Groups and algebras of binary relations
Bulletin of Symbolic Logic 8 (1):38-64 (2002)
| Abstract | In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras, He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jonsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms-called coset relation algebras-that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well, We prove that every atomic relation algebra satisfying a certain measurability condition-a condition generalizing the conditions imposed by Jonsson and Tarski-is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,701 |
| External links |
  Try with proxy. |
| Through your library | Configure |
Similar books and articlesTarek Sayed Ahmed (2008). On Complete Representations of Reducts of Polyadic Algebras. Studia Logica 89 (3):325 - 332.
Vera Stebletsova (2000). Weakly Associative Relation Algebras with Polyadic Composition Operations. Studia Logica 66 (2):297-323.
Vera Stebletsova & Yde Venema (2001). Undecidable Theories of Lyndon Algebras. Journal of Symbolic Logic 66 (1):207-224.
Roger D. Maddux (1992). Relation Algebras of Every Dimension. Journal of Symbolic Logic 57 (4):1213-1229.
Bronisław Tembrowski (1983). The Theory of Boolean Algebras with an Additional Binary Operation. Studia Logica 42 (4):389 - 405.
Roger D. Maddux (1989). Nonfinite Axiomatizability Results for Cylindric and Relation Algebras. Journal of Symbolic Logic 54 (3):951-974.
Robin Hirsch, Ian Hodkinson & Roger D. Maddux (2002). Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory. Journal of Symbolic Logic 67 (1):197-213.
Roger D. Maddux (1994). Undecidable Semiassociative Relation Algebras. Journal of Symbolic Logic 59 (2):398-418.
Roger D. Maddux (1991). The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations. Studia Logica 50 (3-4):421 - 455.
Robert Goldblatt (1985). An Algebraic Study of Well-Foundedness. Studia Logica 44 (4):423 - 437.
Analytics
Monthly downloads
Sorry, there are not enough data points to plot this chart.
|
Added to index2009-01-28Total downloads4 ( #178,748 of 549,124 )Recent downloads (6 months)0How can I increase my downloads? |
My notes
Discussion
