Basises of integers under the multiplace shift operations

A notion of $(k,m)$-basis of $\mathbb Z$ is defined for integers $k,m$ ($0<k<m$, $(k,m)=\nobreakspace1$). Its definition uses an extension operation: a subset $U\subset\mathbb Z$ may be extended to $U\cup\{i,i+k,i+m\}$ if $|U\cap\{i,i+k,i+m\}|=2$ for some $i\in\mathbb Z$. A minimal subset $S\subset\mathbb Z$ is a $(m,k)$-basis if each $z\in\mathbb Z$ belongs to an extension of $S$ obtained by several extension operations. A structure of $(m,k)$-basises is investigated, precise bounds for the number of their elements are obtained.