special elements in a relation algebra

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

  • function element if e-;ei,

  • injective element if it is a function element such that e;e-i,

  • surjective element if e-;e=i,

  • reflexive element if ia,

  • symmetric element if a-a,

  • transitive element if a;aa,

  • subidentity if ai,

  • antisymmetric element if aa- is a subidentity,

  • equivalence element if it is symmetricPlanetmathPlanetmathPlanetmathPlanetmath and transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath (not necessarily reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath!),

  • 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,cA, 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 relationsMathworldPlanetmath 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 IT:={(x,x)xT} where TS
antisymmetric element antisymmetric relation
equivalence element symmetric reflexive relation (not an equivalence relationMathworldPlanetmath!)
domain element dom(R)×S where RS2
range element S×ran(R) where RS2
ideal element
rectangle U×VS2
square U2, where US

 References 1 S.R.Givant,The Structure of Relation Algebras Generated by Relativizations,AmericanMathematicalSociety(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