]> 2009/04/07 OpenCyc Knowledge Base Copyright© 2001-2009 Cycorp, Inc., http://www.cyc.com/, Austin, TX, USA This file contains an OWL representation of information contained in the OpenCyc Knowledge Base. The content of this OWL file is licensed under the Creative Commons Attribution 3.0 license whose text can be found at http://creativecommons.org/licenses/by/3.0/legalcode. The content of this OWL file, including the OpenCyc content it represents, constitutes the "Work" referred to in the Creative Commons license. The terms of this license equally apply to, without limitation, renamings and other logically equivalent reformulations of the content of this OWL file (or portions thereof) in any natural or formal language, as well as to derivations of this content or inclusion of it in other ontologies. Mappings between OpenCyc terms and Wikipedia article names provided by Olena Medelyan and Catherine Legg, University of Waikato, NZ under a Creative Commons Attribution 3.0 license. externalID A unique, language-neutral, variable-sized identifier for a concept that can be used to refer unambiguously to that concept across OWL exports or across Cyc inference engines. label A natural-language representation for a concept that is both human readable and readable by the Cyc inference engine. These terms are not guaranteed to refer to the same concept across time but are guaranteed to be consistent within a particular OWL export. Use 'cycAnnot:externalID' for unambiguously referring to a concept across OWL exports or across Cyc inference engines. PartialOrdering partial ordering The collection of all those <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a>s ORDER in which the ordering relation R is a reflexive, transitive and antisymmetric relation on the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER. R is reflexive on S if and only if for each X in S, R(X X). R is transitive on S if and only if for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). R is antisymmetric on S if and only if for each X and Y in S, R(X Y) and R(Y X) imply X = Y. For example, if you take a set of <a href="http://sw.opencyc.org/concept/Mx4rvtUAU5wpEbGdrcN5Y29ycA" class="cyc_term">List</a>s and take the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation restricted to this set, then you have a <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> because the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation is reflexive, transitive and antisymmetric. Since the ordering relation in each <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is reflexive and transitive, the collection <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is a subcollection of <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a>. Subcollections of <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> include <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a>, <a href="http://sw.opencyc.org/concept/Mx4rv0c0spwpEbGdrcN5Y29ycA" class="cyc_term">TotalOrdering</a> and <a href="http://sw.opencyc.org/concept/Mx4rvft16pwpEbGdrcN5Y29ycA" class="cyc_term">Lattice_LatticeTheoretic</a>. If you want a <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a> in which the ordering relation is irreflexive, see <a href="http://sw.opencyc.org/concept/Mx4rvnPp-5wpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering_Strict</a>. Hasse diagram oset lattice partial order partially ordered set Directed Acyclic Graph ordered set poset DAG partially ordered

Partially ordered set

http://en.wikipedia.org/wiki/Partially_ordered_set A specialization of <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a>. An instance ORDER of <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is also an instance of <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a> just in case the ordering relation R of ORDER orders elements of the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER into a tree-like structure, so that each pair of elements of S has a common 'R-lower-bound' in S (i.e., for each X, Y in S, there is a Z in S such that R(Z X) and R(Z Y)), and the set of 'R-lower-bounds' of each X in S is ordered in a chain by R (i.e., {Y: Y is in S and R(Y X)} is a chain). Note that an instance of <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a> can itself be a chain, i.e., an instance of <a href="http://sw.opencyc.org/concept/Mx4rv0c0spwpEbGdrcN5Y29ycA" class="cyc_term">TotalOrdering</a>. TreeOrdering tree ordering Lattice-LatticeTheoretic The collection of all mathmetical structures called 'lattices' in Lattice Theory (this is not the same concept as the crystalline or grid lattices studied in Crystallography and Group Theory). A <a href="http://sw.opencyc.org/concept/Mx4rvft16pwpEbGdrcN5Y29ycA" class="cyc_term">Lattice_LatticeTheoretic</a> is often defined in different but equivalent ways. To define a lattice using ordering relation, it is a <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> ORDER in which each pair of elements of the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER has an R-smallest upper bound and an R-greatest lower bound, where R is the ordering relation on S. To define a lattice using operations, it is a mathematical structure with two operations MEET and JOIN on the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of the structure that satisfy the commutative laws, the associative