Itisknownthatforallfiniten � 5,therearerelationalgebraswithn-dimensionalrelational basesbutnoweakrepresentations. Weprovethatconversely,therearefiniteweaklyrepresentable relation algebraswithnon-dimensionalrelationalbases. Insymbols: neitheroftheclasses RAnandwRRAcontains theother. Themainresultofthispaperconcernstherelationshipbetweentwoclasseseach ofwhichcontainstheclassofrepresentablerelationalgebras(RRA). Representable relation algebras areisomorphic to genuinefields of binaryrelations with natural set-theoretically defined operators(union, set complement, identity, converseand composition). It is knownthat RRAis badly behavedin a number of ways. RRA cannotbedefinedbyfinitelymanyaxioms(14),norbyanysetofequationsusingonly finitely many variables (8), nor by anyset offirst-order sentences containing only finitelymanynon-canonicalsentences(6). Itisundecidablewhetherafiniterelation algebraisrepresentableornot(4). TheequationaltheoryofRRAisundecidable(16). Consequently, researchers have defined and investigated other classes of relation algebras,notidenticalto RRAbutwithsomecommonfeatures. Oneoftheseclassesistheclassofweaklyrepresentablerelationalgebras(wRRA), which have representations rather like classical representations, but the Boolean sum and negation operators are not required to be interpreted as set union and complementrespectively(7,page459). Clearly wRRARRAanditturnsoutthat theinclusionisproper. Another kind of representation which allows us to generalise RRAis called a relativised representation. The definition is like the definition of a classical repre- sentation,butalloperatorsarerelativisedtotheunitelementintherepresentation, whichissimplyareflexiveandsymmetricbinaryrelationcontainingalltheotherre- lations. Theclassofrelationalgebraswithrelativisedrepresentationsturnsouttobe very well behaved—allrelation algebras haverelativised representations, the class of relation-type algebras with relativised representations isfinitely axiomatisable (10,Theorem5.20),andallfiniterelationalgebrashavefiniterelativisedrepresenta- tions(4,Theorem19.13). Therearemanyvariantsofthisdefinitionwhichimpose dierent restrictions on the type of unit that is allowed. A s et of points in the
['Robin Hirsch', 'Ian M. Hodkinson', 'Roger D. Maddux']
WEAK REPRESENTATIONS OF RELATION ALGEBRAS AND RELATIONAL BASES