A GLOBAL CONVERGENCE THEOREM IN BOOLEAN ALGEBRA

Abstract

Robert has established a global convergence theorem in $\{0,1\}^{n}$: If a map $\hat{F}$ from $\{0,1\}^{n}$ to itself is contracting relative to the boolean vector distance $d$, then there exists a positive integer $p \leq n$ such that $\hat{F}^{p}$ is constant. In other words, $\hat{F}$ has a unique fixed point $\xi$ such that for any $x$ in $\{0,1\}^{n}$, we have $\hat{F}^{p}(x) = \xi$. The structure ($\{0,1\},+,\cdot,-,0,1$) may be regarded as the two-element boolean algebra. In this paper, this result is extended to any map $F$ from the product $X$ of $n$ finite boolean algebras to itself.