Index of operators on ordered sets

From Maths
Jump to: navigation, search


If you are given a set, say [ilmath]X[/ilmath] and any of a:

on that set, then this page indexes various operators that might take such a structured [ilmath]X[/ilmath] as an argument. For example:

  • [ilmath]\Uparrow\!\!A[/ilmath] or [ilmath]\Upsilon\!A[/ilmath], if this is what you're stuck on you're in the right place.

Remember, all posets are presets. So what applies to a preset applies to a poset.

References are used to document where the notation has been seen, although the page that discusses it should also cover that.


The following are related more by topic than by any sort of other order (eg alphabetical) this is because the index ought to be quite short, and there is no unique way to interpret the following notation.

Notation Conditions Meaning Comment
[ilmath]\Uparrow\!\!A[/ilmath][1] Here [ilmath]A[/ilmath] is a preset This is the [ilmath]\Uparrow[/ilmath]-functor applied to an object in the [ilmath]\mathrm{PRE} [/ilmath] category.

This is equivalent to saying that: [ilmath]\Uparrow\!\!A[/ilmath] can be used to denote the Alexandroff topology of a preset, where we get a topological space; the topology on [ilmath]\Uparrow\!\!A[/ilmath] is [ilmath]\Upsilon\!A[/ilmath] - the family of all upper sections of [ilmath]A[/ilmath]

The functor is [ilmath]\Uparrow:\mathrm{PRE}\leadsto[/ilmath][ilmath]\mathrm{TOP} [/ilmath], which takes presets to topological spaces
[ilmath]\Upsilon\!A[/ilmath][1] [ilmath]A[/ilmath] is a preset This denotes the family of all upper sections of [ilmath]A[/ilmath] This is actually a topology on [ilmath]A[/ilmath], called the Alexandroff topology. See also: [ilmath]\Uparrow[/ilmath]-functor



  1. 1.0 1.1 An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition

Template:Index navbox