12,029 research outputs found
Integral Lattices in TQFT
We find explicit bases for naturally defined lattices over a ring of
algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of
odd prime order. Some applications relating quantum invariants to classical
3-manifold topology are given.Comment: 31 pages, v2: minor modifications. To appear in Ann. Sci. Ecole Norm.
Su
The equational theory of the natural join and inner union is decidable
The natural join and the inner union operations combine relations of a
database. Tropashko and Spight [24] realized that these two operations are the
meet and join operations in a class of lattices, known by now as the relational
lattices. They proposed then lattice theory as an algebraic approach to the
theory of databases, alternative to the relational algebra. Previous works [17,
22] proved that the quasiequational theory of these lattices-that is, the set
of definite Horn sentences valid in all the relational lattices-is undecidable,
even when the signature is restricted to the pure lattice signature. We prove
here that the equational theory of relational lattices is decidable. That, is
we provide an algorithm to decide if two lattice theoretic terms t, s are made
equal under all intepretations in some relational lattice. We achieve this goal
by showing that if an inclusion t s fails in any of these lattices, then
it fails in a relational lattice whose size is bound by a triple exponential
function of the sizes of t and s.Comment: arXiv admin note: text overlap with arXiv:1607.0298
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
Relational lattices via duality
The natural join and the inner union combine in different ways tables of a
relational database. Tropashko [18] observed that these two operations are the
meet and join in a class of lattices-called the relational lattices- and
proposed lattice theory as an alternative algebraic approach to databases.
Aiming at query optimization, Litak et al. [12] initiated the study of the
equational theory of these lattices. We carry on with this project, making use
of the duality theory developed in [16]. The contributions of this paper are as
follows. Let A be a set of column's names and D be a set of cell values; we
characterize the dual space of the relational lattice R(D, A) by means of a
generalized ultrametric space, whose elements are the functions from A to D,
with the P (A)-valued distance being the Hamming one but lifted to subsets of
A. We use the dual space to present an equational axiomatization of these
lattices that reflects the combinatorial properties of these generalized
ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that
these equations correspond to combinatorial properties of the dual spaces of
lattices, in a technical sense analogous of correspondence theory in modal
logic. In particular, this leads to an exact characterization of the finite
lattices satisfying these equations.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven,
Netherland
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