ordered-set

Types of Ordered Set

In order theory, different types of ordered set are studied, including:

• Posets, orderings in which not all pairs are comparable

• Preorders, generalization of poset allowing ties (represented as equivalences and distinct from incomparabilities)

• Semiorders, poset determined by comparison of numerical values, in which values that are too close to each other are incomparable; subfamily of posets with certain restrictions

• Total orders, orderings that specify order for all elements, all pairs are comparable

• Well-orders, total orders in which every non-empty subset has a least element

• Well-quasi-orderings, a class of preorders generalizing the well-orders

• Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities)

• Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise

• Lattices, posets in which each pair of elements has a greatest lower bound (GLB) and a least upper bound (LUB).

Chains

• Chain is a totally ordered set or a totally ordered subset of a poset.

• Chain complete is a partially ordered set in which every chain has a least upper bound.

• Antichain is a poset in which no two elements are comparable, i.e. there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.

Monotonicity A monotonic function, or monotone function, is a function between ordered sets that preserves or reverses the given order.

• Order-reversing: a function $f$, antitone, between posets $P$ and $Q$ is a function for which, for all elements $x, y$ of $P$, $x \le y$ (in P) implies $f(y) \le f(x)$ (in Q).
• Monotonically decreasing is the name of such function in the presence of total orders (in math.Analysis).

• Order-preserving or monotone as the dual notion is called.

Bounds

• Bounded poset is one that has a least element and a greatest element.

• Poset is bounded complete if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

• Closure operator on the poset P is a function $C:P \to P$ that is monotone, idempotent, and satisfies $C(x) \ge x$ for all x in P.

Ordered Sets

• a binary relation R on a set S is reflexive if R relates every element of S to itself i.e. sRs or (s,s) ∈ R for all s ∈ S.

• R is symmetric if sRt implies tRs, for all s and t in S.

• R is transitive if sRt AND tRu imply sRu.

• R is antisymmetric if sRt and tRs imply that s = t

• A reflexive and transitive relation R on a set S is called a preorder on S.

• We denote a preorder by s < t "is-less-than" to mean s ≤ t ∧ s ≠ t.

• A preorder (on a set S) that is also antisymmetric is called a partial order

  on S.

• A partial order ≤ is called a total order if it also has the property that,

  for each s and t in S, either s ≤ t or t ≤ s.

• Suppose that ≤ is a partial order on a set S and s and t are elements of S. An element j ∈ S is said to be a join (or least upper bound) of s and t if s ≤ j and t ≤ j, AND for any element k ∈ S with s ≤ k and t ≤ k, we have j ≤ k.

• an element m ∈ S is a meet (or greatest lower bound) of s and t if m ≤ s and m ≤ t, AND for any element n ∈ S with n ≤ s and n ≤ t, we have n ≤ m

• a reflexive, transitive, and symmetric relation on a set S is called

  an equivalence on S.

• let R is a binary relation on a set S. The reflexive closure of R is the smallest reflexive relation R' that contains R. "Smallest" in the sense that if R'' is some other reflexive relation that contains all the pairs in R, then we have R' ⊆ R''

• the transitive closure of R is the smallest transitive relation R' that contains R

• The transitive closure of R is often written R+

• The reflexive and transitive closure of R is the smallest reflexive and transitive relation that contains R, often written R∗.

• Let preorder ≤ on a set S. A decreasing chain in ≤ is a sequence s1,s2,s3,... of elements of S such that each member of the sequence is strictly less than its predecessor: si+1 < si for every i.

• Chains can be either finite or infinite

• Suppose we have a set S with a preorder ≤. We say that ≤ is well

  founded if it contains no infinite decreasing chains. For example, the usual order on the natural numbers, with 0 < 1 < ..., is well founded, but the same order on the integers, ...< −1 < 0 < 1 <... is not. We sometimes omit mentioning ≤ explicitly and simply speak of S as a well-founded set.