Heyting algebra
Noun

Heyting algebra (plural Heyting algebras)

  1. (algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (ab)∧ab and, moreover, that x = ab is the greatest element such that xab (in the sense that if cab then cab).
    • 1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press, page 13 ↗,
      Proposition 1.2.14 should certainly be completed by the observation that the modus ponens holds as well in every Heyting algebra. Since, in the intuitionistic propositional calculus, being a true formula is being a terminal object (see proof of 1.1.3), the modus ponens of a Heyting algebra reduces to
      a=1  and  a\le b  imply  b=1
      which is just obvious.
    • 1997, J. G. Stell, M. W. Worboys, The Algebraic Structure of Sets and Regions, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference, Proceedings, Springer, LNCS 1329, page 163 ↗,
      The main contention of this paper is that Heyting algebras, and related structures, provide elegant and natural theories of parthood and boundary which have close connections to the above three ontologies.
Synonyms
  • (bounded lattice) pseudo-Boolean algebra
Related terms
  • Heyting prealgebra
Translations
  • Italian: algebra di Heyting



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