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 $e^{-};e\le i$,

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

• surjective element if $e^{-};e=i$,

• reflexive element if $i\le a$,

• symmetric element if $a^{-}\le a$,

• transitive element if $a;a\le a$,

• subidentity if $a\le i$,

• 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	$\mathrm{dom}(R)\times S$ where $R\subseteq S^2$
range element	$S\times \mathrm{ran}(R)$ where $R\subseteq S^2$
ideal element
rectangle	$U\times V\subseteq S^2$
square	$U^2$, where $U\subseteq S$