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We examine a naming game with two agents trying to establish a commonvocabulary for n objects. Such efforts lead to the emergence of language thatallows for an efficient communication and exhibits some degree of homonymy andsynonymy. Although homonymy reduces the communication efficiency, it seems tobe a dynamical trap that persists for a long, and perhaps indefinite, time. Onthe other hand, synonymy does not reduce the efficiency of communication, butappears to be only a transient feature of the language. Thus, in our model therole of synonymy decreases and in the long-time limit it becomes negligible. Asimilar rareness of synonymy is observed in present natural languages. The roleof noise, that distorts the communicated words, is also examined. Although, ingeneral, the noise reduces the communication efficiency, it also regroups thewords so that they are more evenly distributed within the available "verbal"space.

Language structure in the n-object naming game