We consider positional numeration systems with negative real base −β, where β>1, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal (−β)-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base β2 with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy (−β)-representation. Such a characterization allows us to study the set of uniquely representable numbers. In the case that β is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy (−β)-representation using a set of forbidden strings.