Branches of Mathematics
Order Theory
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Order theory is the branch of mathematics that studies sets equipped with a binary relation expressing that some elements precede others, drawing out the properties order alone allows independent of the elements' particular nature.
Facts
Central QuestionWhich properties of a system follow from its order structure alone, so that a theorem proved in the abstract for one ordered set carries over to every other set organized the same way. 1 Key DebateHow late the abstract, general study of order came to mathematics compared with its intuitive use. Explicit study of partial orders as objects in their own right is a nineteenth-century development, arriving alongside Georg Cantor's introduction of ordinal numbers as the order types of well-ordered sets, well after mathematicians had used particular orderings, such as the ordering of the integers, for millennia. 1 Cross-Tradition Connections
Sources
1. Wikipedia: Order Theory
Wikimedia FoundationLead sectionQuote, Lead section
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations.
1. Wikipedia: Order Theory
Wikimedia FoundationPartially ordered sets sectionQuote, Partially ordered sets section
A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear.
1. Wikipedia: Order Theory
Wikimedia FoundationHistory sectionQuote, History section
The earliest explicit mentionings of partial orders are probably to be found not before the 19th century. In this context the works of George Boole are of great importance.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaHistory sectionQuote, History section
The earliest explicit mentionings of partial orders are probably to be found not before the 19th century.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaSet theory sectionQuote, Set theory section
Ordinal numbers are then introduced as the order types of well-ordered sets.
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