Template:Binary relations

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Template's default state when transcluded is collapsed. To override, invoke as {{Binary relations|expanded}}.

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

Example call

Calling

{{Binary relations}}

will display:

Transitive binary relations

	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
			Total, Semiconnex					Anti- reflexive
Equivalence relation	Y	✗	✗	✗	✗	✗	Y	✗	✗
Preorder (Quasiorder)	✗	✗	✗	✗	✗	✗	Y	✗	✗
Partial order	✗	Y	✗	✗	✗	✗	Y	✗	✗
Total preorder	✗	✗	Y	✗	✗	✗	Y	✗	✗
Total order	✗	Y	Y	✗	✗	✗	Y	✗	✗
Prewellordering	✗	✗	Y	Y	✗	✗	Y	✗	✗
Well-quasi-ordering	✗	✗	✗	Y	✗	✗	Y	✗	✗
Well-ordering	✗	Y	Y	Y	✗	✗	Y	✗	✗
Lattice	✗	Y	✗	✗	Y	Y	Y	✗	✗
Join-semilattice	✗	Y	✗	✗	Y	✗	Y	✗	✗
Meet-semilattice	✗	Y	✗	✗	✗	Y	Y	✗	✗
Strict partial order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict weak order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict total order	✗	Y	Y	✗	✗	✗	✗	Y	Y
	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$aRa$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by

in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.

Call with alignment

Calling

{{Binary relations|position=left}}

will display:

Transitive binary relations

	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
			Total, Semiconnex					Anti- reflexive
Equivalence relation	Y	✗	✗	✗	✗	✗	Y	✗	✗
Preorder (Quasiorder)	✗	✗	✗	✗	✗	✗	Y	✗	✗
Partial order	✗	Y	✗	✗	✗	✗	Y	✗	✗
Total preorder	✗	✗	Y	✗	✗	✗	Y	✗	✗
Total order	✗	Y	Y	✗	✗	✗	Y	✗	✗
Prewellordering	✗	✗	Y	Y	✗	✗	Y	✗	✗
Well-quasi-ordering	✗	✗	✗	Y	✗	✗	Y	✗	✗
Well-ordering	✗	Y	Y	Y	✗	✗	Y	✗	✗
Lattice	✗	Y	✗	✗	Y	Y	Y	✗	✗
Join-semilattice	✗	Y	✗	✗	Y	✗	Y	✗	✗
Meet-semilattice	✗	Y	✗	✗	✗	Y	Y	✗	✗
Strict partial order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict weak order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict total order	✗	Y	Y	✗	✗	✗	✗	Y	Y
	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$aRa$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by

in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.

Expanded with alignment

Calling

{{Binary relations|expanded|position=left}}

will display:

Transitive binary relations

	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
			Total, Semiconnex					Anti- reflexive
Equivalence relation	Y	✗	✗	✗	✗	✗	Y	✗	✗
Preorder (Quasiorder)	✗	✗	✗	✗	✗	✗	Y	✗	✗
Partial order	✗	Y	✗	✗	✗	✗	Y	✗	✗
Total preorder	✗	✗	Y	✗	✗	✗	Y	✗	✗
Total order	✗	Y	Y	✗	✗	✗	Y	✗	✗
Prewellordering	✗	✗	Y	Y	✗	✗	Y	✗	✗
Well-quasi-ordering	✗	✗	✗	Y	✗	✗	Y	✗	✗
Well-ordering	✗	Y	Y	Y	✗	✗	Y	✗	✗
Lattice	✗	Y	✗	✗	Y	Y	Y	✗	✗
Join-semilattice	✗	Y	✗	✗	Y	✗	Y	✗	✗
Meet-semilattice	✗	Y	✗	✗	✗	Y	Y	✗	✗
Strict partial order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict weak order	✗	Y	✗	✗	✗	✗	✗	Y	Y
Strict total order	✗	Y	Y	✗	✗	✗	✗	Y	Y
	Symmetric	Antisymmetric	Connected	Well-founded	Has joins	Has meets	Reflexive	Irreflexive	Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$aRa$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by

in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.

The above documentation is transcluded from Template:Binary relations/doc.
Editors can experiment in this template's sandbox and testcases pages.
Add categories to the /doc subpage. Subpages of this template.

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Template:Binary relations

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