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: - ∃0 = 0
- ∃
*x* ≥ *x* - ∃(
*x* + *y*) = ∃*x* + ∃*y*; - ∃
*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 univeral quantifier satisfies the identities: - ∀1 = 1
- ∀
*x* ≤ *x* - ∀(
*xy*) = ∀*x*∀*y*; - ∀
*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. |