Spaces which are retracts or cofactors of paratopological groups

In this paper we investigate Tychonoff spaces which are retracts of paratopological groups. A strong necessary condition for that is the existence of a certain binary operation on the space (called a τ-twister), which was introduced in [2,3]. Some general theorems are established which imply that βω is not a retract of a paratopological group. We also notice, using some deep results of V.V. Uspenskij, that the space ω1 of countable ordinals is not a retract of any topological group (see Fact 3).