Quasi-identities and quasiverbal subalgebras (Q1585631)

The author studies classes \(\mathcal L\) of universal algebras with a constant term \(c(x)= c(y)\) for all elements \(x,y\) and \( f(c(x),\dots,c(x))= c(x) = c \) for every basic operation \(f\). Given quasivarieties \(\mathcal {U, V}\), \({\mathcal U}*_{\mathcal L} {\mathcal V}\) is the class of all algebras \( A \in \mathcal L\) such that the subalgebra \(c\theta \in \mathcal U\) where \(\theta \) is the smallest congruence on \(A\) such that \(A/\theta \in \mathcal V\). Given a basis of quasi-identities for the quasivarieties \(\mathcal {U, V}\), the author exhibits a basis for the quasi-identities of the quasivariety \(\mathcal U * \mathcal V\). The author applies this result to the case when \(\mathcal L\) is the class of lattice ordered groups.