Partition monoids and embeddings in 2-generator regular ∗-semigroups

The partition monoid Pn is known to be minimally 4 -generated (for n≥3 ). Modulo some small values of n , we show that: (1) Pn embeds in a 3 -generator subsemigroup of Pn+1 ; (2) Pn does not embed in a 2 -generator subsemigroup of Pn+1 ; and (3) Pn embeds in a 2 -generator subsemigroup of Pn+3 . A consequence of (3) is that every finite semigroup embeds in a finite 2 -generator regular ∗ -semigroup.