## Relation Algebras with n-Dimensional Relational Bases (1999)

Venue: | Annals of Pure and Applied Logic |

Citations: | 9 - 2 self |

### BibTeX

@ARTICLE{Hirsch99relationalgebras,

author = {Robin Hirsch and Ian Hodkinson},

title = {Relation Algebras with n-Dimensional Relational Bases},

journal = {Annals of Pure and Applied Logic},

year = {1999},

volume = {101},

pages = {227--274}

}

### OpenURL

### Abstract

We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of non-associative 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.

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Citation Context ...the algebra of all sets of ultralters of A, with algebraic operations induced from those in A. It embeds A as a subalgebra and we identify A with its image under this embedding. For more details see [=-=JT51]-=-. 3 An atomic relation algebra 3 is said to have an n-dimensional relational basis if 9 can survive forever in this game, whatever challenges 8 makes. We denote the class of all such algebras by B n h... |

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Citation Context ...In the example above, a relevant sentence is 9x 0 ; : : : ; x p V i E(x i ; x j ), where E is the edge relation (playing the role of 6=). A similar result for arbitrary L and r = ! can also be proved =-=[-=-Bar77]. 5.2 The modied game EF p r (A; B) The denition of the Ehrenfeucht{Frasse game EF p r (A; B), above, is quite standard. We need a minor modication of the game to make it appropriate to our purp... |

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Citation Context ...worse, the variety RRA of representable relation algebras is notsnitely axiomatisable. Among several reactions to these negative results, one was to seek an explicit innite axiomatisation of RRA. In [=-=Lyn5-=-0] Lyndon laid down a schema generating an innite sequence Research of the second author partially supported by UK EPSRC grant GR/K54946. Many thanks to Peter Jipsen and Yde Venema for discussing Mad... |

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Citation Context ... can be seen as approximations to RRA. The Lyndon conditions can be expressed naturally in terms of a two-player game. This was done in [HH97d, HH97b], but much the same ideas were used by Maddux, in =-=[Mad82]-=-, for example, and indeed in [Lyn50]. We believe that the explicit use of games concentrates attention on the essential concepts rather than notational details, and so permits more lucid proofs. The g... |

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Citation Context ...d relative to RA n : that is, there is nosrst-order sentence such that A 2 RA l i A 2 RA n and A j= . This is again similar to the position for RRA, which, as we said, is notsnitely axiomatisable [Mo=-=n64]-=-. However, there is some evidence on the other side. We already noted that there is an existential second-order sentence expressing that a non-associative algebra is in B n . For RA n , recent work of... |

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Citation Context ... !, RA l is notsnitely axiomatisable over RA n . 1.1 Approximations to RRA The notion of a relation algebra was dened by Jonsson and Tarski ([JT48]; the early history of the subject is surveyed in [Ma=-=d91b]-=-), and at the time it was hoped that relation algebras would capture the notion of an algebra of binary relations, in the same way that the axioms for boolean algebra capture theselds of sets (algebra... |

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Citation Context ...ave the `nite algebra onsnite base property'. RA ! = RRA certainly does not have the property: for example, the `point algebra', used in the articial planning community and described for example in [H=-=H97d-=-], has eight elements but does not have anysnite !-square relativised representation. We remark that with respect to a slightly dierent notion of n-square relativised representation, the answer to the... |

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Citation Context ...ver loses (a lost position would certainly be one at which 8 has a winning strategy), so this must be a winning strategy for her. (We have simply proved that the game EF p r (A; B) is determined; cf. =-=[GS53]-=-.) 2 5.5 How 9 can win G(A A;B ) In the proof of proposition 43, 8 forced 9 to attempt the game EF p r (A; B) (and he won). He did this by building a red clique from a `base' (c; c 0 ). Definition 45 ... |

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Citation Context ...conjecture of Maddux [Mad83]: for every n; l with 4 ns !, RA l is notsnitely axiomatisable over RA n . 1.1 Approximations to RRA The notion of a relation algebra was dened by Jonsson and Tarski ([JT4=-=8]-=-; the early history of the subject is surveyed in [Mad91b]), and at the time it was hoped that relation algebras would capture the notion of an algebra of binary relations, in the same way that the ax... |

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Citation Context ... CRA for the class of all relation algebras that have a complete representation | one in which all meets and joins existing in the algebra are preserved. It is easily seen that 1 For more details see =-=[HH97b]-=-. 2 such an algebra must be atomic. It was shown in [HH97b] that an atomic relation algebra satis es the Lyndon conditions if and only if it is elementarily equivalent to an algebra that has a complet... |

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Citation Context ...a solution for this problem is at hand. In [Mad83], Maddux dened MA n , standing for `matrix algebra of degree n', to be (in eect) the closure of B n under taking subalgebras. He renamed it RA n in [M=-=ad89]-=- and we will use the more recent and suggestive notation here. He showed that RA n was a variety, the variety generated by B n , and that T n RA n = RRA. Hence we may choose to approximate RRA by the ... |

