Boolean Dimension, Components and Blocks

Abstract

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if $\dim (C)\le d$ for every component C of a poset P, then $\dim (P)\le \max \limits \{2,d\}$; also if $\dim (B)\le d$ for every block B of a poset P, then $\dim (P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) = d for every component C of a poset P, then ldim(P) = d + 2; however, for every d = 4, there exists a poset P with ldim(P) = d and $\dim (B)\le 3$ for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) = d for every component C of P, then bdim(P) = 2 + d + 4 · 2^d; also if bdim(B) = d for every block of P, then bdim(P) = 19 + d + 18 · 2^d.