Properties of the Closure Operator in Peano Algebras

Abstract

A Peano $\sigma$-algebra generated by a set $M$, denoted by $\overline M$, is a set of words over the alphabet $M \cup \{\sigma \}$ satisfying some rules. The set $M$ is the support set of $\overline{M}$. The symbol $\sigma$ is a distinguished symbol to build these words. In this paper we study several properties of the closure operator in Peano algebras. If $M_0$ is a subset of the support set $M$ then the closure operator $f$ is defined by$f(M_0)=\overline {M_0}$. The main results show that $f$ is a monotone operator under inclusion and satisfies the morphism property under intersection. Several properties concerning the layers of the Peano $\sigma$-algebras are also presented.