2017/12/31 by Mészáros, Tamás, Micek, Piotr, Trotter, William T.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1801.00288
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)≤ d for every component C of a poset P, then dim(P)≤ max\2,d\; also if dim(B)≤ d for every block B of a poset P, then dim(P)≤ 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)≤ 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⋅2d; also if bdim(B)≤ d for every block of P, then bdim(P)≤ 19+d+18⋅ 2d.