Meaning of Heyting algebra | Babel Free

Definitions

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).

Equivalents

Examples

“The 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.”

“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.”

CEFR level