A metric minimal PI cascade with 2c minimal ideals

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than (where ) minimal left ideals is proximal isometric (PI). Then we show the existence of various minimal PI-flows with many minimal left ideals, as follows. For the acting group , we construct a metric minimal PI-flow with minimal left ideals. We then use this example and results established in Glasner and Weiss. [On the construction of minimal skew-products. Israel J. Math.34 (1979), 321-336] to construct a metric minimal PI cascade with minimal left ideals. We go on to construct an example of a minimal PI-flow on a compact manifold and a suitable path-wise connected group of a homeomorphism of , such that the flow is PI and has minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade that is PI (of order three) and has minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication 'less than minimal left ideals implies PI', fails.