Games played by Boole and Galois

We define an infinite class of 2-pile subtraction games, where the amount that can be subtracted from both piles simultaneously is an extended Boolean function f of the size of the piles, or a function over GF(2). Wythoff's game is a special case. For each game, the second player winning positions are a pair of complementary sequences. Sample games are presented, strategy complexity questions are discussed, and possible further studies are indicated. The motivation stems from the major contributions of Professor Peter Hammer to the theory and applications of Boolean functions.