A Boolean monoid is a structure A = (A,∨,0, ∧,1,¬,·,e) such that
∧,1,¬) is a Boolean algebra,
(A,·,e) is a monoid,
· is join-preserving in each argument: (x∨y)·z = (x·z)∨(y·z) and x·(y∨z) = (x·y)∨(x·z)
· is normal in each argument: 0·x = 0 and x·0 = 0.
Let A and B be Boolean monoids. A morphism from A to B is a function h : A→B that is a Boolean homomorphism and preserves ·, e: h(x·y) = h(x)·h(y) and h(e) = e.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property||yes|
|Definable principal congruences||no|
|Equationally definable principal congruences||no|
|Strong amalgamation property|
|Epimorphisms are surjective|