On pseudovarieties of monounary algebras (Q2911872)

Algebras with one unary operation are called monounary algebras. A class of finite algebras is called a pseudovariety if it is closed under subalgebras, homomorphic images and direct products of finitely many members. A pseudovariety \({\mathcal P}\) is called equational if there exists a variety \({\mathcal V}\) such that \({\mathcal P}\) consists of all finite members of \({\mathcal V}\). A constructive description of the members of all pseudovarieties of monounary algebras is given. That description uses finite products, homomorphisms and subalgebras. It is shown that every equational pseudovariety of monounary algebras is finitely generated; moreover, if this pseudovariety is not the pseudovariety of all monounary algebras, it can be generated by a single algebra.