A class \(\mathbf K\) of algebras of a similarity type \(\Delta\) is called a generalized variety if it closed under subalgebras, homomorphic images, finite products and all powers, and \(\mathbf K\) is called a pseudovariety if it consists of finite algebras and is closed under subalgebras, homomorphic images and finite products. For a class \(\mathbf K\) of algebras of a similarity type \(\Delta\), let \(L(\mathbf K)\) denote the class of all algebras \( A\) such that any finitely generated subalgebra of \(A\) belongs to \(\mathbf K\), and \(M(\mathbf K)\) denote the class of all algebras \(A\) such that any monogenic subalgebra of \(A\) belongs to \(\mathbf K\). Several results for classes \(\mathbf K\) consisting of unary algebras are presented. Namely, \(L(\mathbf K)\) and \(M(\mathbf K)\) are described for a class \(\mathbf K\) of algebras such that \(\mathbf K\) is a variety of unary algebras or a generalized variety of unary algebras or a pseudovariety of unary algebras.