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Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle X} can equivalently be defined as an equivalence relation on X {\displaystyle X} , together with a partial order on the set of equivalence class.
Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, := < is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ...
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations . The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves ...
A directed set's preorder is called a direction. The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, [2] meaning that every pair of elements is bounded below. [3] [a] Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other ...
Three well-orderings on the set of natural numbers with distinct order types (top to bottom): , +, and +. Every well-ordered set is order-equivalent to exactly one ordinal number , by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified ...
A strict total order on a set is a strict partial order on in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
Besides these two most common order types, brokers may offer a number of other options, such as stop-loss orders or stop-limit orders. Order types differ by broker, but they all have market and ...
The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals.