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Equivalence Relation Relation Surjective Reflexive Non Degenerate Sesquilinear Function Injective Function Operator Reflexive Transitive Symmetric Antisymmetric Type Polyadic Algebra With Equality Relation Algebra

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Special Elements In A Relation Algebra

Let be a relation algebra with operators $(\vee,\wedge,\ ;,',^-,0,1,i)$ of type . 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 (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	binary relation on set
function element	function (on )
injective element	injection
surjective element	surjection
reflexive element	reflexive relation
symmetric element	symmetric relation
transitive element	transitive relation
subidentity	$I_T:=\lbrace (x,x)\mid x\in T\rbrace$ 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	, where $U\subseteq S$

Bibliography

1: S. R. Givant, The Structure of Relation Algebras Generated by Relativizations, American Mathematical Society (1994).

1	Equivalence Relation
1	Reflexive Non Degenerate Sesquilinear
1	Surjective
1	Relation
1	Function
1	Injective Function
1	Operator
2	Antisymmetric
2	Transitive
2	Reflexive
2	Symmetric
3	Type
6	Polyadic Algebra With Equality
13	Relation Algebra