  An integer $k$ is called regular (mod $n$) if there exists an integer $x$
such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$
possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that
$k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular
integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of
Euler's $\phi$ function. We introduce the multidimensional generalization of
$\varrho$, which is the analogue of Jordan's function. We establish identities
for the power sums of regular integers (mod $n$) and for some other finite sums
and products over regular integers (mod $n$), involving the Bernoulli
polynomials, the Gamma function and the cyclotomic polynomials, among others.
We also deduce an analogue of Menon's identity and investigate the maximal
orders of certain related functions.
