A property of the solvable radical in finitely decidable varieties (Q2773238)

A class \(K\) of algebras is said to be finitely definable if the first-order theory of the class of finite members of \(K\) is decidable. In the paper a list of conditions is produced which are shown to be both necessary and sufficient for a finitely generated congruence modular variety to be finitely definable. One of these conditions is \((\dag)\): the centralizer of the monolith of any finite subdirectly irreducible algebra \(A\) in the variety is comparable to every congruence of \(A\). The necessity of \((\dag)\) is proved under the assumption of finite decidability for congruence modular varieties. For such a variety \(K\) the only types that can appear in the finite algebras in \(K\) are \(2\), \(3\) and \(4\). If \(A\) is a finite irreducible algebra whose monolith \(\mu\) has type \(3\) or \(4\) then \((\dag)\) automatically holds. Thus, the only interesting cases are when the type of \(\langle\text{O}_A,\mu\rangle\) is \(1\) or \(2\). The authors prove the type \(2\) case and that in the type \(1\) case the centralizer of the monolith is strongly solvable.