addition in groups of residues of the form ${\ensuremath{\mathbb{Z}}}/2^t{\ensuremath{\mathbb{Z}}}$, $t\ge 2$, which we have called addition with carry.

Theorem \[cor:types\] directly generalizes from base 2 to arbitrary prime bases $p$. In the case of a composite base $b$, we also have the equivalent of ( \[eqn:2-adic\]), as well as some additional direct products arising from the factorization of $b$ into prime powers. Even in the latter cases, only two types of addition of (sub)vectors of digits appear, addition with or without carry.

We have seen that there exist *only two types* of addition for digit vectors, while there exist many different variants of this binary operation, for example by changing the underlying partition sequence. As a consequence, from the duality principle only two types of function systems arise, the Walsh functions and the $b$-adic functions. For those digits that are added without carry, the Walsh system applies, and for those digits that are added with carry, the $b$-adic system of functions is appropriate. These two function systems cover all possible cases.

Notation {#s:notation}
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Throughout this paper, $b$ denotes a positive integer, $b\ge 2$, and $\mathbf{b}=(b_1, \ldots, b_s)$ stands for a vector of not necessarily distinct integers $b_i\ge 2$, $1\le 