Order Theory - Background and Motivation

Background and Motivation

Orders are everywhere in mathematics and related fields like computer science. 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. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Another familiar example of an ordering is the lexicographic order of words in a dictionary.

The above types of orders have a special property: each element can be compared to any other element, i.e. it is greater, smaller, or equal. However, this is not always a desired requirement. For example, consider the subset ordering of sets. If a set A contains all the elements of a set B, then B is said to be smaller than or equal to A. Yet there are some sets that cannot be related in this fashion. Whenever both contain some elements that are not in the other, the two sets are not related by subset-inclusion. Hence, subset-inclusion is only a partial order, as opposed to the total orders given before.

Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order. These insights can then be readily transferred to many less abstract applications.

Driven by the wide practical usage of orders, numerous special kinds of ordered sets have been defined, some of which have grown into mathematical fields of their own. In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate functions between them. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found.