Monadic Boolean algebra
In abstract algebra, a monadic Boolean algebra is an algebraic structure having signature
- ⟨A, ·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,
where
- ⟨A, ·, +, ', 0, 1⟩
is a Boolean algebra. ∃, denoting the existential quantifier, is a prefixed unary operator 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 monadic quantifiers that Boolean algebras play for ordinary propositional logic.
The dual algebra begins with the universal quantifier ∀ defined as ∀x = (∃x ' ) '. By the principle of duality, the universal 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 monadic Boolean algebras may be formulated dually 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 (1)-(4) above.
Monadic Boolean algebras can be regarded as a special case of interior algebras, with the universal (existential) quantifier interpreting the interior (closure) operators. Monadic Boolean algebras are then interior algebras in which every open element is closed (and dually, every closed element is open). Monadic Boolean algebras thus form a variety, and are also the semisimple interior algebras.
The modal logic S5, viewed as a theory in S4, is a model of monadic Boolean algebras in the same way that S4 is a model of interior algebra. Moreover, monadic Boolean algebras provide the algebraic semantics for S5. Hence S5-algebras and "monadic Boolean algebras" are synonymous.
Monadic Boolean algebras are to monadic predicate logic what polyadic algebras are to first order logic. We owe both of these algebras Paul Halmos; his papers on the subject are reprinted in Halmos (1962).
Reference
- Paul Halmos, 1962. Algebraic Logic. New York: Chelsea.