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2010
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The Lattice Structure of Sets of Surjective Hyper-Operations

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The Lattice Structure of Sets of Surjective Hyper-Operations
Abstract. We study the lattice structure of sets (monoids) of surjective hyper-operations on an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order (fo) logic without equality. Specifically, for a countable set of relations (forming the finite-domain structure) B, the set of relations definable over B in positive fo logic without equality consists of exactly those relations that are invariant under the surjective hyper-endomorphisms (shes) of B. The evaluation problem for this logic on a fixed finite structure is a close relative of the quantified constraint satisfaction problem (QCSP). We study in particular an inverse operation that specifies an automorphism of our lattice. We use our results to give a dichotomy theorem for the evaluation problem of positive fo logic without equality on structures that are she-complementative, i.e. structures B whose set of shes is closed u...
Barnaby Martin
Added 10 Feb 2011
Updated 10 Feb 2011
Type Journal
Year 2010
Where CP
Authors Barnaby Martin
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