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Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions.
Relations among tables are mapped in the same way: by convention. For instance, to create a one-to-many relationship between two tables one assigns an array to the property bearing the name of the target table. This automatically creates the table as well as the required columns. Code example, demonstrating a simple CRUD operation and a relation:
A prefix order is a binary relation "≤" over a set P which is antisymmetric, transitive, reflexive, and downward total, i.e., for all a, b, and c in P, we have that: a ≤ a (reflexivity); if a ≤ b and b ≤ a then a = b (antisymmetry); if a ≤ b and b ≤ c then a ≤ c (transitivity); if a ≤ c and b ≤ c then a ≤ b or b ≤ a ...
Order-dual. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse. Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q). Order isomorphism.
A base relation representing this kind of entity is called, informally, an associative table. An associative entity (using Chen notation) As mentioned above, associative entities are implemented in a database structure using associative tables, which are tables that can contain references to columns from the same or different database tables ...
The incomparability relation is always symmetric and it will be reflexive if and only if < is an irreflexive relation (which is assumed by the above definition). Consequently, a strict partial order < is a strict weak order if and only if its induced incomparability relation is an equivalence relation.
A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric. Among other ways of defining wqo's, one is to say that they are quasi-orderings which do not contain infinite strictly decreasing sequences (of the form x 0 > x 1 > x 2 > ⋯ {\displaystyle x_{0}>x_{1}>x_{2}>\cdots } ) [1] nor infinite ...
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...