Free-lattice functors weakly preserve epi-pullbacks

Abstract

Suppose p(x, y, z) and q(x, y, z) are terms. If there is a common “ancestor” term \(s(z_{1},z_{2},z_{3},z_{4})\) specializing to p and q through identifying some variables

$$\begin{aligned} p(x,y,z)&\approx s(x,y,z,z)\\ q(x,y,z)&\approx s(x,x,y,z), \end{aligned}$$

then the equation

$$\begin{aligned} p(x,x,z)\approx q(x,z,z) \end{aligned}$$

is a trivial consequence. In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, a converse is true, too. Given terms p, q,  and an equation

where \(\{u_{1},\ldots ,u_{m}\}=\{v_{1},\ldots ,v_{n}\},\) there is always an “ancestor term” \(s(z_{1},\ldots ,z_{r})\) such that \(p(x_{1},\ldots ,x_{m})\) and \(q(y_{1},\ldots ,y_{n})\) arise as substitution instances of s,  whose unification results in the original equation (\(*\)). In category theoretic terms the above proposition, when restricted to lattices, has a much more concise formulation:Free-lattice functors weakly preserve pullbacks of epis. Finally, we show that weak preservation is all that we can hope for. We prove that for an arbitrary idempotent variety \({{\mathcal {V}}}\) the free-algebra functor \(F_{{\mathcal {V}}}\) will not preserve pullbacks of epis unless \({{\mathcal {V}}}\) is trivial (satisfying \(x\approx y\)) or \({{\mathcal {V}}}\) contains the “variety of sets” (where all operations are implemented as projections).