# special elements in a relation algebra

Let $A$ be a relation algebra with operators $(\vee,\wedge,\ ;,^{\prime},^{-},0,1,i)$ of type $(2,2,2,1,1,0,0,0)$. Then $a\in A$ is called a

• function element if $e^{-}\ ;e\leq i$,

• injective element if it is a function element such that $e\ ;e^{-}\leq i$,

• surjective element if $e^{-}\ ;e=i$,

• reflexive element if $i\leq a$,

• symmetric element if $a^{-}\leq a$,

• transitive element if $a\ ;a\leq a$,

• subidentity if $a\leq i$,

• antisymmetric element if $a\wedge a^{-}$ is a subidentity,

• equivalence element if it is symmetric and transitive (not necessarily reflexive!),

• domain element if $a\ ;1=a$,

• range element if $1\ ;a=a$,

• ideal element if $1\ ;a\ ;1=a$,

• rectangle if $a=b\ ;1\ ;c$ for some $b,c\in A$, and

• square if it is a rectangle where $b=c$ (using the notations above).

These special elements are so named because they are the names of the corresponding binary relations on a set. The following table shows the correspondence.

element in relation algebra $A$ binary relation on set $S$
function element function (on $S$)
injective element injection
surjective element surjection
reflexive element reflexive relation
symmetric element symmetric relation
transitive element transitive relation
subidentity $I_{T}:=\{(x,x)\mid x\in T\}$ where $T\subseteq S$
antisymmetric element antisymmetric relation
equivalence element symmetric reflexive relation (not an equivalence relation!)
domain element $\operatorname{dom}(R)\times S$ where $R\subseteq S^{2}$
range element $S\times\operatorname{ran}(R)$ where $R\subseteq S^{2}$
ideal element
rectangle $U\times V\subseteq S^{2}$
square $U^{2}$, where $U\subseteq S$

${{{{}\end{center}\inner@par\thebibliography\bibitem{sg}S.R.Givant,\emph{The % Structure of Relation Algebras Generated by Relativizations},% AmericanMathematicalSociety(1994).\endthebibliography\begin{flushright}\begin{% tabular}[]{|ll|}\hline Title&special elements in a relation algebra\\ Canonical name&SpecialElementsInARelationAlgebra\\ Date of creation&2013-03-22 17:48:43\\ Last modified on&2013-03-22 17:48:43\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&9\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 03G15\\ Defines&function element\\ Defines&injective element\\ Defines&surjective element\\ Defines&reflexive element\\ Defines&symmetric element\\ Defines&transitive element\\ Defines&equivalence element\\ Defines&domain element\\ Defines&range element\\ Defines&ideal element\\ Defines&rectangle\\ Defines&square\\ Defines&antisymmetric element\\ Defines&subidentity\\ \hline}\end{tabular}}}\end{flushright}\end{document}$