Notes on Free Monadic Boolean Algebras

If F B(2n − 1) denotes the Boolean algebra with 2n − 1 free generators and P(2n) is the Cartesian product of 2n Boolean algebras all equal to F B(2n − 1), we define on P(2n) an existential quantifier ∃ by means of a relatively complete Boolean subalgebra of P(2n) and we prove that (P(2n),∃) is the monadic Boolean algebra with n free generators. Every element of P(2n) is a 2n-tuple whose coordinates are in F B(2n − 1); in particular, so are the n generators of P(2n). We indicate in this work the coordinates of the n generators of P(2n).