## Step by Step - Building Representations in Algebraic Logic (1995)

Venue: | Journal of Symbolic Logic |

Citations: | 28 - 15 self |

### BibTeX

@ARTICLE{Hirsch95stepby,

author = {Robin Hirsch and Ian Hodkinson},

title = {Step by Step - Building Representations in Algebraic Logic},

journal = {Journal of Symbolic Logic},

year = {1995},

volume = {62},

pages = {225--279}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...

### Citations

2385 | Maintaining knowledge about temporal intervals
- Allen
- 1983
(Show Context)
Citation Context ...tation. PROBLEM 3 Is it possible to find a set of axioms that characterise the finite relation algebras with universal representations? 6 Some examples of Relation Algebras The Allen interval algebra =-=[All83]-=- is a finite relation algebra with 13 atoms, representing the 13 possible relations that two non-empty intervals of a linear order can bear to each other. It is not hard to show [Hir94] that the atomi... |

973 | Temporal constraint networks
- Dechter, Meiri, et al.
- 1991
(Show Context)
Citation Context ...re !-categorical; see [Hir94] for more details. The point algebra and the Allen interval algebra can both be shown to be !-categorical (first proved in [VK86] and [vBC90]). The metric linear order of =-=[DMP91]-=- has only one countable, square representation: the rationals. 8 Ultrafilter extensions Up to now, we have mainly dealt with complete representations. The corresponding networks have been labelled by ... |

469 |
Kei s l e r, Model Theory
- Chang, J
- 1973
(Show Context)
Citation Context ...ion algebras. Note An infinite relation algebra cannot be (strongly or weakly) !-categorical. This is a corollary of the Ryll-Nardzewski--Engeler--Svenonius theorem on model-theoretic !-categoricity (=-=[CK90]-=-, theorem 2.3.13). PROBLEM 9 Can an infinite, atomic relation algebra be categorical in any uncountable cardinal (under the obvious definition)? November 30, 1995 29 PROBLEM 10 Find an infinite axioma... |

384 | Constraint propagation algorithms for temporal reasoning
- Vilain, Kautz
- 1986
(Show Context)
Citation Context ... that some of the best known relation algebras are !-categorical; see [Hir94] for more details. The point algebra and the Allen interval algebra can both be shown to be !-categorical (first proved in =-=[VK86]-=- and [vBC90]). The metric linear order of [DMP91] has only one countable, square representation: the rationals. 8 Ultrafilter extensions Up to now, we have mainly dealt with complete representations. ... |

218 | On the calculus of relations - Tarski - 1941 |

189 |
Boolean algebras with operators
- Jónsson, Tarski
- 1952
(Show Context)
Citation Context ...ion to A of the canonical projectionsj : Q i2I B i ! B j is surjective. ffl Any relation algebra A is isomorphic to a subdirect product of simple relation algebras (called the simple components of A) =-=[JT52]-=-. A is representable if and only if each of its simple components is representable. ffl A representation of a general atomic relation algebra partitions the domain into sets S i (i 2 I) and each atom ... |

125 | S.: A Formalization of Set Theory without Variables - Tarski, Givant - 1987 |

84 | Modal Logic
- VENEMA, OTHER
- 2001
(Show Context)
Citation Context ...ear flows with constant domains (in predicate temporal logic each moment in time is associated with a domain and there is quantification over that domain). Another related example of the method is in =-=[Ven93]-=- where a characterisation and completeness theorem are given for the class of so-called `finite dimensional cube-frames' This type of construction is closely related to Robinson forcing [Hod85, Rob79,... |

80 |
Exact and approximate reasoning about temporal relations
- Beek, Cohen
- 1990
(Show Context)
Citation Context ...of the best known relation algebras are !-categorical; see [Hir94] for more details. The point algebra and the Allen interval algebra can both be shown to be !-categorical (first proved in [VK86] and =-=[vBC90]-=-). The metric linear order of [DMP91] has only one countable, square representation: the rationals. 8 Ultrafilter extensions Up to now, we have mainly dealt with complete representations. The correspo... |

