By Jürgen Müller

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**Extra resources for Algebraic combinatorics**

**Example text**

Hence let N be a ﬁnite set of cardinality n ∈ N0 . Since for L ⊆ M ⊆ N we have [L, M ] ∼ = [L \ L, M \ L] ∼ = [{}, {1, . . , |M | − |L|}] as partially ordered sets, we conclude that it suﬃces to compute µn := µ(∅, N ). By induction on n ∈ N0 we show that µn = (−1)n : For n = 0 we have µ0 = 1; and for n ≥ 1 the set N has nk subset of cardinality k, for all k ∈ {0, . . , n}, n−1 hence we get µn = − k=0 (−1)k nk = (−1)n . b) We show that the principle of inclusion-exclusion is a special case of M¨ obius inversion: Let N be an arbitrary set again.

A) Let X := {∅, {1}, {2}, {2, 3}, {1, 2, 3}}, partially ordered by set-theoretic inclusion ⊆. Then by inspection X is a lattice, with meet M ∩ M ∈ X and join M ∪ M ∈ X, for M, M ∈ X, and has ∅ and {1, 2, 3} as its zero and one elements, respectively. But both ∅ ⊂ {1} ⊂ {1, 2, 3} and ∅ ⊂ {2} ⊂ {2, 3} ⊂ {1, 2, 3} are saturated chains, of lengths 2 and 3, respectively, thus X is not graded. b) The set N0 is a partially ordered set with respect to the ≤ relation. It is a locally ﬁnite chain, thus in particular a lattice, with meet min{i, j} and join max{i, j}, for i, j ∈ N0 ; it has zero element 0, but no maximal element.

From e2 M0 = e2 , . . , ea C and e1 M0 = e1 , ea+1 , . . , en C we conclude Cn×1 = e2 M0 ⊕ e1 M0 , hence M0 has Jordan normal form Ja−1 ⊕ Jb+1 . 1) Incidence algebras. a) Let X be a locally ﬁnite partially ordered set, and let A(X) := {f ∈ Maps(X × X, Q); f (x, y) = 0 if x ≤ y}. Then A(X) is a Q-vector space with respect to pointwise addition (f +g)(x, y) := f (x, y)+g(x, y) and pointwise scalar multiplication (cf )(x, y) := c · f (x, y), for all f, g ∈ A(X) and x, y ∈ X and c ∈ Q. Convolutional multiplication on A(X) is well-deﬁned by letting (f g)(x, y) := z∈X f (x, z)g(z, y), for all f, g ∈ A(X) and x, y ∈ X: If x ≤ y then all summands vanish, hence (f g)(x, y) = 0, while if x ≤ y we get (f g)(x, y) = x≤z≤y f (x, z)g(z, y), which by local ﬁniteness is a ﬁnite sum.

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