Ideally Exact Categories, Varieties of Universal Algebras and Multi-valued Logics

Abstract

We study the notions of coherent and ideal actions in the setting of ideally exact varieties of universal algebras. We prove that, if V is a semi-abelian variety and U is obtained from V by freely adding nullary operations together with suitable identities, then the ideally exact context determined by the associated free–forgetful monadic adjunction with cartesian unit has a good theory of actions if and only if a compatibility condition between coherent actions and the added identities is satisfied. Our setting applies in particular to certain categories of interest in algebraic logic, including MV-algebras and product algebras.