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Glossary of order theory
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== S == * '''Saturated chain'''. A [[Total order#Chains|chain]] in a poset such that no element can be added ''between two of its elements'' without losing the property of being totally ordered. If the chain is finite, this means that in every pair of successive elements the larger one covers the smaller one. See also maximal chain. * '''[[Scattered order|Scattered]]'''. A total order is scattered if it has no densely ordered subset. * '''[[Scott-continuous]]'''. A monotone function ''f'' : ''P'' β ''Q'' between posets ''P'' and ''Q'' is Scott-continuous if, for every directed set ''D'' that has a supremum sup ''D'' in ''P'', the set {''fx'' | ''x'' in ''D''} has the supremum ''f''(sup ''D'') in ''Q''. Stated differently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent to being [[Continuity (topology)|continuous]] with respect to the [[Scott topology]] on the respective posets. * '''[[Scott domain]]'''. A Scott domain is a partially ordered set which is a [[bounded complete]] [[Algebraic poset|algebraic]] [[Complete partial order|cpo]]. * '''Scott open'''. See ''Scott topology''. * '''Scott topology'''. For a poset ''P'', a subset ''O'' is '''Scott-open''' if it is an [[upper set]] and all directed sets ''D'' that have a supremum in ''O'' have non-empty intersection with ''O''. The set of all Scott-open sets forms a [[topology]], the '''Scott topology'''. * '''[[Semilattice]]'''. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-empty meets (infima) exist. Accordingly, one speaks of a '''join-semilattice''' or '''meet-semilattice'''. * '''Smallest element'''. See ''least element''. * [[Sperner property of a partially ordered set]] * [[Sperner poset]] * [[Strictly Sperner poset]] * [[Strongly Sperner poset]] * '''[[Strict order]]'''. See {{em|strict partial order}}. * '''[[Strict partial order]]'''. A strict partial order is a [[Homogeneous relation|homogeneous binary relation]] that is [[Transitive relation|transitive]], [[Irreflexive relation|irreflexive]], and [[Antisymmetric relation|antisymmetric]]. * '''[[Strict preorder]]'''. See {{em|strict partial order}}. * '''[[Supremum]]'''. For a poset ''P'' and a subset ''X'' of ''P'', the [[least element]] in the set of [[upper bound]]s of ''X'' (if it exists, which it may not) is called the '''supremum''', '''join''', or '''least upper bound''' of ''X''. It is denoted by sup ''X'' or <math>\bigvee</math>''X''. The supremum of two elements may be written as sup{''x'',''y''} or ''x'' ∨ ''y''. If the set ''X'' is finite, one speaks of a '''finite supremum'''. The dual notion is called ''infimum''. * '''Suzumura consistency'''. A binary relation R is Suzumura consistent if ''x'' R<sup>∗</sup> ''y'' implies that ''x'' R ''y'' or not ''y'' R ''x''.<ref name=BosSuz/> * '''[[Symmetric relation]]'''. A [[homogeneous relation]] ''R'' on a set ''X'' is symmetric, if ''x R y'' implies ''y R x'', for all elements ''x'', ''y'' in ''X''.
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