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1: Generative Effects - Orders and Adjunctions

Last updated

Jan 19, 2021
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- 1.1: What is Order?

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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

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