71 |
Cylindric Algebras (Part I
- Henkin, Monk, et al.
- 1971
(Show Context)
Citation Context ...of all cylindric set algebras of dimension ff is denoted Cs ff . ffl A cylindric algebra of dimension ff is defined to be a structure C = (C; ; \Gamma; 0; 1; cs; ds) ;!ff obeying the following axioms =-=[HMT71]-=- for every x; y 2 C; ; ;s! ff: 1. (C; ; \Gamma; 0; 1) is a boolean algebra 2. cs0 = 0 3. xscsx 4. cs(xscsy) = csxscsy 5. cscsx = cscsx 6. ds= 1 7. ifs6= ; , then ds= cs(dsds) 8. ifs6= , then cs(dsx)sc... |

66 |
The representation of relational algebras
- Lyndon
- 1956
(Show Context)
Citation Context ... over the class PRA. Hence, RRA is simply the closure of PRA under isomorphism. It is natural to ask whether the axioms defining RA are complete over PRA --- i.e., whether they axiomatise RRA. Lyndon =-=[Lyn50]-=- showed that the answer was negative 3 : there are (abstract) relation algebras, i.e. structures satisfying axioms 1 to 9 above, which are not isomorphic to any concrete algebra of binary relations. L... |

61 | The Origin of Relation Algebras
- Maddux
- 1991
(Show Context)
Citation Context ...tep by Step method Examples of this technique are numerous. The Henkin construction for proving the completeness of first-order logic is an important early example. The method is used by Roger Maddux =-=[Mad69]-=- to show that every semi-associative relation algebra has a representation as a subalgebra of a relativized proper relation algebra. Maddux went on in [Mad82] to show that every weakly associative rel... |

49 |
Some varieties containing relation algebras
- Maddux
- 1982
(Show Context)
Citation Context ...ample. The method is used by Roger Maddux [Mad69] to show that every semi-associative relation algebra has a representation as a subalgebra of a relativized proper relation algebra. Maddux went on in =-=[Mad82]-=- to show that every weakly associative relation algebra can be represented as a subalgebra of a proper relation algebra relativized to a symmetric, reflexive relation. (See section 11 below.) A consid... |

46 |
Algebraic logic
- Andréka, Németi, et al.
(Show Context)
Citation Context ...nd so completeness proofs over irreflexive flows of 2 References to relation algebra include [Tar41, JT48, TG87, Mad91, Giv94, N ' 95] and to cylindric algebra there is [HMT71, HMT85, Mon93, N ' 95]. =-=[AMN91]-=- contains a variety of articles on algebraic logic. November 30, 1995 4 time require an additional section in the proof where reflexive clusters are bulldozed to make them irreflexive (see, for exampl... |

45 |
On representable relation algebras
- Monk
- 1964
(Show Context)
Citation Context ... conditions as well as the proof of completeness. 3 McKenzie [McK70] page 286, provides the simplest example of a relation algebra, with just four atoms, which does not possess a representation. Monk =-=[Mon64]-=- proves that the class RRA cannot be finitely axiomatised at all. J'onsson has shown further that the representable relation algebras cannot be characterised by any set of axioms using a fixed number ... |

40 |
Sur l’extension aux relations de quelques proprietés des ordres
- Fraı̈ssé
- 1954
(Show Context)
Citation Context ...JEP and AP. 2 Definition The countable, universal, homogeneous, square representation constructed in the last part of the preceding theorem is called the Fraiss'e limit of the amalgamation class. See =-=[Fra54]-=-. 5 First-order characterisation of the finite, representable relation algebras Recall that the language L r of relation algebras has constants 1; 0; Id and function symbols ; \Gamma; ; ;, and, as pre... |

40 | The origin of relation algebras in the development and axiomatization of the calculus of relations - Maddux - 1991 |

36 | Algebraizations of quantifier logics: an introductory overview”, Studia Logica 50 - Németi - 1991 |

25 |
Infinite games with perfect information, Contributions to the theory of games
- Gale, Stewart
- 1953
(Show Context)
Citation Context ...formula more explicitly, let us consider what these winning strategies are. First, since Gn is of finite length it is a determined game, meaning that if 9 does not have a winning strategy then 8 does =-=[GS53]-=-. So what is a strategy for 8? It is a set of instructions telling him how to play each round of the game and the instructions for the ith round of the game (0si ! n) must depend only on player 9's pr... |

