Infinite partition monoids

Let X and X be the partition monoid and symmetric group on an infinite set X. We show that X may be generated by X together with two (but no fewer) additional partitions, and we classify the pairs β X for which X is generated by X ∪ {β}. We also show that X may be generated by the set X of all idempotent partitions together with two (but no fewer) additional partitions. In fact, X is generated by X ∪ {β} if and only if it is generated by X ∪ X ∪ {β}. We also classify the pairs β X for which X is generated by X ∪ {β}. Among other results, we show that any countable subset of X is contained in a 4-generated subsemigroup of X, and that the length function on X is bounded with respect to any generating set. (Note: Some of the scientific symbols can not be represented correctly in the abstract. Please read with caution and refer to the original publication.)