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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 9 (6 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.
Finite Variable Logics
, 1993
"... In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this conne ..."
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Cited by 7 (0 self)
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In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for L ! 1! . Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout.
P.: The number of L∞κ–equivalent nonisomorphic models for κ weakly compact
 Fundam. Math
, 2002
"... 718 revision:20020112 modified:20020112 For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are L∞,κequivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of ..."
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718 revision:20020112 modified:20020112 For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are L∞,κequivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ 1 1definable over Vκ. By [SV] it is possible to have a generic extension, where the possible numbers of equivalence classes of Σ 1 1equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of No(M) for models of weakly compact cardinality. 1 1
A CONSTRUCTION OF MANY UNCOUNTABLE RINGS USING SFP DOMAINS AND ARONSZAJN TREES
, 1991
"... The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those ..."
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The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those rings whose indices are related in the ordering. We then take the subdirect product consisting of those elements of the direct product having finite support in the sense of this domain structure. In the special case where the homomorphisms are isomorphisms of a fixed ring S, our construction reduces to taking the Boolean power of 5 by a Boolean algebra canonically associated with the SFP domain. We examine the ideals of a ring obtainable in this way, showing for instance that each ideal is determined by its projections onto the factor rings. We give conditions on the underlying SFP domain that ensure that the ring is atomless. We examine the relationship between the L«,,0theory of the ring and that of the SFP domain. In Part II we prove a 'nonstructure theorem ' by exhibiting 2 N  pairwise nonembeddable Loo(uequivalent rings of cardinality K, with various higherorder properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality K,.