25 |
Algebraic Relativization and Arrow Logic
- Marx
- 1995
(Show Context)
Citation Context ...ativized to a symmetric, reflexive relation. (See section 11 below.) A considerable generalisation of this work on relativized relation algebra, also based on the step by step method, can be found in =-=[Mar95]-=-. The step by step method appears explicitly in a proof by Andr'eka of the Resek--Thompson theorem [AT88] to prove that their finite axiom set \Sigma is complete over a certain class of relativized cy... |

24 |
Representation problems for relation algebras
- Jónsson, Tarski
- 1192
(Show Context)
Citation Context ...lent to OE 0 n over simple relation algebras. We'll show that the equations feq n : n ! !g exactly characterise the class of all representable relation algebras, using the facts that RRA is a variety =-=[JT48]-=-, and so closed under subdirect products and homomorphic images; that every relation algebra A is a subdirect product of simple relation algebras A ( 2 ), each of which is a homomorphic image of A (th... |

22 | Cylindric set algebras - Henkin, Monk, et al. - 1981 |

19 | Complete representations in algebraic logic
- Hirsch, Hodkinson
- 1997
(Show Context)
Citation Context ...urther work Roger Maddux constructs an atomic, non-integral, representable relation algebra with no complete representation ([Mad69], page 154--173). It is related to an example of Lyndon [Lyn50]. In =-=[HH96]-=-, an integral, atomic representable relation algebra with no complete representation is constructed, and used to show that the class of relation algebras that have a complete representation is not ele... |

19 | Non-finite axiomatizability results for cylindric and relation algebras - Maddux - 1989 |

18 |
Representations of integral relation algebras
- McKenzie
- 1970
(Show Context)
Citation Context ...algebras. Later, we will approach this problem using games, which will give us a quite considerable simplification in the definition of the conditions as well as the proof of completeness. 3 McKenzie =-=[McK70]-=- page 286, provides the simplest example of a relation algebra, with just four atoms, which does not possess a representation. Monk [Mon64] proves that the class RRA cannot be finitely axiomatised at ... |

15 | Relation algebras of intervals
- Hirsch
- 1996
(Show Context)
Citation Context ...terval algebra [All83] is a finite relation algebra with 13 atoms, representing the 13 possible relations that two non-empty intervals of a linear order can bear to each other. It is not hard to show =-=[Hir94]-=- that the atomic networks of the Allen interval algebra form an amalgamation class. The last part of theorem 5 now tells us that it has a countable, universal, homogeneous, square representation, and ... |

14 |
Relation algebras and projective geometries
- Lyndon
- 1961
(Show Context)
Citation Context ...with each pair of distinct points related by #. Still, D has only one countable, square representation (up to base-isomorphism) and thus is `weakly !-categorical' (see section 7). ffl Lyndon Algebras =-=[Lyn61]-=- Lets4. The th Lyndon Algebra is an atomic relation algebra A with atoms Id; a 1 ; : : : ; a \Gamma1 . Each atom is self-converse, and composition of atoms is given by a i ; a i = Idsa i (0 ! i ! ) an... |

14 |
The representation of relation algebras
- McKenzie
- 1966
(Show Context)
Citation Context ...somorphism. Thus N is a homomorphism if and only if it is a representation. 2 The following theorem is due to J. D. Monk. Here we give an alternative, simpler proof than the original one (reported in =-=[McK66]-=-, theorem 2.12). November 30, 1995 31 THEOREM 22 (J. D. Monk) A relation algebra A is representable if and only if its ultrafilter extension A + has a complete representation. 14 PROOF: ( Since A can ... |

12 |
Axioms for tense logic I: "since" and "until
- Burgess
- 1982
(Show Context)
Citation Context ...-Thompson theorem [AT88] to prove that their finite axiom set \Sigma is complete over a certain class of relativized cylindric set algebras. An interesting example from temporal logic is the proof in =-=[Bur82]-=- where a set of axioms in the temporal language with Until and Since is shown to be complete over linear flows of time. The proof differs from the standard Henkin construction in the following way. In... |

9 | The Algebra of Convex Time Intervals
- Ladkin, Maddux
- 1987
(Show Context)
Citation Context ... is tempting to draw the conclusion that a relation algebra is `!-categorical' (that all its countable representations are base-isomorphic --- a theorem first proved for the Allen interval algebra in =-=[LM87]-=- 12 ) if its atomic networks form an amalgamation class. However, this conclusion would be far too hasty, and to illustrate this we define a relation algebra whose atomic networks form an amalgamation... |

