"Linear order" redirects here. Not to be confused with Linear order (linguistics).
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In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in :
(reflexive).
If and then (transitive).
If and then (antisymmetric).
or (strongly connected, formerly called total).
Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.[1]
Total orders are sometimes also called simple,[2]connex,[3] or full orders.[4]
A set equipped with a total order is a totally ordered set;[5] the terms simply ordered set,[2]linearly ordered set,[3][5] and loset[6][7] are also used. The term chain is sometimes defined as a synonym of totally ordered set,[5] but refers generally to some sort of totally ordered subsets of a given partially ordered set.
An extension of a given partial order to a total order is called a linear extension of that partial order.
^Halmos 1968, Ch.14.
^ abBirkhoff 1967, p. 2.
^ abSchmidt & Ströhlein 1993, p. 32.
^Fuchs 1963, p. 2.
^ abcDavey & Priestley 1990, p. 3.
^Strohmeier, Alfred; Genillard, Christian; Weber, Mats (1 August 1990). "Ordering of characters and strings". ACM SIGAda Ada Letters (7): 84. doi:10.1145/101120.101136. S2CID 38115497.
^Ganapathy, Jayanthi (1992). "Maximal Elements and Upper Bounds in Posets". Pi Mu Epsilon Journal. 9 (7): 462–464. ISSN 0031-952X. JSTOR 24340068.
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