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Distributions on distributive lattices and profinite groups. (English) Zbl 0699.06010

In the first section the author investigates the category DL, whose objects are the distributive lattices, and full subcategories of DL.
Some dualities between these categories are proved. B. Mazur introduced a definition of distribution on a directed projective system of discrete finite sets [B. Mazur, Analyse p-adique, Bourbaki report (1972); B. Mazur and P. Swinnerton-Dyer, Invent. Math. 25, 1-61 (1974; Zbl 0281.14016)]. The author formulates the following more general definition. Let L be a distributive lattice with 0, and A be an abelian group. A map \(\phi: L\to A\) such that \(\phi(a\vee b)=\phi(a)+\phi(b)\) for \(a,b\in L\) with \(a\wedge b=0\) is called a distribution on L with values in B.
Let \({\mathcal D}_{L,R}\) be the category whose objects are the distributions \(\phi: L\to A\) with values in R-modules. It is proved that the category \({\mathcal D}_{L,R}\) has an initial object u which is called the universal distribution on L. Let G be a profinite group. A distributive lattice L with 0 together with a continuous action \(G\times L\to L: (g,a)\mapsto ga\) of G on the discrete distributive lattice L, is called a G-lattice. Given a G-lattice L and a discrete G-module A, a distribution \(\phi: L\to A\) is called a G-distribution if \(\phi(ga)=g\phi(a)\) for \(g\in G\), \(a\in L.\)
Let us denote by \({\mathcal D}_{G,L}\) the category whose objects are G- distributions on the G-lattice L. Let H be a closed subgroup of G. A covariant functor \(St^ H_ G: {\mathcal D}_{H,L}\to {\mathcal D}_{G,L}\) is introduced and investigated. It is shown that there exists a natural connection between certain profinite groups and certain lattice ordered monoids.
A definition of ordinary distribution of weight w defined on a set X with values in R-modules is formulated. The definition is too complicated to be given here. It is a generalization of the definition of an ordinary distribution [S. Lang, Cyclotomic fields (1978; Zbl 0395.12005)] and that of a distribution of weight n [D. Kubert, Bull. Soc. Math. France 107, 179-202 (1979; Zbl 0409.12021)].
Let us denote by \({\mathcal D}_{X,R,w}\) the category whose objects are ordinary distributions of weight w defined on a set X with values in R- modules. It is shown that the category has an initial object \(\phi_ u\) which is called the universal ordinary distribution assigned to the triple \((X,R,w)\). A structure theorem for the R-module \(u=u_{X,R,w}\) is proved. The theorem extends a theorem of Kubert.
Reviewer: G.Pestov

MSC:

06D99 Distributive lattices
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