Distributors and Wallman locales

• Autores: Marcel Erné
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 34, Nº 1, 2008, págs. 69-98
• Idioma: inglés
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• Resumen

  • A distributor in an m-semilattice (a join-semilattice with an isotone multiplication) is a nonempty upper set containing both the join of a and c and the join of b and c iff it contains the join of ab and c. Distributors provide a far-reaching extension of the filter theory for distributive lattices, quantales and similar objects to structures where no distributive laws are assumed a priori. The closure system of all distributors is a universal locale over the given m-semilattice, and the principal distributors form its universal distributive lattice quotient. Moreover, distributors are a helpful tool for the spectral theory of m-semilattices and various related kinds of (ordered) algebras. We present diverse alternative characterizations of Scott-open distributors in complete m-semilattices, for example as kernels of join-preserving homomorphisms onto compact locales, and we establish a one-to-one correspondence between Scott-open distributors and those nuclei whose range is a Wallman locale, the pointfree analogue of a compact T1-topology