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[parent] special elements in a relation algebra (Definition)

Let $ A$ be a relation algebra with operators $ (\vee,\wedge,\ ;,',^-,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:=\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 $ U^2$, where $ U\subseteq S$

Bibliography

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



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Also defines:  function element, injective element, surjective element, reflexive element, symmetric element, transitive element, equivalence element, domain element, range element, ideal element, rectangle, square, antisymmetric element, subidentity

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Cross-references: equivalence relation, relation, antisymmetric, transitive relation, symmetric relation, reflexive relation, surjection, injection, function, binary relations, Reflexive, transitive, symmetric, type, operators, relation algebra
There are 76 references to this entry.

This is version 6 of special elements in a relation algebra, born on 2008-02-15, modified 2008-02-16.
Object id is 10275, canonical name is SpecialElementsInARelationAlgebra.
Accessed 831 times total.

Classification:
AMS MSC: 03G15 (Mathematical logic and foundations :: Algebraic logic :: Cylindric and polyadic algebras; relation algebras)
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