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A set equipped with a total order is a totally ordered set; [5] the terms simply ordered set, [2] linearly ordered set, [3] [5] toset [6] and loset [7] [8] are also used. The term chain is sometimes defined as a synonym of totally ordered set, [5] but generally refers to a totally ordered subset of a given partially ordered set.
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" {<}
A chain is a totally ordered set or a totally ordered subset of a poset. See also total order. Chain complete. A partially ordered set in which every chain has a least upper bound. 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.
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a: left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,
An ordered set in which every pair of elements is comparable is called totally ordered. Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse. One and the same set may be equipped with different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.
Order theory, study of various binary relations known as orders; Order topology, a topology of total order for totally ordered sets; Ordinal numbers, numbers assigned to sets based on their set-theoretic order; Partial order, often called just "order" in order theory texts, a transitive antisymmetric relation
In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via ...