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. % CUT
Here 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.out
Independence 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_2
The 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