We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars with arbitrarily many registers.

翻译：半线性和理性的组合可以自然地扩展至这一设置,允许轨道-无限结合,而不是仅允许有限结合。我们证明,单登记自成一体的自成一体的自成一体的语文的混合图像并不总是半线性,但总是合理。我们还将后一种结果提升为语法:单登记自成一体的不使用背景语言的通感图像是合理的,因此,与登记自成一体的自成一体的自成一体的语法相仿。我们用任意的注册方式将自成一体的自成一体的和语法的类似结果进行推断。