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- Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See also total order.
- Closure operator. A closure operator on the poset P is a function C : P → P that is monotone, idempotent, and satisfies C(x) ≥ x for all x in P.
- Compact. An element x of a poset is compact if it is way below itself, i.e. x<<x. One also says that such an x is finite.
- Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x.
- Comparability graph. The comparability graph of a poset (P, ≤) is the graph with vertex set P in which the edges are those pairs of distinct elements of P that are comparable under ≤ (and, in particular, under its reflexive reduction <).
- A complete Boolean algebra is a Boolean algebra that is a complete lattice.
- Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts frame and locale.
- Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.
- Complete partial order. A complete partial order, or cpo, is a directed complete partial order (q.v.) with least element.
- Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in the article on completeness (order theory), any poset for which either all suprema or all infima exist is already a complete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of a complete lattice. In other cases, complete (meet-) semilattices are defined to be bounded complete cpos, which is arguably the most complete class of posets that are not already complete lattices.
- Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.
- Completion. A completion of a poset is an order-embedding of the poset in a complete lattice.
- Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every element x of P is the supremum of a directed set contained in {y in B | y<<x}.
- Continuous function. See Scott-continuous.
- Converse. The converse <° of an order < is that in which x <° y whenever y < x.
- Cover. An element y of a poset P is said to cover an element x of P (and is called a cover of x) if x < y and there is no element z of P such that x < z < y.
- cpo. See complete partial order.
Read more about this topic: Glossary Of Order Theory