View source for Template:Binary relations

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  ! style="text-align:center;padding-left:0.2em;padding-right:0.2em;font-size:90%<!--(for [show/hide] link)-->;" | {{large|[[Transitive relation|Transitive]]&nbsp;[[binary relation]]s}} {{Navbar|Binary relations|mini=y|brackets=y|style=float:right; padding-left: 15px; padding-right: 5px;}}
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| {{Aligned table
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  |
  | [[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 |
<!-------------------------------------------------------------------------------------------------
      sym   antisym   total  wellfound joins    meets   reflex  irreflex  asym
-------------------------------------------------------------------------------------------------->
  | {{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&}}
<!--
  | | | | | | | | | |
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-->
  | [[Strict partial order]] 
  | {{n&}} | {{aye}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{n&}} | {{aye}} | {{aye}}

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

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

<!-------------------------------------------------------------------------------------------------
      sym   antisym   total  wellfound joins    meets   reflex  irreflex  asym
-------------------------------------------------------------------------------------------------->
  |
  | [[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 always true for the row's term (at the very left), while {{n&}} 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 {{aye}} in the "Symmetric" column and {{n&}} in the "Antisymmetric" column, respectively. <br>
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> <br>
A term's definition may require additional properties that are not listed in this table.
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