  We study the ordering kinetics of a generalization of the voter model with
long-range interactions, the $p$-voter model, in one dimension. It is defined
in terms of boolean variables $S_{i}$, agents or spins, located on sites $i$ of
a lattice, each of which takes in an elementary move the state of the majority
of $p$ other agents at distances $r$ chosen with probability $P(r)\propto
r^{-\alpha}$. For $p=2$ the model can be exactly mapped onto the case with
$p=1$, which amounts to the voter model with long-range interactions decaying
algebraically. For $3\le p<\infty$, instead, the dynamics falls into the
universality class of the one-dimensional Ising model with long-ranged coupling
constant $J(r)=P(r)$ quenched to small finite temperatures. In the limit $p\to
\infty$, a crossover to the (different) behavior of the long-range Ising model
quenched to zero temperature is observed.
