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 prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities:
- ∃0 = 0
- ∃x ≥ x
- ∃(x + y) = ∃x + ∃y;
- ∃x∃y = ∃(x∃y)
∃x is the existential closure of x. Dual to &exist is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x' )'.
Monadic Boolean algebras have a dual formulation that takes &forall' as primitive and ∃ as defined, so that ∃x := (∀x ' ) . Hence the algebra dual to a monadic Boolean algebra has signature ⟨A, ·, +, ', 0, 1, ∀⟩ where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra and ∀ satisfies the duals of the above identities, namely (1)-(4) below:
- ∀1 = 1
- ∀x ≤ x
- ∀(xy) = ∀x∀y;
- ∀x + ∀y = ∀(x + ∀y)
∀x is the universal closure of x.
Discussion
Monadic Boolean algebras are a special case of interior algebras, with the universal (existential) quantifier interpreting the interior (closure) operators. Hence monadic Boolean algebras are interior algebras in which every open element is closed (and dually, every closed element is open). Hence monadic Boolean algebras form a variety and are 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 algebras while working on polyadic algebras; Halmos (1962) reprints the relevant papers. See Halmos and Givant (1998) for a gentler treatment.
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.
See also
References
- Paul Halmos, 1962. Algebraic Logic. New York: Chelsea.
- ------ and Steven Givant, 1998. Logic as Algebra. Mathematical Association of America.