special elements in a relation algebra (original) (raw)

Let A be a relation algebra with operators (∨,∧,;,′,-,0,1,i) of type (2,2,2,1,1,0,0,0). Then a∈A is called a

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  injective element if it is a function element such that e;e-≤i,
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  surjective element if e-;e=i,
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  reflexive element if i≤a,
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  transitive element if a;a≤a,
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  subidentity if a≤i,
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  antisymmetric element if a∧a- is a subidentity,
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  domain element if a;1=a,
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  range element if 1;a=a,
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  ideal element if 1;a;1=a,
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  rectangle if a=b;1;c for some b,c∈A, and
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  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.

References 1 ⁢S.R.G⁢i⁢v⁢a⁢n⁢t,The Structure of Relation Algebras Generated by Relativizations,A⁢m⁢e⁢r⁢i⁢c⁢a⁢n⁢M⁢a⁢t⁢h⁢e⁢m⁢a⁢t⁢i⁢c⁢a⁢l⁢S⁢o⁢c⁢i⁢e⁢t⁢y⁢(1994).Titlespecial elements in a relation algebraCanonical nameSpecialElementsInARelationAlgebraDate of creation2013-03-22 17:48:43Last modified on2013-03-22 17:48:43OwnerCWoo (3771)Last modified byCWoo (3771)Numerical id9AuthorCWoo (3771)Entry typeDefinitionClassificationmsc 03G15Definesfunction elementDefinesinjective elementDefinessurjective elementDefinesreflexive elementDefinessymmetric elementDefinestransitive elementDefinesequivalence elementDefinesdomain elementDefinesrange elementDefinesideal elementDefinesrectangleDefinessquareDefinesantisymmetric elementDefinessubidentity