Exercise 14, § 6

Let $L$ be a finite ordered set and $A$ a commutative ring. In the ring $M$ of square matrices over $A$ with index set $L$, the set of matrices $m = (m_{uv})$ such that $m_{uu} = 0$ except when $u \leq v$ is a subring $B$.

a) An element $m = (m_{uv})$ of $B$ is invertible in $B$ if and only if $m_{uu}$ is invertible in $A$ for all $u \in L$.

b) Suppose henceforth that $A = \mathbf{Z}$. Let $\zeta$ be the element of $B$ defined by $\zeta(u, v) = 1$ if $u \leq v$ and $\zeta(u, v) = 0$ otherwise. Show that $S$ has an inverse $\mu$ (also written $\mu_L$) in $B$ (called the Mobius function of $L$). Let $f$ and $g$ be two mappings of $L$ into $\mathbf{Z}$. Show that the relations

and are equivalent.

c) Assume that $L$ is the product of a finite family of ordered sets $L_1, \ldots, L_r$. Show that

for $x = (x_1, \ldots, x_r)$ and $y = (y_1, \ldots, y_r)$ such that $x \leq y$.

d) Let $m \geq 1$ be an integer and let $D$ be the set of divisors of $m$, equipped with the order relation « $x$ divides $y$ ». Show that $\mu_D(x, y) = (-1)^s$ where $s$ is the cardinal of the set of prime numbers $p$ dividing $y/x$ when $y/x$ is not divisible by the square of an integer, and $\mu_D(x, y) = 0$ otherwise. (By applying c) reduce to the case where $m$ is a power of a prime number.)

e) Suppose that the ordered set $L$ is lattice-ordered. Let $a, b, c$ be distinct elements of $L$ such that $a \leq b \leq c$, and let $\Phi$ be the set of $x \in L$ such that $a \leq x$ and $\sup(x, b) = c$, and $\Psi$ the corresponding set defined by $y \leq c$ and $\inf(y, b) = a$. Show that we have the relation

f) Let $T$ be a finite set and $\mathscr{R}(T)$ the set of equivalence relations on $T$, ordered by the relation « $R$ is finer than $S$ », written $R \leq S$. Show that $\mathscr{R}(T)$ is lattice-ordered. Let $R, S$ in $\mathscr{R}$ be such that $R \leq S$; let $U_1, \ldots, U_k$ be the equivalence classes under $S$ and for $1 \leq i \leq k$, let $\alpha_i$ be the number of equivalence classes under $R$ contained in $U_i$. Show that

(if $A_i$ is the set of equivalence classes under $R$ contained in $U_i$, observe that the interval $(R, S)$ of $\mathscr{R}(T)$ is isomorphic to $\mathscr{R}(A_1) \times \ldots \times \mathscr{R}(A_r)$, apply c) and reduce to the case where $R$ is the least element of $\mathscr{R}(T)$ and $S$ the greatest ; treat this last case by using e)).