Kybernetika - Article detailKybernetikaInternational journal of Institute of Information Theory and AutomationGreedy and lazy representations in negative base systems1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-\beta)$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base $\beta^2$ with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy $(-\beta)$-representation. Such a characterization allows us to study the set of uniquely representable numbers. In the case that $\beta$ is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy $(-\beta)$-representation using a set of forbidden strings.]]>K. DajaniCh. KalleTransformations generating negative $\beta$-expansions.Integers 11B (2011), A5, 1-18.K. DajaniC. KraaikampFrom greedy to lazy expansions and their driving dynamics.Exposition. Math. 20 (2002), 4, 315-327.M. de VriesV. KomornikUnique expansions of real numbers.Adv. Math. 221 (2009), 2, 390-427.P. Erd{ö}sI. Jo{ó}V. KomornikCharacterization of the unique expansions $1=\sum^\infty_{i=1}q^{-n_i}$ and related problems.Bull. Soc. Math. France 118 (1990), 3, 377-390.S. ItoT. SadahiroBeta-expansions with negative bases.Integers 9 (2009), A22, 239-259.Ch. KalleW. SteinerBeta-expansions, natural extensions and multiple tilings associated with Pisot units.Trans. Amer. Math. Soc. 364 (2012), 2281-2318.W. ParryOn the $\beta $-expansions of real numbers.Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.M. PediciniGreedy expansions and sets with deleted digits.Theoret. Comput. Sci. 332 (2005), 1-3, 313-336.A. RényiRepresentations for real numbers and their ergodic properties.Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493.K. SchmidtOn periodic expansions of Pisot numbers and Salem numbers.Bull. London Math. Soc. 12 (1980), 4, 269-278.W. ThurstonGroups, tilings, and finite state automata.AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989.