Heyting algebra

Heyting algebra

A Heyting algebra is a complete, bounded lattice that satisfies the axioms of intuitionistic logic, where the operations of conjunction and implication coincide and the implication is the residuum of the meet. It can also be viewed as a CIBRL (complete, infinite, distributive, bounded, lattice with a relative complement) structure. Heyting algebras are used in the study of fuzzy logic and have a category theoretical perspective.From: Soft Computing and Its Applications, Volume One [2014], Fragments of quasi-Nelson: residuation [2023]

The use of diagrammatic reasoning in teaching economics for the digital economy

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Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018

James Juniper

Heyting algebras, which conform to intuitionist rather than classical propositional logic, play an important role in algebraic topology. Caterina and Gangle (2013) establish this relationship between logic and topology in a lemma reproduced below, immediately after the definition of an “upper set” of G, a notion required by the lemma concerned: The upper set for any g ∈ G, is defined as g ↑ = {h ∈ G: g →h} (i.e. g ↑ represents the collection of “the futures” of g). A subset U of G is an upper closed set if g ↑ ⊆ U for any g ∈ U.

Fragments of quasi-Nelson: residuation

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Published in Journal of Applied Non-Classical Logics, 2023

U. Rivieccio

On every CIBRL , the presence of the constant 0 allows us to define a negation operation () given by . A Heyting algebra can be viewed as a CIBRL where the operations and coincide (hence, the implication ⇒ is the residuum of the meet ).

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The use of diagrammatic reasoning in teaching economics for the digital economy

Fragments of quasi-Nelson: residuation