Abstract
LetIdenote the interval [0,1] in its usual topology and letS=S(I)be the semigroup of continuous mappings ofIinto itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties ofSare still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing thatShas infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5:Shas infinitely many distinct congruences. We believe thatSmust have c distinct ideals, but we have been unable to prove this.
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