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On Representable Relation Algebras

Journal of Symbolic Logic 31 (3):508-508 (1966)

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(1 other version)Relational dual tableau decision procedures and their applications to modal and intuitionistic logics.Joanna Golińska-Pilarek, Taneli Huuskonen & Emilio Muñoz-Velasco - 2014 - Annals of Pure and Applied Logic 165 (2):409-427.details This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) Download Export citation Bookmark 3 citations
Notions of density that imply representability in algebraic logic.Hajnal Andréka, Steven Givant, Szabolcs Mikulás, István Németi & András Simon - 1998 - Annals of Pure and Applied Logic 91 (2-3):93-190.details Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...) Download Export citation Bookmark 7 citations
Complexity of equations valid in algebras of relations part I: Strong non-finitizability.Hajnal Andréka - 1997 - Annals of Pure and Applied Logic 89 (2):149-209.details 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, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary 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: (...) Download Export citation Bookmark 23 citations
Complexity of equations valid in algebras of relations part II: Finite axiomatizations.Hajnal Andréka - 1997 - Annals of Pure and Applied Logic 89 (2-3):211-229.details 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, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary 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: (...) Download Export citation Bookmark 4 citations
Finite, integral, and finite-dimensional relation algebras: a brief history.Roger D. Maddux - 2004 - Annals of Pure and Applied Logic 127 (1-3):117-130.details Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order. Download Export citation Bookmark
Relation algebras from cylindric algebras, I.Robin Hirsch & Ian Hodkinson - 2001 - Annals of Pure and Applied Logic 112 (2-3):225-266.details We characterise the class S Ra CA n of subalgebras of relation algebra reducts of n -dimensional cylindric algebras by the notion of a ‘hyperbasis’, analogous to the cylindric basis of Maddux, and by representations. We outline a game–theoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of S Ra CA n. Download Export citation Bookmark 6 citations
A sequent calculus for relation algebras.Roger Maddux - 1983 - Annals of Pure and Applied Logic 25 (1):73-101.details Download Export citation Bookmark 19 citations
Atom structures of cylindric algebras and relation algebras.Ian Hodkinson - 1997 - Annals of Pure and Applied Logic 89 (2):117-148.details For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not.Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur (...) Download Export citation Bookmark 25 citations

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