Concept
Complete Heyting algebra
Overview
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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In the lowest grade of eternal objects are to be placed those objects whose individual essences are simple. This is the grade of zero complexity. Next consider any set of such objects, finite or infinite as to the number of its members.
Alfred North Whitehead, Science and the modern world
States what Complete Heyting algebra is.
Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid--corpus solidum--and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
Friedrich Engels, Landmarks of Scientific Socialism
States what Complete Heyting algebra is.
Call it {alpha}. There will be one moment of the temporal series of {alpha} which covers the given event-particle. Thus the position of the event-particle in the temporal series {alpha} is defined by this moment, which we will call M. The position of the particle in the space of M is then fixed in the ordinary way by three levels which intersect in it and in it only. This procedure of fixing the position of an…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Complete Heyting algebra is.
Related ideas
- Mathematics
- Power set
- Morphism
- Topological space
- Material conditional
- Cambridge University Press
- Heyting algebra
Place in the tradition
Part of Many-valued logic.