abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

, a monadic Boolean algebra is an

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

''A'' with

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...

:⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨''A'', ·, +, ', 0, 1⟩ is a

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

. The monadic/

unary operator In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...

∃ denotes the

existential quantifier In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...

, which satisfies the identities (using the received

prefix A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. Particul ...

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

∀, the

universal quantifier In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...

, 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''.

Discussion

Monadic Boolean algebras have an important connection to

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

. If ∀ is interpreted as the

interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are d ...

of topology, (1)–(3) above plus the axiom ∀(∀''x'') = ∀''x'' make up the axioms for an

interior algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordin ...

. But ∀(∀''x'') = ∀''x'' can be proved from (1)–(4). Moreover, an alternative axiomatization of monadic Boolean algebras consists of the (reinterpreted) axioms for an

, plus ∀(∀''x'')' = (∀''x'')' (Halmos 1962: 22). Hence monadic Boolean algebras are the

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

interior/ closure algebras such that: *The universal (dually, existential) quantifier interprets the interior ( closure) operator; *All open (or closed) elements are also

clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...

. A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(''x''∨∀''y'') = ∀''x''∨∀''y'' (Halmos 1962: 21). This axiomatization obscures the connection to topology. Monadic Boolean algebras form a

variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...

. They are to monadic predicate logic what

s are to

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

, and what polyadic algebras are to

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...

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 Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...

. 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. Likewise, monadic Boolean algebras supply the algebraic semantics for ''S5''. Hence S5-algebra is a

synonym A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are al ...

for monadic Boolean algebra.

References

, 1962. ''Algebraic Logic''. New York: Chelsea. * ------ and Steven Givant, 1998. ''Logic as Algebra''. Mathematical Association of America. Algebraic logic Boolean algebra Closure operators {{logic-stub

Discussion

See also

References