9 |
Axiomatising First-order Temporal Logic: Until and Since over Linear Time
- Reynolds
- 1996
(Show Context)
Citation Context ...rk the model is built step by step and only universal-existential conditions need be satisfied. Restrictions on space prevent us from giving more details here. The same step by step method is used in =-=[Rey92]-=- to extend this result to predicate temporal logic by showing the completeness of some axioms over the class of all linear flows with constant domains (in predicate temporal logic each moment in time ... |

8 | Completing theories by forcing - BARWISE, ROBINSON - 1970 |

8 | The structure of relation algebras generated by relativizations - Givant - 1994 |

7 | Weakly representable but not representable relation algebras - Andréka - 1994 |

6 | Cylindric Algebras Part II - Henkin, Monk, et al. - 1985 |

4 | The representation of relation algebras, ii - Lyndon - 1956 |

4 | Finite integral relation algebras, Universal Algebra and Lattice Theory, Springer-Verlag - Maddux - 1985 |

3 |
A Stone type representation theorem for algebras of relations of higher rank
- Andr'eka, Thompson
- 1988
(Show Context)
Citation Context ... work on relativized relation algebra, also based on the step by step method, can be found in [Mar95]. The step by step method appears explicitly in a proof by Andr'eka of the Resek--Thompson theorem =-=[AT88]-=- to prove that their finite axiom set \Sigma is complete over a certain class of relativized cylindric set algebras. An interesting example from temporal logic is the proof in [Bur82] where a set of a... |

3 |
Building Models by Games. Number 2
- Hodges
- 1985
(Show Context)
Citation Context ... An elegant method of determining whether a relation algebra has a representation and, if so, what properties its representations might have, is to try to construct a representation using a game (see =-=[Hod85]-=-). The crucial conjunction lemma (below) will allow us to avoid lengthy book-keeping arguments in the proofs. Let A be an atomic relation algebra and let K be a class of finite atomic A-networks. Our ... |

3 | Finite integral relation algebras - Maddux - 1985 |

2 |
Some interconnections between modern algebra and mathematical logic
- Henkin
- 1953
(Show Context)
Citation Context ...versal sentences OE 0 n exactly characterise the representable relation algebras. By an argument of Henkin this result carries over from countable to arbitrary relation algebras. A theorem of Henkin (=-=[Hen53]-=- theorem 1, essentially compactness) implies that if we can show that every finitely generated relation algebra satisfies fOE 0 n : n ! !g if and only if it is representable then the result holds for ... |

1 |
A nonpermutational integral relation algebra
- Andr'eka, Duntsch, et al.
- 1992
(Show Context)
Citation Context ...ctively permutational) representation. In [McK70] it is shown that the subalgebras of group relation algebras are not finitely axiomatisable over the permutational relation algebras. Andr'eka et. al. =-=[ADN92]-=- go on to show that the permutational relation algebras are not finitely axiomatisable over the integral relation algebras. They do this by constructing an infinite set of integral non-permutational r... |

1 |
Reasoning in and about Time
- Blackburn
- 1989
(Show Context)
Citation Context ...ntains a variety of articles on algebraic logic. November 30, 1995 4 time require an additional section in the proof where reflexive clusters are bulldozed to make them irreflexive (see, for example, =-=[Bla89]-=-). The Burgess proof works in a different way. To prove completeness a consistent set of formulas is extended to a MCS. So we start with just a single point in the structure. New points, labelled by M... |

1 |
Projective Planes. Graduate Texts in Mathematics
- Hughes, Piper
- 1973
(Show Context)
Citation Context ... 11, respectively. This is possible, as 2+3+5+7+11 = 28, while l has 30 elements. Then oe has order 2:3:5:7:11 = 2310. But oe can't be induced by any collineation, as we can show (cf. theorem 13.1 of =-=[HP73]-=-) that no collineation of G has order greater than the number of points of G, viz., jGj = n 2 + n + 1 = 871. To see this, let \Gamma be the group generated by any collineation of G (or, more generally... |

1 | Selected Papers: Model Theory and Algebra - Robinson - 1979 |

1 | Barwise and A Robinson. Completing theories by forcing - J - 1970 |