In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a monadic Boolean algebra is an algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

 with signature

Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...

⟨A, ·, +, ', 0, 1, ∃⟩ of type

Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...

  ⟨2,2,1,0,0,1⟩,

where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.

The prefix

Prefix

A prefix is an affix which is placed before the root of a word. Particularly in the study of languages,a prefix is also called a preformative, because it alters the form of the words to which it is affixed.Examples of prefixes:...

ed 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

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...

 to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x' )'.

A monadic Boolean algebra has a dual

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...

 formulation that takes ∀ as primitive and ∃ as defined, so that ∃x := (∀x ' )' . Hence the dual algebra has signature ⟨A, ·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...

 version of the above identities:

1. ∀1 = 1
2. ∀x ≤ x
3. ∀(xy) = ∀x∀y
4. ∀x + ∀y = ∀(x + ∀y).

∀x is the universal closure of x.

Discussion

Monadic Boolean algebras have an important connection to topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. 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

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 ordinary propositional logic...

. 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

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 ordinary propositional logic...

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

Semisimple algebra

In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...

 interior/closure algebras such that:

• The universal (dually, existential) quantifier interprets the interior (closure
  Closure operator
  In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
  
  ) operator;
• All open (or closed) elements are also clopen
  Clopen set
  In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
  
  .

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 (universal algebra)

In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...

. They are to monadic predicate logic what Boolean algebras are to propositional logic, and what polyadic algebras are to first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

. Paul Halmos

Paul Halmos

Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...

 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

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

. The modal logic S5

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

, viewed as a theory in S4, is a model of monadic Boolean algebras in the same way that S4

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

 is a model of interior algebra. Likewise, monadic Boolean algebras supply the algebraic semantics for S5. Hence S5-algebra is a synonym

Synonym

Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...

 for monadic Boolean algebra.

See also

• monadic logic
• modal logic
  Modal logic
  Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
  
  
• interior algebra
  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 ordinary propositional logic...
  
  
• Kuratowski closure axioms
  Kuratowski closure axioms
  In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
  
  
• clopen set
  Clopen set
  In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...