laws, the idempotent laws and the absorption laws. Note that the correspondence of these two different ways of defining lattices is characterized by the following: for all X, Y in S, Y = (JOIN X Y) <=> R(X Y) <=> X = (MEET X Y). See <a href="http://sw.opencyc.org/concept/Mx4rvz2YXJwpEbGdrcN5Y29ycA" class="cyc_term">meetFunctionOnLattice</a> and <a href="http://sw.opencyc.org/concept/Mx4rv__iHZwpEbGdrcN5Y29ycA" class="cyc_term">joinFunctionOnLattice</a>. lattice order METT-TC-Ordering METT TC Ordering The 'standard,' or 'expected' ordering of METT-TC principles. PartialOrdering partial ordering The collection of all those <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a>s ORDER in which the ordering relation R is a reflexive, transitive and antisymmetric relation on the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER. R is reflexive on S if and only if for each X in S, R(X X). R is transitive on S if and only if for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). R is antisymmetric on S if and only if for each X and Y in S, R(X Y) and R(Y X) imply X = Y. For example, if you take a set of <a href="http://sw.opencyc.org/concept/Mx4rvtUAU5wpEbGdrcN5Y29ycA" class="cyc_term">List</a>s and take the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation restricted to this set, then you have a <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> because the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation is reflexive, transitive and antisymmetric. Since the ordering relation in each <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is reflexive and transitive, the collection <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is a subcollection of <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a>. Subcollections of <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> include <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a>, <a href="http://sw.opencyc.org/concept/Mx4rv0c0spwpEbGdrcN5Y29ycA" class="cyc_term">TotalOrdering</a> and <a href="http://sw.opencyc.org/concept/Mx4rvft16pwpEbGdrcN5Y29ycA" class="cyc_term">Lattice_LatticeTheoretic</a>. If you want a <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a> in which the ordering relation is irreflexive, see <a href="http://sw.opencyc.org/concept/Mx4rvnPp-5wpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering_Strict</a>. Pretty String (<a href="http://sw.opencyc.org/concept/Mx4rwLSVCpwpEbGdrcN5Y29ycA" class="cyc_term">prettyString</a> TERM STRING) means that STRING is the English word or expression (sequence of words) commonly used to refer to TERM. The predicate <a href="http://sw.opencyc.org/concept/Mx4rwLSVCpwpEbGdrcN5Y29ycA" class="cyc_term">prettyString</a> is used by the code which generates CycL to English paraphrases, but its applicability is not restricted to this use. prettyString the concept corresponding to ORDERED-SET in SENSUS-Information1997 (MeaningInSystemFn SENSUS-Information1997 "ORDERED-SET") Wikipedia Article URL (<a href="http://sw.opencyc.org/concept/Mx4rNv0nbm4TTjOp7yhmnzOyqg" class="cyc_term">wikipediaArticleURL</a> THING URL) means that in <a href="http://sw.opencyc.org/concept/Mx4rtqXA6OC8QdiWC72DuLJdUw" class="cyc_term">Wikipedia_WebSite</a> THING is described by an article located at URL wikipediaArticleURL PartialOrdering partial ordering The collection of all those <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a>s ORDER in which the ordering relation R is a reflexive, transitive and antisymmetric relation on the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER. R is reflexive on S if and only if for each X in S, R(X X). R is transitive on S if and only if for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). R is antisymmetric on S if and only if for each X and Y in S, R(X Y) and R(Y X) imply X = Y. For example, if you take a set of <a href="http://sw.opencyc.org/concept/Mx4rvtUAU5wpEbGdrcN5Y29ycA" class="cyc_term">List</a>s and take the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation restricted to this set, then you have a <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> because the <a href="http://sw.opencyc.org/concept/Mx4rwQxEhZwpEbGdrcN5Y29ycA" class="cyc_term">subLists</a> relation is reflexive, transitive and antisymmetric. Since the ordering relation in each <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is reflexive and transitive, the collection <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> is a subcollection of <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a>. Subcollections of <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> include <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a>, <a href="http://sw.opencyc.org/concept/Mx4rv0c0spwpEbGdrcN5Y29ycA" class="cyc_term">TotalOrdering</a> and <a href="http://sw.opencyc.org/concept/Mx4rvft16pwpEbGdrcN5Y29ycA" class="cyc_term">Lattice_LatticeTheoretic</a>. If you want a <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a> in which the ordering relation is irreflexive, see <a href="http://sw.opencyc.org/concept/Mx4rvnPp-5wpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering_Strict</a>. ObjectType type of object The collection of all collections that are "object-like" in at least one respect. A collection <code>COL</code> is object-like just in case there is some sense of 'part' according to which any (or nearly any) given