2,339 research outputs found
A survey of clones on infinite sets
A clone on a set X is a set of finitary operations on X which contains all
projections and which is moreover closed under functional composition. Ordering
all clones on X by inclusion, one obtains a complete algebraic lattice, called
the clone lattice. We summarize what we know about the clone lattice on an
infinite base set X and formulate what we consider the most important open
problems.Comment: 37 page
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
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Stable domination and independence in algebraically closed valued fields
We seek to create tools for a model-theoretic analysis of types in
algebraically closed valued fields (ACVF). We give evidence to show that a
notion of 'domination by stable part' plays a key role. In Part A, we develop a
general theory of stably dominated types, showing they enjoy an excellent
independence theory, as well as a theory of definable types and germs of
definable functions. In Part B, we show that the general theory applies to
ACVF. Over a sufficiently rich base, we show that every type is stably
dominated over its image in the value group. For invariant types over any base,
stable domination coincides with a natural notion of `orthogonality to the
value group'. We also investigate other notions of independence, and show that
they all agree, and are well-behaved, for stably dominated types. One of these
is used to show that every type extends to an invariant type; definable types
are dense. Much of this work requires the use of imaginary elements. We also
show existence of prime models over reasonable bases, possibly including
imaginaries
Algebraic Properties of Valued Constraint Satisfaction Problem
The paper presents an algebraic framework for optimization problems
expressible as Valued Constraint Satisfaction Problems. Our results generalize
the algebraic framework for the decision version (CSPs) provided by Bulatov et
al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties
and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP
languages to weighted algebras. We show that the difficulty of VCSP depends
only on the weighted variety generated by the associated weighted algebra.
Paralleling the results for CSPs we exhibit a reduction to cores and rigid
cores which allows us to focus on idempotent weighted varieties. Further, we
propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the
hardness direction and verify that it agrees with known results for VCSPs on
two-element sets [Cohen et al. 2006], finite-valued VCSPs [Thapper and Zivny
2013] and conservative VCSPs [Kolmogorov and Zivny 2013].Comment: arXiv admin note: text overlap with arXiv:1207.6692 by other author
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
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