Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

1: Generative Effects - Orders and Adjunctions

( \newcommand{\kernel}{\mathrm{null}\,}\)

  • 1.1: What is Order?
    Above we informally spoke of two different ordered sets: the order on system connectivity and the order on booleans false ≤ true. Then we related these two ordered sets by means of Alice’s observation Φ. Before continuing, we need to make such ideas more precise. We begin with a review of sets and relations and give the definition of a preorder—short for preordered set—and a good number of examples.
  • 1.2: Meets and Joins
    As we have said, a preorder is a set P endowed with an order ≤ relating the elements. With respect to this order, certain elements of P may have distinctive characterizations, either absolutely or in relation to other elements. We have discussed joins before, but we discuss them again now that we have built up some formalism.
  • 1.3: Galois Connections
    The preservation of meets and joins, and in particular issues concerning generative effects, is tightly related to the theory of Galois connections, which is a special case of a more general theory we will discuss later, namely that of adjunctions. We will use some adjunction terminology when describing Galois connections.
  • 1.4: Summary and further reading
  • 1.5: More than the sum of their parts


This page titled 1: Generative Effects - Orders and Adjunctions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Brendan Fong & David I. Spivak (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?