proper part of an instance of <code>COL</code> is not itself an instance of <code>COL</code>; when this happens <code>COL</code> is said to be object-like with respect to that sense of 'part'. More precisely, for a collection <code>COL</code> to be an instance of <a href="http://sw.opencyc.org/concept/Mx4rvVirnZwpEbGdrcN5Y29ycA" class="cyc_term">ObjectType</a> it is sufficient that there be some specialization <code>PARTPRED</code> of <a href="http://sw.opencyc.org/concept/Mx4rwgfukKs8QdePzLB9nLNpTw" class="cyc_term">properParts</a> such that, for any <code>OBJ1</code> and <code>OBJ2</code>, if <code>(<a href="http://sw.opencyc.org/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a> OBJ1 COL)</code> and <code>(PARTPRED OBJ1 OBJ2)</code> both hold, then <code>(<a href="http://sw.opencyc.org/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a> OBJ2 COL)</code> does not hold. (Also sufficient for <code>COL</code>'s being an object-type is that there be some specialization <code>INVPARTPRED</code> of the inverse of <a href="http://sw.opencyc.org/concept/Mx4rwgfukKs8QdePzLB9nLNpTw" class="cyc_term">properParts</a> (see <a href="http://sw.opencyc.org/concept/Mx4rvWHsNJwpEbGdrcN5Y29ycA" class="cyc_term">genlInverse</a>) such that <code>(INVPARTPRED OBJ2 OBJ1)</code>, with everything else remaining the same as above.) Note that neither of the above sufficient conditions for <code>COL</code>'s being an object-type is strictly necessary: some exceptions are allowed. Thus as long as either one of the above conditionals holds in nearly all cases, <code>COL</code> should be considered an instance of <a href="http://sw.opencyc.org/concept/Mx4rvVirnZwpEbGdrcN5Y29ycA" class="cyc_term">ObjectType</a>. Here are two examples. Consider <a href="http://sw.opencyc.org/concept/Mx4rvViVwZwpEbGdrcN5Y29ycA" class="cyc_term">Automobile</a>. Take an instance of that, say my car. Now consider one of the proper <a href="http://sw.opencyc.org/concept/Mx4rvVj5FpwpEbGdrcN5Y29ycA" class="cyc_term">physicalParts</a> of my car, say the steering wheel. The steering wheel is not an instance of <a href="http://sw.opencyc.org/concept/Mx4rvViVwZwpEbGdrcN5Y29ycA" class="cyc_term">Automobile</a>. And the same would be true for any proper physical part of any car. So <a href="http://sw.opencyc.org/concept/Mx4rvViVwZwpEbGdrcN5Y29ycA" class="cyc_term">Automobile</a> is an <a href="http://sw.opencyc.org/concept/Mx4rvVirnZwpEbGdrcN5Y29ycA" class="cyc_term">ObjectType</a>. Consider <a href="http://sw.opencyc.org/concept/Mx4rvVjyV5wpEbGdrcN5Y29ycA" class="cyc_term">CalendarYear</a>. No proper <a href="http://sw.opencyc.org/concept/Mx4rvWn4OZwpEbGdrcN5Y29ycA" class="cyc_term">timeSlices</a> of a year is itself a year. So <a href="http://sw.opencyc.org/concept/Mx4rvVjyV5wpEbGdrcN5Y29ycA" class="cyc_term">CalendarYear</a> is an <a href="http://sw.opencyc.org/concept/Mx4rvVirnZwpEbGdrcN5Y29ycA" class="cyc_term">ObjectType</a>. See <a href="http://sw.opencyc.org/concept/Mx4rvVir35wpEbGdrcN5Y29ycA" class="cyc_term">StuffType</a> for the contrasting (though not disjoint) notion of being stuff-like. The collection of all those <a href="http://sw.opencyc.org/concept/Mx4rvztTgpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrdering</a>s ORDER in which the ordering relation R is a reflexive and transitive relation on the <a href="http://sw.opencyc.org/concept/Mx4rvlGiF5wpEbGdrcN5Y29ycA" class="cyc_term">baseSet</a> S of ORDER, i.e., for each X in S, R(X X) holds, and for each X, Y and Z in S, if R(X Y) and R(Y Z) then R(X Z). For example, if you take the set of all people in the states today, and take the relation '__ is at least as tall as ...' (i.e., either __ is as tall as ... or __ is taller than ...) on this set, you get a <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a> because this relation is reflexive and transitive on the set of all people in the states today. Note that the <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a> in this example is neither a <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a> nor a <a href="http://sw.opencyc.org/concept/Mx4rvnPp-5wpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering_Strict</a>. Subcollections of <a href="http://sw.opencyc.org/concept/Mx4rwUJHwJwpEbGdrcN5Y29ycA" class="cyc_term">QuasiOrdering</a> include <a href="http://sw.opencyc.org/concept/Mx4rwTWq1ZwpEbGdrcN5Y29ycA" class="cyc_term">PartialOrdering</a>, <a href="http://sw.opencyc.org/concept/Mx4rvrKTw5wpEbGdrcN5Y29ycA" class="cyc_term">TreeOrdering</a>, <a href="http://sw.opencyc.org/concept/Mx4rv0c0spwpEbGdrcN5Y29ycA" class="cyc_term">TotalOrdering</a> and <a href="http://sw.opencyc.org/concept/Mx4rvft16pwpEbGdrcN5Y29ycA" class="cyc_term">Lattice_LatticeTheoretic</a>. QuasiOrdering quasi-ordering relational structures math topic RelationalStructures-Math-Topic A <a href="http://sw.opencyc.org/concept/Mx4rAmoSCGJbQdiSXZJvYiNhkQ" class="cyc_term">CycVocabularyTopic</a> and a <a href="http://sw.opencyc.org/concept/Mx4rtGXkHpNaEdqAAAACs0uFOQ" class="cyc_term">KBDependentCollection</a>. wikipediaArticleName (<a href="http://sw.opencyc.org/concept/Mx4rTv-jk9SPTXa991kk5mAvHg" class="cyc_term">wikipediaArticleName</a> THING NAME) means that in <a href="http://sw.opencyc.org/concept/Mx4rtqXA6OC8QdiWC72DuLJdUw" class="cyc_term">Wikipedia_WebSite</a> THING is described by an article with the title NAME Wikipedia Article Name