Ordered set is used with distinct meanings in order theory. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying
- A set with a binary relation R on its elements that is reflexive (for all a in the set, aRa), antisymmetric (if aRb and bRa, then a=b) and transitive (if aRb and bRc, then aRc) is described as a partially ordered set or poset. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement
- If the binary relation is antisymmetric, transitive and also total (for all a and b in the set, aRb or bRa) then the set is a totally ordered set. In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation
- If every non-empty subset has a least element then the set is a well-ordered set. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every
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