On Local Properties of Unary Algebras

摘要

If every finitely generated subalgebra of an algebra A has a property P, then A is said to have P as a local property. In this paper we study classes L(K) and M(K), where K is a class of unary algebras, L(K) consists of all algebras such that every finitely generated subalgebra is in K, and M(K) is the class of all algebras with every monogenic subalgebra in K. In particular, the cases where K is a variety, a generalized variety or a pseudovariety, are considered. It is also shown that the monogenic closure of a variety or a pseudovariety equals the regularization of the class. Finally, we note that some of our central concepts are derived from the theory of finite automata and hence many of the results can beinterpreted in that theory.