Extending maps to profinite completions in finitely generated quasivarieties

We consider the problem of extending maps from algebras to their profinite completions in finitely generated quasivarieties. Our developments are based on the construction of the profinite completion of an algebra as its natural extension. We provide an extension which is a multi-map and we study its continuity properties, and the conditions under which it is a map

Novel Crohn Disease Locus Identified by Genome-Wide Association Maps to a Gene Desert on 5p13.1 and Modulates Expression of PTGER4

To identify novel susceptibility loci for Crohn disease (CD), we undertook a genome-wide association study with more than 300,000 SNPs characterized in 547 patients and 928 controls. We found three chromosome regions that provided evidence of disease association with p-values between 10(−6) and 10(−9). Two of these (IL23R on Chromosome 1 and CARD15 on Chromosome 16) correspond to genes previously reported to be associated with CD. In addition, a 250-kb region of Chromosome 5p13.1 was found to contain multiple markers with strongly suggestive evidence of disease association (including four markers with p < 10(−7)). We replicated the results for 5p13.1 by studying 1,266 additional CD patients, 559 additional controls, and 428 trios. Significant evidence of association (p < 4 × 10(−4)) was found in case/control comparisons with the replication data, while associated alleles were over-transmitted to affected offspring (p < 0.05), thus confirming that the 5p13.1 locus contributes to CD susceptibility. The CD-associated 250-kb region was saturated with 111 SNP markers. Haplotype analysis supports a complex locus architecture with multiple variants contributing to disease susceptibility. The novel 5p13.1 CD locus is contained within a 1.25-Mb gene desert. We present evidence that disease-associated alleles correlate with quantitative expression levels of the prostaglandin receptor EP4, PTGER4, the gene that resides closest to the associated region. Our results identify a major new susceptibility locus for CD, and suggest that genetic variants associated with disease risk at this locus could modulate cis-acting regulatory elements of PTGER4

A Duality for Boolean Contact Algebras

The well-known de Vries duality, established by H. de Vries in 1962, states that the category of compact Hausdorff spaces is dually equivalent to that of de Vries algebras. The notion of Boolean contact algebra (BCA) was developed independently in the context of region-based theory of space. Düntsch and Winter established a representation theorem for BCAs, showing that every BCA is isomorphic to a dense subalgebra of the regular closed sets of a T_1 weakly regular space. It appears that BCAs are a direct generalization of de Vries algebras, and that the representation theorem for complete BCAs generalizes de Vries duality for objects. During a conference, Vakarelov raised the question of dualizing morphisms. We answer this question using concepts similar to those of modal logic's neighborhood semantics