By Giovanni Landi (auth.)

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**Additional info for An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment**

**Example text**

The point corresponding to Yn (k) at level n of the diagram is linked to the point corresponding to Yn+1 (j) at level n + 1, if and only if Yn (k) ∩ Fn+1 (j) = ∅. The multiplicity of the embedding is always 1. Thus, the partial embeddings of the diagram are given by (n, k) p (n + 1, j) , with p = 1 if Yn (k) ∩ Fn+1 (j) = ∅ , p = 0 otherwise . 69) above. 36). We shall construct a similar correspondence between closed subsets F ⊆ P and the ideals IF in the AF-algebra A with subdiagram ΛF ⊆ D(A).

13 is seen to coincide with the diagram of Fig. 11. As we have previously said, the latter corresponds to the AF-algebra A∨ = CP1 + K(H) + CP2 , H = H1 ⊕ H2 . 79) Example 10. Another interesting example is provided by the lattice P4 (S 1 ) for the one-dimensional sphere in Fig. 1. This topological space contains six closed sets: K0 = {x1 , x2 , x3 , x4 } , K1 = {x1 , x3 , x4 } , K2 = {x3 } , K3 = {x4 } , K4 = {x2 , x3 , x4 } , K5 = {x3 , x4 } = K2 ∪ K3 . 4 Noncommutative Lattices 49 01 11 21 31 41 s ❅ ❅ ❅s13 s ❅ ❅ ❅s s s23 22 ❅ ❅ ❅s s s33 32 ❅ ❅ ❅s s s43 41 ❅ ..

48) B ∈ A. Moreover, the dimensions (p1 , p2 ) and (q1 , q2 ) are related N11 p1 + N12 p2 = q1 , N21 p1 + N22 p2 = q2 . 4 Noncommutative Lattices n1 j=1 k=1 Nkj Md(1) (C) . 51) Nkj Md(1) (C) with a subalgebra of Md(2) (C). The j nonnegative integers Nkj satisfy the condition n1 j=1 (1) Nkj dj k (2) = dk . 52) One says that the algebra Md(1) (C) is partially embedded in Md(2) (C) with j k multiplicity Nkj . A useful way of representing the algebras A1 , A2 and the embedding A1 → A2 is by means of a diagram, the so called Bratteli diagram (1) [17], which can be constructed out of the dimensions dj , j = 1, .