Ralph Freese

Supposep(x, y, z) andq(x, y, z) are terms. If there is a common “ancestor” term $$s(z_{1},z_{2},z_{3},z_{4})$$s(z1,z2,z3,z4) specializing topandqthrough identifying some variables $$\begin{aligned} p(x,y,z)&\approx s(x,y,z,z)\\ q(x,y,z)&\approx s(x,x,y,z), \end{aligned}$$p(x,y,z)≈s(x,y,z,z)q(x,y,z)≈s(x,x,y,z), then the equation $$\begin{aligned} p(x,x,z)\approx q(x,z,z) \end{aligned}$$p(x,x,z)≈q(x,z,z) 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 termsp, q, and an equation where $$\{u_{1},\ldots ,u_{m}\}=\{v_{1},\ldots ,v_{n}\},$${u1,…,um}={v1,…,vn}, there is always an “ancestor term” $$s(z_{1},\ldots ,z_{r})$$s(z1,…,zr) such that $$p(x_{1},\ldots ,x_{m})$$p(x1,…,xm) and $$q(y_{1},\ldots ,y_{n})$$q(y1,…,yn) arise as substitution instances ofs, 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 pullbacksof epis. Finally, we show thatweakpreservation is all that we can hope for. We prove that for an arbitrary idempotent variety $${{\mathcal {V}}}$$V the free-algebra functor $$F_{{\mathcal {V}}}$$FV will notpreservepullbacks of epis unless $${{\mathcal {V}}}$$V is trivial (satisfying $$x\approx y$$x≈y ) or $${{\mathcal {V}}}$$V contains the “variety of sets” (where all operations are implemented as projections).