Homomorphisms

Abstract

A Boolean homomorphism is a mapping f from a Boolean algebra B, say, to a Boolean algebra A,such that

$$ f(p \wedge q) = f(p) \wedge f(q), $$

(1)

$$ f(p \vee q) = f(p) \vee f(q), $$

(2)

$$ f(p') = (f(p))', $$

(3)

whenever p and q are in B. In a somewhat loose but brief and suggestive phrase, a homomorphism is a structure-preserving mapping between Boolean algebras. A convenient synonym for “homomorphism from B to A” is “A-valued homomorphism on B”. Such expressions will be used most frequently in case A = 2.