The reducts of equality up to primitive positive interdefinability

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

[1]  Simon Thomas,et al.  Reducts of the random graph , 1991, Journal of Symbolic Logic.

[2]  M. Pinsker.  Sublattices of the lattice of local clones , 2008, 0801.2392.

[3]  Heribert Vollmer,et al.  Playing with Boolean Blocks , Part II : Constraint Satisfaction Problems 1 , 2004 .

[4]  Manuel Bodirsky,et al.  Quantified Equality Constraints , 2007, LICS.

[5]  Reinhard Pöschel,et al.  Funktionen- und Relationenalgebren , 1979 .

[6]  Richard Statman,et al.  Logic for computer scientists , 1989 .

[7]  Martin Goldstern,et al.  A survey of clones on infinite sets , 2006 .

[8]  Michael Pinsker.  More Sublattices of the Lattice of Local Clones , 2010, Order.

[9]  Markus Junker,et al.  The 116 reducts of (ℚ, <, a) , 2008, Journal of Symbolic Logic.

[10]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[11]  E. C. Milner.  Basic WQO- and BQO-Theory , 1985 .

[12]  Analytic clones , 2004, math/0404214.

[13]  Markus Junker,et al.  The 116 reducts of (Q, <, a) , 2008, J. Symb. Log..

[14]  Manuel Bodirsky,et al.  The complexity of temporal constraint satisfaction problems , 2008, STOC.

[15]  P. Cameron.  Transitivity of permutation groups on unordered sets , 1976 .

[16]  The Minimal Clones above the Permutations , 2005, math/0512367.

[17]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[18]  Manuel Bodirsky,et al.  The Complexity of Equality Constraint Languages , 2006, Theory of Computing Systems.

[19]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory A.

[20]  M. Pinsker.  THE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET , 2004, math/0410406.

[21]  Jaroslav Nesetril,et al.  Constraint Satisfaction with Countable Homogeneous Templates , 2003, J. Log. Comput..

[22]  Friedhelm Meyer auf der Heide,et al.  Theory of Computing Systems: Guest Editors' Foreword , 2003 .

[23]  Emil L. Post.  The two-valued iterative systems of mathematical logic , 1942 .

[25]  Ivo G. Rosenberg,et al.  Locally Maximal Clones , 1982, J. Inf. Process. Cybern..

[26]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

[27]  L. Heindorf.  The maximal clones on countable sets that include all permutations , 2002 .

[28]  Ágnes Szendrei,et al.  Clones in universal algebra , 1986 .

[29]  M. Pinsker.  Maximal clones on uncountable sets that include all permutations , 2004, math/0401103.

[30]  P. Cameron,et al.  Oligomorphic permutation groups , 1990 .

[31]  D. Lau,et al.  Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory (Springer Monographs in Mathematics) , 2006 .

[32]  I. G. Rosenberg,et al.  Finite Clones Containing All Permutations , 1994, Canadian Journal of Mathematics.