The following problem was posed by \textit{J. D. Lawson} [Semigroup Forum 12, 265-280 (1976; Zbl 0327.22003)]. Let I be the interval [0,1], provided with the ''min''-multiplication. Is it true that every semitopological action of I on a compact space is in fact a topological action? It is now well-known that the answer to this question is ''yes''. Actually, this assertion is true for any compact totally ordered space with the ''min''-multiplication. In the present paper we shall investigate the above problem for ''generalized I-semigroups'', a class of compact semigroups comprising both the ''min''-semigroups associated with compact totally ordered spaces and the I-semigroups.