Heyting algebra

English

Etymology

After Dutch mathematician Template:W, who developed the theory as a way of modelling his Template:W.

Noun

1. Template:Lb 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 (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).
   • Template:Quote-text
   • 1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Template:W, 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, Template:W 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.

Usage notes

• The symbols for the lattice operations join (∨) and meet (∧) and for the partial order relation (≤) are reinterpreted as logical connectives: ∨ becomes or, ∧ becomes and and ≤ becomes proves (⊢).
  • Thus, (a→b)∧a ≤ b (the defining condition for →) becomes (a→b), a ⊢ b, which is modus ponens. The qualifying condition c∧a ≤ b ⇒ c ≤ a→b becomes c, a ⊢ b ⇒ c ⊢ a→b, which is the deduction theorem.
• The Template:M of a, denoted ¬a, is defined as a→0, and a→b is called the Template:M of a with respect to b
• A Heyting algebra in which a∨¬a = 1 (the law of excluded middle) is a Boolean algebra. In this sense, Heyting algebras generalise Boolean algebras, which model (propositional) classical logic.

Synonyms

• Template:Sense Template:L

Hypernyms

• Template:Sense Template:L, Template:L, Template:L

Hyponyms

• Template:Sense Template:L, Template:L, finite distributive lattice

Derived terms

• Template:L, Template:L
• Template:L, Template:L
• Template:L

Related terms

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Translations

• Italian: Template:T

See also

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Further reading

• Template:Pedia
• Template:Pedia
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• Template:Pedia
• Template:Pedia
• Heyting algebra on Template:W
• Pseudo-Boolean algebra on Template:W