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In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets /, J and S\U are finite, where U is the ideal of S generated by the entries of P.1991 Mathematics subject classification: 20M05.

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