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Citation Context ...RA n is not closed under completions (and hence, by a result of Venema [Ven97], not Sahlqvist-axiomatisable). This can be shown by a slight adaptation of the example used to prove the same for RRA in =-=[Hod97]-=-. RA 4 (RA) is closed under completions, by a result of Monk [Mon70]; so is RA 3 (SA), as Maddux showed [Mad91c, theorem 36]. Problem 2 Is RA 5 closed under completions? 41 Questions related to thesni... |

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Citation Context ...ts of n-dimensional cylindric algebras obtained in this way. It can be shown that SRaCA 3 SRaCA 4 ; and T n SRaCA n = RRA, so again we are dealing with a series of approximations to RRA. In [HH98=-=a]-=- we prove an analogue of theorem 24 for the classes SRaCA n , and give them a `hyper-basis' characterisation similar to relational bases. We also recursively axiomatise these classes. It is further sh... |

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Citation Context ...ma [Ven97], not Sahlqvist-axiomatisable). This can be shown by a slight adaptation of the example used to prove the same for RRA in [Hod97]. RA 4 (RA) is closed under completions, by a result of Monk =-=[Mon70-=-]; so is RA 3 (SA), as Maddux showed [Mad91c, theorem 36]. Problem 2 Is RA 5 closed under completions? 41 Questions related to thesnite base property are also raised. For n 3, the equational theory o... |

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Citation Context ...o RA l (ns!) is. In a similar way, we obtain the result discussed earlier: Theorem 2 For any n with 5 ns!, CRA n is not elementary. Our construction in the proofs of these two results originated in [=-=Hir95-=-] and has been used in dierent forms in [HH97b, Hod97, HH99c]. The version presented here is in a fairly general form (theorem 39) and may be useful in other applications. We will discuss it in more d... |

6 |
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Citation Context ...ases We still have to investigate the relative strength of the n-dimensional-basis approximation to RRA, for varying n. Clearly, RA 3 RA 4 RA ! . The inclusions were shown to be strict in [Mad92]. Maddux conjectured [Mad83, page 90] that if 4 ns ! then RA l is notsnitely based relative to RA n : that is, there is nosrst-order sentence such that A 2 RA l i A 2 RA n and A j= . This is ag... |

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Citation Context ...nd other results such as proposition 18) be generalised to B ; CRA for uncountable ? It is worth mentioning that for n 6, RA n is not closed under completions (and hence, by a result of Venema [Ven=-=97]-=-, not Sahlqvist-axiomatisable). This can be shown by a slight adaptation of the example used to prove the same for RRA in [Hod97]. RA 4 (RA) is closed under completions, by a result of Monk [Mon70]; s... |

5 |
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Citation Context ...spect to a slightly dierent notion of n-square relativised representation, the answer to the problem is positive for all n; this can be shown by Herwig's theorem [Her95, Her98b], using techniques of [=-=AHN9-=-9]. Here is a vague question. Consider the following evidence: CRA 3 and CRA 4 are elementary, consisting of the atomic algebras in RA n ; a similar result holds for boolean algebras. But for n 5, C... |

5 |
Model Theory. Vol. 42 of Encyclopedia of Mathematics and its Applications
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(Show Context)
Citation Context ... N 1 , 3. If (x; y); (y; z); (x; z) 2 dom(N 2 ) then (N 2 (x; y) ; N 2 (y; z)) N 2 (x; z) 6= 0. 7 A more general theorem providing explicit recursive universal axiomatisations of `P C 0 ' classes [H=-=od9-=-3] satisfying some restrictions is proved in [HH98a]. Classes such as RAn ; SRaCAn ; RRA; etc., are covered. 23 Subnetworks We write (N 1 ; N 2 ) (N 0 1 ; N 0 2 ) if (i) N 1 N 0 1 , and (ii) wheneve... |

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(Show Context)
Citation Context ...sense of denition 6 below; this algorithm accepts A if and only if A 2 B n , if and only if A + 2 B n (because A issnite, so that A = A + ), if and only if A 2 RA n , as required.) It was shown in [H=-=H99c]-=- that the corresponding problem for RRA is undecidable, so this is an advantage of using RA n instead. 1.9 Complete relativised representations Now that we have a notion of representation, we can look... |

2 |
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Citation Context ...1 Questions related to thesnite base property are also raised. For n 3, the equational theory of the variety RA n is undecidable: see for example, [AKN + 96, theorem 3.1(i)], and, for RA 3 ( = SA), [=-=Mad94]-=-. Now we gave a recursive axiomatisation of RA n in theorem 35. So the validities of RA n are recursively enumerable. Hence, it cannot be that every equation that is not valid in RA n fails in somesni... |

1 | Decidable versions of logic - Nemeti - 1995 |