This paper contributes to the study of positional determinacy of infinite duration games played on potentially infinite graphs with neutral transitions. Recently, [Ohlmann, TheoretiCS 2023] established that positionality of prefix-independent objectives is preserved by finite lexicographic products. We propose two different notions of infinite lexicographic products indexed by arbitrary ordinals, and extend Ohlmann's result by proving that they also preserve positionality. In the context of one-player positionality, this extends positional determinacy results of [Grädel and Walukiewicz, Logical Methods in Computer Science 2006] to edge-labelled games and arbitrarily many priorities for both Max-Parity and Min-Parity. Moreover, we show that the Max-Parity objectives over countable ordinals are complete for the infinite levels of the difference hierarchy over $Σ^0_2$ and that Min-Parity is complete for the class $Σ^0_3$. We obtain therefore positional languages that are complete for all those levels, as well as new insights about closure under unions and neutral letters.

翻译：本文研究了在具有中性转移的潜在无限图上进行的无限时长博弈中位置确定性的问题。近期，[Ohlmann, TheoretiCS 2023] 指出，前缀独立目标的位置性在有限字典序乘积下得以保持。我们提出了两种由任意序数索引的无限字典序乘积的不同定义，并扩展了Ohlmann的结果，证明这些乘积同样保持了位置性。在单人位置性的背景下，这将[Grädel and Walukiewicz, Logical Methods in Computer Science 2006] 中的位置确定性结果扩展到了边标记博弈以及Max-Parity和Min-Parity的任意多个优先级。此外，我们证明了可数序数上的Max-Parity目标对于$Σ^0_2之上差分层级的无限层级是完备的，而Min-Parity对于类$Σ^0_3$是完备的。因此，我们获得了在这些所有层级上完备的位置语言，以及关于并运算和中性字母封闭性的新见解。