Half-positionally determined winning conditions

Basic definitions: games, half-positional determinacy. - Half-positional winning conditions and omega-regular languages. How to check whether the given omega-regular winning condition is finitely half-positional? - Positional/suspendable conditions (PS). Definitions and examples. Half-positional determinacy of a countable union on PS conditions, and of the winning conditions in the class XPS (extended positional/suspendable conditions).