Congruence semimodular varieties. II: Regular varieties (Q1337154)

[Part I is reviewed above.] An equation \(p= q\) is regular (or normal) if \(p\) and \(q\) have the same free variables. A variety \(V\) of algebras is regular if it can be axiomatized by regular equations. If \(A\) is an algebra of a regular variety \(V\) and \(p\) a unary polynomial of \(A\), then \(p\) is called permissible if there is an \((n+1)\)-ary term \(t\) which depends on all of its variables in \(V\) and an \(n\)-tuple \(\overline a\in A^ n\) such that \(p(x)= t^ A(x,\overline a)\). Define \(\prec_ A\) as the transitive closure of \(\{\langle p(a), a\rangle| a\in A\) and \(p\) a permissible polynomial of \(A\}\) and \(\approx_ A\) as the symmetric part of the quasi-order \(\prec_ A\). The main result of the paper under review states that a regular variety \(V\) is congruence semimodular iff for all \(A\in V\) and all nonzero \(\alpha\in \text{Con }A\) one has \(\alpha\not\subseteq\approx_ A\).