By Jürgen Müller

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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.

1 b) For example, let the exponential series be deﬁned as exp := n≥0 n! Xn ∈ 1 + XQ[[X]] ⊆ Q[[X]]. Then for the associated Taylor series we have exp(x) = exp(x), for all x ∈ C, hence we have exp ∈ C[[X]]∞ . Moreover, we have ∂ 1 1 n−1 = n≥0 n! X n = exp ∈ Q[[X]]. n≥1 (n−1)! X ∂X exp = If K is a ﬁeld of characteristic 0, then for f ∈ XK[[X]] we have exp(f ) := 1 n ∈ 1 + XK[[X]] ⊆ K[[X]], fulﬁlling the identity exp(f + g) = n≥0 n! (n−k)! 1 g j ) = n≥0 n! exp(f ) · exp(g) ∈ K[[X]], for all f, g ∈ XK[[X]], hence exp(f )−1 = exp(−f ) ∈ K[[X]].

We consider the basic example from number theory: the set N, partially ordered by divisibility | . Then N is a locally ﬁnite lattice, with meet gcd(c, d) and join lcm(c, d), for c, d ∈ N; it 1 as its zero element, but does not have a one element. Indeed, for n ∈ N, the interval in N between 1 and n is given as Tn := {d ∈ N; d | n}, that is the set of divisors of n . n ∈ N, and any interval in N, between m | n ∈ N say, is isomorphic to T m All saturated chains in Tn are of the form 1 = d0 | d1 | · · · | dt = n, where di di−1 ∈ N is a prime, for all i ∈ {1, .