''Jordan–Dedekind chain condition, of a partially ordered set''
See [[Jordan–Dedekind lattice]]; [[Partially ordered set]]; [[Chain]].
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...ion]]; [[Partially ordered set|Partially ordered set]]; [[Order (on a set)|Order (on a set)]].
[[Category:Order, lattices, ordered algebraic structures]]
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...tally ordered set|Totally ordered set]]; [[Partially ordered set|Partially ordered set]].
[[Category:Order, lattices, ordered algebraic structures]]
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''order-preserving mapping''
...rdered sets (considered as algebraic systems with a single relation, cf. [[Algebraic system]]). An isotone mapping is also called a monotone mapping.
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...rtially ordered group (cf. [[Ordered group|Ordered group]]) is a naturally ordered semi-group.
* {{Ref|a1}} L. Fuchs, "Partially ordered algebraic systems", Pergamon (1963) {{ZBL|0137.02001}}
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...[[antitone mapping]] of a [[partially ordered set]] $ A $ into a partially ordered set $ B $, for which the [[inverse mapping]] is also antitone, i.e., a [[In
[[Category:Order, lattices, ordered algebraic structures]]
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For the concept of order-preserving maps on the real numbers, see [[Monotone function]].
[[Category:Order, lattices, ordered algebraic structures]]
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''of a partially ordered set''
...ition not involving the notion of interval is: A subset $A$ of a partially ordered set is convex if $a\leq b\leq c$ and $a,c\in A$ imply $b\in A$.
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* {{MSCwiki|06}} [[:Category:Order, lattices, ordered algebraic structures]] [See also {{MSCwiki|18B35}}]
* {{MSCwiki|08}} [[:Category:General algebraic systems]]
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...erence is [[#References|[a1]]]. See also [[Partially ordered set|Partially ordered set]]; [[Chain|Chain]].
[[Category:Order, lattices, ordered algebraic structures]]
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''of a partially ordered set''
...lements $a\leq b$ such that there are no other elements in the [[partially ordered set]] between them, i.e.
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A [[totally ordered set]] $X$ all proper cuts of which are [[Dedekind cut]]s, i.e. in any parti
...tinuous set is a [[conditionally-complete lattice]] which is a dense total order. The phrase "continuous set" is not used in the Western literature.
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''order relation''
...nt $a$, if $a\triangleleft b$. For examples and references see [[Partially ordered set]].
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...l element or $B$ has a minimal element. A [[continuous set]] is a totally ordered set in which all cuts are Dedekind cuts.
See also: [[Completion, MacNeille (of a partially ordered set)|Dedekind–MacNeille completion]].
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* T.S. Blyth, "Lattices and Ordered Algebraic Structures", Springer (2005) {{ISBN|1852339055}} {{ZBL|1073.06001}}
* Ivan Rival (ed.), "Algorithms and Order", Kluwer (1989) {{ISBN|940107691X}} {{ZBL|0709.68004}}
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...e theory of [[partially ordered set]]s. If $(P,{\sqsubseteq})$ is a [[pre-order]]ed set, then define $O_x = \{ y \in P : y \sqsubseteq x \}$ for $x \in P$.
...in O_x$, defines a pre-order on $X$, the [[specialization of a point]] pre-order.
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...directed if $\leq$ (respectively, the opposite order $\geq$) is a directed order. For example, the set of all open coverings $\{\gamma\}$ of a topological s
...)]]). In computer science, data structures are often modelled by partially ordered sets in which every upwards directed subset has a least upper bound (though
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[[Category:Order, lattices, ordered algebraic structures]]
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Many authors require a directed order to be a [[Partial order|partial order]] (i.e. to satisfy the condition that $x\leq y$ and $y\leq x$ together impl
[[Category:Order, lattices, ordered algebraic structures]]
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...pothesis]] was stated. A topological space (a Boolean algebra, a partially ordered set) satisfies the Suslin condition if and only if every family of non-empt
...hain condition, and this usage is also (misleadingly) applied to partially ordered sets.
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