Horn sentences with (W) and weak Mal'cev conditions
A second order sentence \(U:(\exists f_ 1,...,f_ m)(\forall x_ 1,...,x_ n)M\) where M is a finite conjunction of equations in function variables \(f_ 1,...,f_ m\) and individual variables \(x_ 1,...,x_ n\) is called a strong Mal'cev condition. Any disjunction of strong Mal'cev conditions \(U_ n\), \(n\geq 1\), where \(U_ n\) implies \(U_{n+1}\) for all n, is called a Mal'cev condition. A countably (continuously, resp.) weak Mal'cev condition is a conjunction of countably (at most continuously, resp.) many Mal'cev conditions. A universal lattice Horn sentence (i.e. a first order sentence \(\chi:(\forall\alpha_ 0,...,\alpha_ t)((\bigwedge^{k}_{i=1}p_ i=q_ i)\Rightarrow p\leq q)\) where \(p_ i,q_ i,p\), and q are lattice terms of the variables \(\alpha_ 0,...,\alpha_ t\) and \(k\geq 0)\) is said to satisfy (W) if the finitely presented lattice \(FL(\{\alpha_ 0,...,\alpha_ t\};p_ i(\alpha_ 0,...,\alpha_ t)=q_ i(\alpha_ 0,...,\alpha_ t),\quad i\leq k)\) satisfies the Whitman condition. The paper solves a problem of B. Jónsson whether for any universal lattice Horn sentence \(\chi\) there exists a countably weak Mal'cev condition which holds in a variety \({\mathcal V}\) of algebras iff \(\chi\) is satisfied in Con \({\mathfrak A}\), \({\mathfrak A}\in {\mathcal V}\). The authors present an algorithm which associates appropriate continuously weak Mal'cev conditions with Horn sentences satisfying (W). Applying the main theorem the classical Pixley-Wille algorithm as well as a simpler version of the first author's former result are easily obtained.
scientific article; zbMATH DE number 3875361