Complete partial order

Order and Logic

Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016

1.1.1 Definition Let (D, ⊑) be a partially ordered set. We call (D, ⊑) an ωcomplete partial order or an ω-cpo if ⊔ A exists in D for each ωchain A in D, and D has an element ⊥, called the least element or bottom element, satisfying ⊥ ⊑ x for all x ∈ D.We call (D, ⊑) chain complete if every chain in D has a supremum.We call (D, ⊑) a complete partial order or a cpo if ⊔ A exists in D for each directed subset A of D, and D has a bottom element.We call (D, ⊑) a complete upper semi-lattice if H A exists in D for each directed subset A of D, and ⊓ A exists for each subset A of D.We call (D, ⊑) a complete lattice if H A and ⊓ A exist in D for every subset A of D.

A mosaic of Chu spaces and Channel Theory I: Category-theoretic concepts and tools

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Journal Information

Published in Journal of Experimental & Theoretical Artificial Intelligence, 2019

Chris Fields, James F. Glazebrook

Note that (1) above allows ‘single-object concepts’; these will become important in Fields and Glazebrook (2018) as representations of object tokens (cf. Fields, 2012), as well as for a hierarchical iteration of the idea. Definition 3.2. A complete algebraic lattice (henceforth, an algebraic lattice) is a partial order which is both a complete lattice and a directed complete partial order (DCPO).

Complete partial order

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Order and Logic

A mosaic of Chu spaces and Channel Theory I: Category-theoretic concepts and tools