Heyting algebra en · NOUN
Etymology
After Dutch mathematician Arend Heyting, who developed the theory as a way of modelling his intuitionistic logic.
Meanings
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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).
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.
which is just obvious.
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<b imply b=1The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.
1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Heyting algebras | Number=Plur | lexicographic |
Deriveds
complete Heyting algebra · coHeyting algebra · bi-Heyting algebra · co-Heyting algebra · biHeyting algebra
Hypernyms
bicartesian closed category (bounded lattice) · distributive lattice (bounded lattice) · residuated lattice (bounded lattice)
Hyponyms
complete Heyting algebra (bounded lattice) · finite distributive lattice (bounded lattice) · Boolean algebra (bounded lattice)
Relateds
relative pseudo-complement · pseudo-complement · Heyting prealgebra · residuated lattice
Synonyms
pseudo-Boolean algebra (bounded lattice)
Translations (1)
it algebra di Heyting (bounded lattice equipped with operation called implies)