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In the memory of Professor A. Kert&zFree inverse semigroups became a subject of intense studies in the last few years. Their existence was proved long ago: as algebras with two operations (binary multiplication and unary involution) inverse semigroups may be characterized by a finite system of identities, i.e. they form a variety of algebras [10]. Therefore, free inverse semigroups do exist.A construction of a free algebra in a variety of algebras (as a quotient algebra of an absolutely free word algebra) is well known. Free inverse semigroups in such a form were considered by V. V. VAO_' ,,~ER [14] who found certain properties of such semigroups. A monogenic free inverse semigroup (i.e. a free inverse semigroup with one generator) was described by L. M. GLUSKIN [2]. Later this semigroup was described by H. E. SCHEIBLICH in a slightly different form [8]. The most essential progress in this direction was made in a paper [9] by H. E. SCHEIBLlCrI who described arbitrary free inverse semigroups. A relevant paper [1] by C. EBERHART and J. SELDEN should be mentioned. There are papers on some special properties of free inverse semigroups. N. R. REILLV described free inverse subsemigroups of free inverse semigroups [7], results in this direction were obtained also by W. D. MUNN and L. O'CARROLL.Let ff'-'cx denote the free inverse semigroup with the set X of free generators. A monogenic free inverse semigroup will be denoted ffd 1. Time and then we will write ~r instead of YJx. We do not consider ~Jo, a one-element inverse semigroup.This paper contains two main results. The first one coincides with the title, the second consists in a description of flee inverse semigroups (if a free inverse semigroup is presented as a quotient algebra of a free involuted semigroup, then each element of ffJ is a c]ass of equivalent words, we give a canonical form of the words). Certain corollaries witb~ properties of free inverse semigroups follow.All results of the paper were reported by the author at a meeting of the seminnar "Semigroups" in the Saratov State University on October 21, ~971.THEOREM I. Free inverse semigroups are not finitely presentable either as semigroups or as involuted semigroups.The proof of the theorem is subdivided in a series of lemmas.LEMMA 1. A semigroup F generated by two elements u and v satisfying the infinite list of defining relations: I) uvu=u, vuv=v; Am,,) umv"+"u"=v"u"+"v'~ for all natural m and n, is a free inverse semigroup. Ac~a Mathematica Academiae Scientiarum Hungaricae 26, 197~

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