  We propose a generalization of positional games, supplementing them with a
restriction on the order in which the elements of the board are allowed to be
claimed. We introduce poset positional games, which are positional games with
an additional structure -- a poset on the elements of the board. Throughout the
game play, based on this poset and the set of the board elements that are
claimed up to that point, we reduce the set of available moves for the player
whose turn it is -- an element of the board can only be claimed if all the
smaller elements in the poset are already claimed.
  We proceed to analyse these games in more detail, with a prime focus on the
most studied convention, the Maker-Breaker games. First we build a general
framework around poset positional games. Then, we perform a comprehensive study
of the complexity of determining the game outcome, conditioned on the structure
of the family of winning sets on the one side and the structure of the poset on
the other.
