Indicwiki: 2 revisions imported from :alpha:Template:Binary_relations

2022-12-16T08:33:53Z

2 revisions imported from alpha:Template:Binary_relations

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alpha>Indicwiki at 06:58, 16 December 2022

2022-12-16T06:58:35Z

← Older revision		Revision as of 12:28, 16 December 2022
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	[[Category:Mathematics sidebar templates]]		[[Category:Mathematics sidebar templates]]
	}}</noinclude>		}}</noinclude>

			[[Category:Mathematics sidebar templates]]

alpha>Sarika: Created page with "{| class="wikitable collapsible {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="table-layout:auto; width:1%; {{#switch:{{{position|}}} |cen..."

2022-12-14T11:06:31Z

Created page with "{| class="wikitable collapsible <includeonly>{{#ifeq:{{{1|}}}|expanded | |collapsed}}</includeonly>" style="table-layout:auto; width:1%; {{#switch:{{{position|}}} |cen..."

New page

{| class="wikitable collapsible <includeonly>{{#ifeq:{{{1|}}}|expanded | |collapsed}}</includeonly>" style="table-layout:auto; width:1%; {{#switch:{{{position|}}} |center|centre=margin-left:auto; margin-right:auto; |#default=float:{{{position|right}}};}} margin-top:0; padding-top:0;"
|-
! style="padding-left:0.5em; padding-right:0.5em;" | [[Transitive relation|Transitive]] [[binary relation]]s
|-
| {{Aligned table
| cols = 10
| style = text-align:center;
| rowstyle = vertical-align:middle;
| col1align = right
| col1style = padding-left:0.6em; padding-right:1.2em; font-weight:bold;
| row1style = padding-left:0.3em; padding-right:0.3em;

|
| [[Symmetric relation|Symmetric]]
| [[Antisymmetric relation|Antisymmetric]]
| [[Connected relation|Connected]]
| [[Well-founded relation|Well-founded]]
| [[Join and meet|Has joins]]
| [[Join and meet|Has meets]]
| [[Reflexive relation|Reflexive]]
| [[Reflexive relation#Irreflexive|Irreflexive]]
| [[Asymmetric relation|Asymmetric]]
| | | | Total, Semiconnex | | | | | Anti-<br/>reflexive |

| {{nowrap|[[Equivalence relation]]}}
| {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| {{nowrap|[[Preorder|Preorder {{smaller|(Quasiorder)}}]]}}
| {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Partial order]]
| {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Total preorder]]
| {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Total order]]
| {{n&}} | {{aye}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Prewellordering]]
| {{n&}} | {{n&}} | {{aye}} | {{aye}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| {{nowrap|[[Well-quasi-ordering]]}}
| {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Well-order]]ing
| {{n&}} | {{aye}} | {{aye}} | {{aye}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Lattice (order)|Lattice]]
| {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{aye}} | {{aye}} | {{aye}} | {{n&}} | {{n&}}

| [[Join-semilattice]]
| {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{aye}} | {{n&}} | {{n&}}

| [[Meet-semilattice]]
| {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{aye}} | {{n&}} | {{n&}}

| [[Strict partial order]]
| {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{aye}}

| [[Weak ordering#Strict weak orderings|Strict weak order]]
| {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{aye}}

| [[Strict total order]]
| {{n&}} | {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{aye}}


|
| [[Symmetric relation|Symmetric]]
| [[Antisymmetric relation|Antisymmetric]]
| [[Connected relation|Connected]]
| [[Well-founded relation|Well-founded]]
| [[Join and meet|Has joins]]
| [[Join and meet|Has meets]]
| [[Reflexive relation|Reflexive]]
| [[Reflexive relation#Irreflexive|Irreflexive]]
| [[Asymmetric relation|Asymmetric]]

| Definitions, for all <math>a, b</math> and <math>S\neq\varnothing :</math>
| <math>\begin{align}&aRb \\ \Rightarrow{} &bRa\end{align}</math>
| <math>\begin{align}aRb\text{ and }&bRa \\ \Rightarrow a ={} &b\end{align}</math>
| <math>\begin{align}a \neq{} &b \Rightarrow \\ aRb\text{ or }&bRa\end{align}</math>
| <math>\begin{align}\min S \\ \text{exists}\end{align}</math>
| <math>\begin{align}a \vee b \\ \text{exists}\end{align}</math>
| <math>\begin{align}a \wedge b \\ \text{exists}\end{align}</math>
| <math>aRa</math>
| <math>\text{not }aRa</math>
| <math>\begin{align}aRb \Rightarrow \\ \text{not }bRa\end{align}</math>
}}
|-
| style="text-align:center;" | {{aye}} indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric. {{n&}} indicates that the property may, or may not hold. All definitions tacitly require the [[homogeneous relation]] <math>R</math> be [[transitive relation|transitive]]: for all <math>a, b, c,</math> if <math>aRb</math> and <math>bRc</math> then <math>aRc,</math> and there are [[homogeneous relation|additional properties]] that a homogeneous relation may satisfy.
|}<noinclude>
{{Documentation
| content =
Template's default state when [[Wikipedia:Transclusion|transcluded]] is collapsed. To override, invoke as <code><nowiki>{{Binary relations |expanded}}</nowiki></code>.

To change the template's position from the default shown, add the parameter ''position'' with the value "left", "center", "centre" or "right".

[[Category:Mathematics sidebar templates]]
}}</noinclude>

Template:Binary relations - Revision history

Indicwiki: 2 revisions imported from :alpha:Template:Binary_relations

alpha>Indicwiki at 06:58, 16 December 2022

alpha>Sarika: Created page with "{| class="wikitable collapsible {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="table-layout:auto; width:1%; {{#switch:{{{position|}}} |cen..."