The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): ∃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′)′.
Moreover, ∀ satisfies the following dualized version of the above identities: ∀x is the universal closure of x. Monadic Boolean algebras have an important connection to topology.
Halmos and Givant (1998) includes an undergraduate treatment of monadic Boolean algebra.
Monadic Boolean algebras also have an important connection to modal logic.