Math. Log. Quart. 57, No. 1, 87 – 94 (2011) / DOI 10.1002/malq.200910120 Existence of partial transposition means representability in cylindric algebras Mikl´ os Ferenczi ∗ Department of Algebra, Budapest University of Technology and Economics, H–1521 Budapest, Hungary Received 9 September 2009, revised 31 March 2010, accepted 6 April 2010 Published online 18 January 2011 Key words Cylindric relativized algebra, polyadic algebra, representation, transposition. MSC (2010) 03G15 We show that the representability of cylindric algebras by relativized set algebras depends on the scope of the operation transposition which can be defined on the algebra. The existence of “partial transposition” assures this kind of representability of the cylindric algebra (while the existence of transposition assures polyadic representation). Further we characterize those cylindric algebras in which the operator transposition can be introduced. c  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction As is well known, cylindric algebras are not representable in the usual universal algebraic sense, i.e. as subdirect products of cylindric set algebras. But, by the Resek–Thompson’s representation theorem, if the system of cylindric axioms is extended by the merry-go-round properties (for short, MGR properties), then the algebra is representable by a cylindric relativized set algebra in CA α (see [7, 3.2.88]). By representability, in this paper, we mean representability by relativized set algebra. MGR properties seem to be more complex properties than the original cylindric axioms. The question arises: why exactly the MGR properties assure representability, what is the background of these assumptions? We are going to answer this question. For the sake of simplicity throughout the paper we deal with weakly commutative cylindric algebras (algebras in CA - α ) rather than cylindric algebras, that is the commutativity of the singles substitutions are assumed instead that of cylindrifications. However, all the results can be extended to cylindric algebras, too. On the one hand, it is known that in every cylindric algebra the transposition operator (operator which inter- changes the ith and j th dimensions) can be defined for elements with at least two closed (bound) dimensions (see [7, 1.5.12]). On the other hand, CA - α having transposition operator (i.e. transposition algebras) is representable ([6, Theorem 1]). But, for the representability of a CA - α it is enough to require less. We introduce the concept of partial transposition algebra (PTA α ). Here the operator transposition works only for dimension complemented elements, i.e. for elements with at least one closed (bound) dimension. We are going to show that a CA - α is representable (thus it satisfies MGR) if and only if it can be extended to a partial transposition algebra (Theorem 3.5). We prove in general, that partial transposition algebras are repre- sentable (Theorem 3.9). Finally, we consider transposition algebras instead of partial transposition algebras. As an application of our results we characterize those cylindric algebras which are reducts of transposition algebras (Theorem 3.10). These CA - α ’s are representable as reduct of certain polyadic relativized set algebras. ∗ e-mail: ferenczi@math.bme.hu c  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim