Classes of structures containing $B_n(x,y)=\binom{x+y}x\mod
n$ or related relations are considered. Definability of
usual arithmetical operations and operations modulo $n$ in
such classes is investigated. Further, for every infinite
set $X$ of positive integers the elementary theory of the
class $\ClassN{B_n}{n\in X}$ is shown to be undecidable
(although it may happen that every its element has decidable
theory).