Free-lattice functors weakly preserve epi-pullbacks

Abstract

AbstractSupposep(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$${{\mathcal {V}}}$$Vthe free-algebra functor$$F_{{\mathcal {V}}}$$FVwill notpreservepullbacks of epis unless$${{\mathcal {V}}}$$Vis trivial (satisfying$$x\approx y$$x≈y) or$${{\mathcal {V}}}$$Vcontains the “variety of sets” (where all operations are implemented as projections).