In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature
⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,
where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.
The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃):
∃0 = 0
∃x ≥ x
∃(x + y) = ∃x + ∃y
∃x∃y = ∃(x∃y).
∃x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x′)′.
A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that ∃x := (∀x′)′. (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra A has signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities:
∀1 = 1
∀x ≤ x
∀(xy) = ∀x∀y
∀x + ∀y = ∀(x + ∀y).
∀x is the universal closure of x.
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