G.C. Rota. The Valuation Ring of a Distributive Lattice. Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. 574--628.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....such that m(0) 0; m(1) 1 and m(b 1 ) m(b 2 ) m(b 1 b 2 ) m(b 1 b 2 ) Theorem 2.5. v : B V (B) is a universal valuation: if m : B E is any valuation, there exists a unique map I : V (B) E such that I ffi v = m. The importance of this construction is stressed in the work of Rota [14]. The elements of V (B) may be thought of as simple random variables on a probability space, while I is the expectation operator. 2.5. Dedekind oe complete Riesz Spaces. We now give a point free version of some results of [17] and present the spectral decomposition of an element of a Dedekind ....

G.C. Rota. The Valuation Ring of a Distributive Lattice. Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. 574--628.

....symbols v(b) b 2 B with the relations vb 1 vb 2 = v(b 1 b 2 ) v(b 1 b 2 ) v1 = 1 v0 = 0 We define 0 a iff a can be written a = Sigmar i vb i with r i 0. The multiplication is defined by v(b 1 )v(b 2 ) v(b 1 b 2 ) The importance of this construction is stressed in the work of Rota [14]. 3. Representation Theory 3.1. Riesz Space. Following [16] we can see any Riesz space E with a strong unit as a set of continuous functions over a compact Hausdorff space X. It may be interesting to give a direct point free description of this space. Theorem 3.1. The following geometrical ....

G.C. Rota. The Valuation Ring of a distributive lattice. Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. 574--628.

....topologically rather than via the more natural blend of order and topology devised by Priestley. As Rota put it, Stone s representation theorem of 1936 for distributive lattices closely imitated his representation theorem for Boolean algebras, and as a consequence turned out to be too contrived. [Rot73] Priestley simply equips the schedule with both Birkhoff s partial order and Stone s topology to make it a partially ordered Stone space, a set bearing two structures, that of a partial order and that of a topology, Stone s in this case. The partial order deals with the absence of complementation, ....

G.-C. Rota. The valuation ring of a distributive lattice. In Proc. Univ. of Houston Lattice Theory Conf. Dept. of Math., Univ. of Houston, 1973.

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