We study matrix identities involving multiplication and unary operations such as transposition or Moore–Penrose inversion. We prove that in many cases such identities admit no finite basis.
Varieties whose algebras have no idempotent element were characterized by B. Csákány by the property that no proper subalgebra of an algebra of such a variety is a congruence class. We simplify this result for permutable varieties and we give a local version of the theorem for varieties with nullary operations.