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Canonical Varieties with No Canonical Axiomatisation
 Trans. Amer. Math. Soc
, 2003
"... We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every ..."
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We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...
COMPLEXITY OF EQUATIONS VALID IN ALGEBRAS OF RELATIONS  Part II: Finite axiomatizations.
"... We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well ..."
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Cited by 18 (2 self)
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We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of nary relations, RPEAn of polyadic equality algebras of nary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 ! n ! !. Completely analogous statement holds for the case n !. This improves Monk's famous nonfinitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...
Relation Algebras of Intervals
 ARTIFICIAL INTELLIGENCE
, 1994
"... Given a representation of a relation algebra we construct relation algebras of pairs and of intervals. If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original rel ..."
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Cited by 16 (3 self)
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Given a representation of a relation algebra we construct relation algebras of pairs and of intervals. If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original relation algebra is !categorical we show that the interval algebra is too. The complexity of relation algebras is studied and it is shown that every pair algebra with infinite representations is intractable. Applications include constructing an interval algebra that combines metric and interval expressivity.
Notions Of Density That Imply Representability In Algebraic Logic
, 1998
"... Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra) . This theorem and its analogues for quasipolyadic algebras with and without equality are formulated in HenkinMonkTarski [1985]. We introduce a natur ..."
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Cited by 15 (1 self)
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Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra) . This theorem and its analogues for quasipolyadic algebras with and without equality are formulated in HenkinMonkTarski [1985]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasipolyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasipolyadic algebras and substitutioncylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the HenkinMonkTarski representation theorem for atomic equalityfree quasipolyadic algebras with rectangular atoms. The er...
Relation Algebras with nDimensional Relational Bases
 Annals of Pure and Applied Logic
, 1999
"... We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , ..."
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We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , and use it to give an explicit (hence recursive) equational axiomatisation of RAn , and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete representation. Then we prove that whenever 4 n < l !, RA l is not nitely axiomatisable over RAn . This con rms a conjecture of Maddux. We also prove that Bn is elementary for n = 3; 4 only.
Relation Algebras from Cylindric Algebras, I
, 1999
"... We characterise the class SRaCAn of subalgebras of relation algebra reducts of ndimensional cylindric algebras (for finite n 5) by the notion of a `hyperbasis', analogous to the cylindric basis of Maddux, and by relativised representations. A corollary is that SRaCAn = SRa(CAn " Crs n ) ..."
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Cited by 11 (8 self)
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We characterise the class SRaCAn of subalgebras of relation algebra reducts of ndimensional cylindric algebras (for finite n 5) by the notion of a `hyperbasis', analogous to the cylindric basis of Maddux, and by relativised representations. A corollary is that SRaCAn = SRa(CAn " Crs n ) = SRa(CAn " Gn ). We outline a gametheoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of SRaCAn .
Provability with Finitely Many Variables
"... For every finite n>=4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); &apo ..."
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For every finite n>=4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); 'n has a proof in firstorder logic with equality that contains exactly n variables, but no proof containing only n \Gamma 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) firstorder binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that 'n has a proof with only n variables. To show that 'n has no proof with only n \Gamma 1 variables we use alternative semantics in place of the usual, standard, settheoretical semantics of firstorder logic.
Axiomatising Various Classes of Relation and Cylindric Algebras
, 1997
"... We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. ..."
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Cited by 8 (5 self)
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We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras.