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Register a new userMoorhouses question on locally finite generalized quadrangles, part 1 -- the countable case
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K. Thas
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We settle a question posed by G. Eric Moorhouse on the model theory and existence of locally finite generalized quadrangles. In this paper, we completely handle the case in which the generalized quadrangles have a countable number of elements.
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We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
We solve a problem posed by Cardinali and Sastry [2] about factorization of $2$-covers of finite classical generalized quadrangles. To that end, we develop a general theory of cover factorization for generalized quadrangles, and in particular we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semipartial geometries coming from $theta$-covers, and consider related problems.
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $sgeq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdos--Renyi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measures on the space of such graphs. The main idea of our construction is based on the new notions of {it measurable universal}, and {it topologically universal} graphs, which are interesting themselves. The realization of the construction can be regarded as two-step randomization for universal measurable graph : {it randomization in vertices} and {it randomization in edges}. For $K_s$-free, $sgeq 3$ there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by D. Aldous about $S_{infty}$-invariant matrices, which we reformulate in appropriate way.
Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant $land$-representation by $land$-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in details properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.
Locally finite omega languages were introduced by Ressayre in [Journal of Symbolic Logic, Volume 53, No. 4, p.1009-1026]. They generalize omega languages accepted by finite automata or defined by monadic second order sentences. We study here closure properties of the family LOC_omega of locally finite omega languages. In particular we show that the class LOC_omega is neither closed under intersection nor under complementation, giving an answer to a question of Ressayre.
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