Free-lattice functors weakly preserve epi-pullbacks

DOI10.1007/S00012-022-00774-5arXiv2103.09566MaRDI QIDQ6363058

Ralph Freese, H. Peter Gumm

Publication date: 17 March 2021

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 �egin{align*} p(x,y,z) & approx s(x,y,z,z)\ q(x,y,z) & approx s(x,x,y,z), end{align*} then the equation [ p(x,x,z)approx q(x,z,z) ] is trivially obtained by syntactic unification of $s(x,y,z,z)$ with $s(x,x,y,z).$ In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, the converse is true, too. Given terms $p,q,$ and an equation �egin{equation} p(u_{1},ldots,u_{m})approx q(v_{1},ldots,v_{n})label{eq:p_eq_q} end{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 preserves pullbacks of epis.

Mathematics Subject Classification ID

Equational logic, Mal'tsev conditions (08B05) Free algebras (08B20) Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads (18C15) Free lattices, projective lattices, word problems (06B25)

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