The Theory of Representations for Boolean Algebras

Stone, M. H. (1936). The Theory of Representations for Boolean Algebras. Transactions of the American Mathematical Society, 40(1), 37-111.

Type

Journal article

Intellectual base

Primary research

Year

1936

DOI

10.1090/s0002-9947-1936-1501865-8

Link

https://doi.org/10.1090/S0002-9947-1936-1501865-8

Cited from

mathematics

Cited by

5 citations across 5 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Associativity
  • Formal frameworks for associativity appear in universal algebra (Birkhoff (1935) on the structure of abstract algebras codifies associative laws as identities ), Stone duality for Boolean algebras (Stone (1936) pairs Boolean algebras — whose join and meet operations are associative — with their dual topological spaces)

    This sourceEstablishes the Stone representation theorem, pairing Boolean algebras (whose join and meet are associative) with their dual Stone spaces

• Completeness
  • Stone's 1937 representation theorem establishes that every Boolean algebra is isomorphic to a field of sets, providing a completeness-style universal representation result for Boolean lattices.

    This sourceStone's representation theorem, establishing that every Boolean algebra is isomorphic to a field of sets — a completeness-style universal representation result for Boolean lattices.

• Duality
  • … hard in primal form can be recast and solved in dual form (where the Lagrangian dual and the LP dual are the canonical industrial cases), logical operations in one vocabulary (∧, ∀) translate mechanically to the other (∨, ∃) via De Morgan's laws, topological and algebraic categories (Boolean algebras ↔ Stone spaces

    This sourcePairs Boolean algebras with Stone spaces (totally disconnected compact Hausdorff spaces). Follow-up: "Applications of the Theory of Boolean Rings to General Topology." Trans. AMS 41, no. 3 (1937): 375–481. Modern treatment: Johnstone, Stone Spaces (Cambridge UP, 1982).

Domain-specific

• Boolean algebra
  • … between constitutive identity and a convenient observable transfers from A canonical example satisfies the full defining rule of Boolean algebra with assumptions and conventions explicit. to A careful use of Boolean algebra tests the constitutive rule and nearest confusable rather than relying on the label alone..
• Stone duality
  • Classical Stone duality is the contravariant equivalence between Boolean algebras and Stone spaces, where a Boolean algebra maps to its compact Hausdorff zero-dimensional ultrafilter space and a Stone space maps to its Boolean algebra of clopen subsets.

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• https://doi.org/10.2307/1989664 ×1

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