Exercise 22, § 11

Algebra, III, 652

Let E be an ordered set in which every interval $[a, b]$ is finite. Let S be the subset of $E \times E$ consisting of the ordered pairs $(x, y)$ such that $x \leq y$ . Show that a coassociative cogebra structure is defined on the A-module $A^{(S)}$ of formal linear combinations of the elements of S by taking as coproduct

c((x, y)) = \sum_{x \leq z \leq y} (x, z) \otimes (z, y).

For this cogebra the linear form $\gamma$ such that

\gamma((x, y)) = 0 \quad \text{if} \quad x \neq y \text{ in } E, \qquad \gamma((x, x)) = 1

is a counit.

Solution

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