Here is presented an algebraic model of emergence of natural language, which defines language as the limit of a communication process in a society of agents. The heterogeneous model consists of semantic algebra Obj created by the set of attributes (elementary perceptions) and syntactic algebra L. Objects are compositional, determined by their attributes and sub-objects. Every agent a maintains its own language La⊆L, which is developed in the communication process using a meaning morphism ma:La→Obj, the speech morphism spa:Obj→La and a naming function na:Obj→Na, where Na is a set of names (subjective attributes, Na∩La=Ø). The model is investigated using computer modelling (language game). In the communication process agents add new words to their languages (at the beginning all La=Ø) and improve their meaning and speech functions using inference and disambiguation of semantics when objects are presented in different contexts. At the first stage of language creation agents use grounded messages (the message's object is included, e.g. pointed at), but on later stages they can also use ungrounded messages. Communication allows errors and several random features. If in the process of sending-receiving messages agents understanding of each other improves, i.e. error rate in recreating message's object decreases, then at the limit they create common language. It is shown, how compositionality (structure) in semantic domain creates elementary compositionality (structured denotations) also in the language. Compositionality of denotations follows from a very simple algorithm of agent's behaviour and does not require any pre-defined word categories or syntactic rules.
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