Editing Monadic Boolean algebra

In [[abstract algebra]], a '''monadic Boolean algebra''' is an [[algebraic structure]] ''A'' with [[signature (logic)|signature]] 

:{{math|1=&lang;·, +, ', 0, 1, ∃&rang;}}   of [[signature (logic)|type]]   &lang;2,2,1,0,0,1&rang;,

where &lang;''A'', ·, +, ', 0, 1&rang; is a [[Boolean algebra (structure)|Boolean algebra]].

The [[monadic (arity)|monadic]]/[[unary operator]] ∃ denotes the [[existential quantifier]], which satisfies the identities (using the received [[prefix]] notation for ∃):
* {{math|1=∃0 = 0}}
* {{math|1=∃''x'' ≥ ''x''}}
* {{math|1=∃(''x'' + ''y'') = ∃''x'' + ∃''y''}}
* {{math|1=∃''x''∃''y'' = ∃(''x''∃''y''). }}
{{math|1=∃''x''}} is the ''existential closure'' of ''x''. [[duality (order theory)|Dual]] to ∃ is the [[unary operator]] ∀, the [[universal quantifier]], defined as {{math|1=∀''x'' := (∃''{{prime|x}}''){{prime}}}}. 

A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that {{math|1=∃''x'' := (∀''{{prime|x}}''){{prime}}}}. (Compare this with the definition of the [[duality (order theory)|dual]] Boolean algebra.) Hence, with this notation, an algebra ''A'' has signature {{math|1=&lang;·, +, ', 0, 1, ∀&rang;}}, with &lang;''A'', ·, +, ', 0, 1&rang; a Boolean algebra, as before. Moreover, ∀ satisfies the following [[duality (order theory)| dualized]] version of the above identities:
# {{math|1=∀1 = 1}}
# {{math|1=∀''x'' ≤ ''x''}}
# {{math|1=∀(''xy'') = ∀''x''∀''y''}}
# {{math|1=∀''x'' + ∀''y'' = ∀(''x'' + ∀''y'')}}.
{{math|1=∀''x''}} is the ''universal closure'' of ''x''.

==Discussion==
Monadic Boolean algebras have an important connection to [[topology]]. If ∀ is interpreted as the [[interior operator]] of topology, (1)–(3) above plus the axiom ∀(∀''x'') = ∀''x'' make up the axioms for an [[interior algebra]]. But ∀(∀''x'') = ∀''x'' can be proved from (1)–(4). Moreover, an alternative axiomatization of monadic Boolean algebras consists of the (reinterpreted) axioms for an [[interior algebra]], plus ∀(∀''x'')' = (∀''x'')' (Halmos 1962: 22). Hence monadic Boolean algebras are the [[Semisimple algebra|semisimple]] interior/[[closure algebra]]s such that:
*The universal (dually, existential) quantifier interprets the [[interior operator|interior]] ([[closure operator|closure]]) operator;
*All open (or closed) elements are also [[clopen set|clopen]].
A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(''x''&or;∀''y'') = ∀''x''&or;∀''y'' (Halmos 1962: 21). This axiomatization obscures the connection to topology.

Monadic Boolean algebras form a [[variety (universal algebra)|variety]]. They are to [[monadic logic|monadic predicate logic]] what [[Lindenbaum–Tarski algebra|Boolean algebra]]s are to [[propositional logic]], and what [[polyadic algebra]]s are to [[first-order logic]]. [[Paul Halmos]] discovered monadic Boolean algebras while working on polyadic algebras; Halmos (1962) reprints the relevant papers. Halmos and Givant (1998) includes an undergraduate treatment of monadic Boolean algebra.

Monadic Boolean algebras also have an important connection to [[modal logic]]. The modal logic [[modal logic|S5]], viewed as a theory in ''S4'', is a model of monadic Boolean algebras in the same way that [[modal logic|S4]] is a model of interior algebra. Likewise, monadic Boolean algebras supply the algebraic semantics for ''S5''. Hence '''S5-algebra''' is a [[synonym]] for monadic Boolean algebra.

==See also==
*[[Clopen set]]
*[[Cylindric algebra]]
*[[Interior algebra]]
*[[Kuratowski closure axioms]]
*[[Łukasiewicz–Moisil algebra]]
*[[Modal logic]]
*[[Monadic logic]]

==References==
* [[Paul Halmos]], 1962. ''Algebraic Logic''. New York: Chelsea.
* ------ and Steven Givant, 1998. ''Logic as Algebra''. Mathematical Association of America.

[[Category:Algebraic logic]]
[[Category:Boolean algebra]]
[[Category:Closure operators]]

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