One gives a formula for the calculation of the local intersection multiplicity index of analytic varieties, analogous to the Bézout formula in the case of algebraic varieties in the projective space, in the case of normal crossings. One obtains also a recurrent process for the calculation of the local intersection multiplicity index of plane analytic curves.

We give Lambek-Carlitz type characterization for completely multiplicative reduced incidence functions in Möbius categories of full binomial type. The $q$-analog of the Lambek-Carlitz type characterization of exponential series is also established.

Rings of formal power series $k\left[\right[C\left]\right]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k\left[\right[C\left]\right]$ : for every $\sigma$ in $k\left[\right[C\left]\right]$ and $c$ in $C$, we let $v(c,\sigma )$ be the first element of the support of $\sigma$ which is greater than or equal to $c$. Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in $k\left[\right[C\left]\right]$. We prove that a cyclically valued ring is a subring of a power series ring $k\left[\right[C,\theta \left]\right]$ with...