A 4-basis for Boolean algebra (BA) in terms of join (v), meet (^), and complement (c).

x v (y v z) = y v (x v z). % AJ x ^ y = c(c(x) v c(y)). % DM x v c(x) = y v c(y). % ONE (x v c(y)) ^ (x v y) = x. % CUTHere are proofs of distributivity, modularity, CC, and B1 from the BA 4-basis.

otter < BA1.in > BA1.out otter < BA2.in > BA2.out otter < BA3.in > BA3.outIndependence of the BA 4-basis is open. In particular, we have not been able to find a proof or countermodel of

{ AJ, DM, CUT } => ONE.The simplest multiequation basis we know of for BA in terms of join and complement is the following, due to C.A. Meredith [13].

c(c(x) v y) v x = x. % MER_1 c(c(x) v y) v (z v y) = y v (z v x). % MER_2The Robbins 3-basis for BA (in terms of join and complement) is the following [12].

(x v y) v z = x v (y v z). % AJ2 x v y = y v x. % CJ c(c(x v c(y)) v c(x v y)) = x. % Robbins