F preserves preimages or (weak) pullbacks.
This provides only the backdrop and the motivation for us to study a class of functors arising from classical Universal Algebra as "free-algebra-functors" where $F_V(X)$ is taken as the base set of the free algebra over $X$ in a variety $V$. We show amongst other things, that $F_V(-)$ preserves preimages if and only if $V$ can be defined by essentially balanced (also called regular) equations. Regarding preservation of kernel pairs, the picture is still fragmented. For instance, if $V$ is a Malcev variety then $F_V(-)$ weakly preserves kernel pairs, and if $V$ is n-permutable and $F_V(-)$ weakly preserves kernel pairs, then $V$ is Malcev.
In joint work with R. Freese we study the case when $V$ is a variety $L$ of lattices, with further order preserving operations permitted. It turns out that $F_L(-)$ always weakly preserves kernel pairs. In purely lattice theoretical terms, this shows that in any lattice variety $L$ any balanced equation
$p(u_1,\ldots,u_m) = q(v_1,\ldots,v_n)$, where $\{u_1,\ldots,u_m\} = \{v_1,\ldots,v_n\}$.
must have an obvious reason, namely that both $p$ and $q$ are substitution instances of a common "ancestor"-term $s$, from which the said equation arises by further identification of variables, resulting in syntactically identical terms. For arbitrary nontrivial idempotent varieties $V$ of algebras we additionally show, using Olšák's theorem, that (strong) preservation of pullbacks is impossible.
Free algebra functors are special instances of copower functors which we can define in any concrete and cocomplete category. For instance, if $M$ is a (commutative) monoid, the functor $M[X]$ arising as $X$-fold sum of $M$, preserves preimages iff $M$ is positive, and weakly preserves pullbacks iff $M$ is (refinable) equidivisible.
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