Monadic Boolean Algebra

In abstract algebra, a monadic Boolean algebra is an algebraic structure with signature

〈A, ·, +, ', 0, 1, ∃〉 of type 〈2,2,1,0,0,1〉,

where 〈A, ·, +, ', 0, 1〉 is a Boolean algebra.

The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃):

• ∃0 = 0
• ∃x ≥ x
• ∃(x + y) = ∃x + ∃y
• ∃x∃y = ∃(x∃y).

∃x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x' )'.

A monadic Boolean algebra has a dual formulation that takes ∀ as primitive and ∃ as defined, so that ∃x := (∀x ' )' . Hence the dual algebra has signature 〈A, ·, +, ', 0, 1, ∀〉, with 〈A, ·, +, ', 0, 1〉 a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities:

1. ∀1 = 1
2. ∀x ≤ x
3. ∀(xy) = ∀x∀y
4. ∀x + ∀y = ∀(x + ∀y).

∀x is the universal closure of x.