Monadic Boolean algebra

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

<A, ·, +, ', 0, 1, ∃>

where

<A, ·, +, ', 0, 1>

is a Boolean algebra and ∃ is a unary operator, called the existential quantifier, satisfying the identities:

1. ∃0 = 0
2. ∃x ≥ x
3. ∃(x + y) = ∃x + ∃y;
4. ∃x∃y = ∃(x∃y)

∃x is called the existential closure of x. Monadic Boolean algebras play the same role for the monadic logic of quantification that Boolean algebras play for ordinary propositional logic.

The dual of the existential quantifier is the universal quantifier ∀ defined by ∀x = (∃x ' ) '. By the principle of duality, the universal quantifier satisfies the identities:

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

∀x is called the universal closure of x. The universal quantifier is recoverable from the existential quantifier via the identity ∃x = (∀x ' ) '. Thus the theory of monadic Boolean algebras may be formulated using the universal quantifier instead of the existential. In this formulation one considers algebraic structures of the form <A, ·, +, ', 0, 1, ∀> where <A, ·, +, ', 0, 1> is a Boolean algebra and ∀ satisfies the properties of a universal quantifier listed above.

Monadic Boolean algebras can be regarded as a specical case of interior algebras the interior operator is the universal quantifier and the closure operator is the existential quantifier. They are then precisely the interior algebras in which every open element is closed, and dually every closed element is open, and form a variety. These are also precisely the semisimple interior algebras. Interior algebras provide the algebraic semantics for the modal logic S4 and in this context the monadic Boolean algebras are the variety corresponding to the modal logic S5 regarded as a theory in S4'. They are thus also referred to as S5-algebras.