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.
The prefixed unary operator ∃ denotes the existential quantifier, which asatisfies the identities:
- ∃0 = 0
- ∃x ≥ x
- ∃(x + y) = ∃x + ∃y;
- ∃x∃y = ∃(x∃y)
∃x is called the existential closure of x. The dual of ∃x is ∀x := (∃x' )', where ∀ is the universal quantifier.
The dual algebra takes the universal quantifier ∀ as primitive, then defines ∃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. Hence monadic Boolean algebras may be formulated dually using the universal quantifier, as follows: ⟨A, ·, +, ', 0, 1, ∀⟩ where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra and ∀ satisfies the properties (1)-(4) above.
Discussion
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.
Monadic Boolean algebras are to monadic predicate logic what Boolean algebras are to propositional logic, and what polyadic algebras are to first order logic. Paul Halmos discovered monadic Boolean algebra while working on polyadic algebras; Halmos (1962) reprints the relevant papers.
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 supply the algebraic semantics for S5. Hence S5-algebras and "monadic Boolean algebras" are synonymous.
Reference
- Paul Halmos, 1962. Algebraic Logic. New York: Chelsea.