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Example text

Since we have distributivity, and c(f g) = (cf )g = f (cg) ∈ A(X), for all c ∈ Q, we conclude that A(X) is a non-commutative Q-algebra called the incidence algebra associated with the partially ordered set X. If X is finite, then letting X = {x1 , . . , xn }, where n := |X| ∈ N, we may assume that xi ≤ xj only if i ≤ j ∈ {1, . . , n}; note that this amounts to refining the given partial order on X to a total order. Then letting Tn (Q) := {A = [aij ]ij ∈ Qn×n ; aij = 0 if i > j} be the set of upper triangular (n × n)-matrices over Q, the map A(X) → Tn (Q) : f → [f (xi , xj )]ij is an injective homomorphism of Q-algebras, which is surjective if and only if X is totally ordered anyway.

N}; i=1 µi = i=1 λi }, thus 1 ≤ r < s ≤ n. Hence we have µr < λr , and λr ≤ λr−1 = µr−1 if r > 1, as well as µs > λs ≥ λs+1 ≥ µs+1 . This yields µ ν := [µ1 , . . , µr−1 , µr + 1, µr+1 , . . , µs−1 , µs − 1, µs+1 , . . , µn ] λ, hence ν = λ. It remains to show µr = µs whenever s > r + 1: Assume to the contrary that µr > µs , and let r < t := min{i ∈ {r + 1, . . , s}; µi−1 > µi } ≤ s. If t = s then µ [µ1 , . . , µr−1 , µr +1, µr+1 , . . , µs−2 , µs−1 −1, µs , . . , µn ] ν = λ, while if t < s then µ [µ1 , .

Then we have d | n ϕ( nd ) = n n d | n |{k ∈ {1, . . , d }; gcd(k, d ) = 1}| = d | n |{k ∈ {1, . . , n}; gcd(k, n) = d}| = | d | n {k ∈ {1, . . , n}; gcd(k, n) = d}| = n, implying that ζ ∗ ϕ = id, and thus ϕ = µ ∗ id = ϕ ∈ F (N). k We determine ϕ explicitly, using prime factorisations: For n = i=1 pni i ∈ N, where p1 , . . , pk ∈ N are pairwise distinct primes, n1 , . . ,k}; k −1 i=1 ( pi ) i i=1 i =1}| =n· ·( k i=1 (1 k i=1 − pni i − i ), thus implying ϕ(n) = n · 1 pi ) = k i=1 pni i −1 (pi − 1).

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Algebraic combinatorics by Jürgen Müller


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