We consider the additive structure of ordinals augmented with right multiplication by \(\omega \). We prove that two terms in the algebra are semantically equal if and only if they take the same value on all elements of a set containing 0 and at least one ordinal of each finite degree. Among these (so-called) test sets the most natural one is that obtained by considering 0 along with all \(\omega \)-powers less than \( \omega ^{\omega }\). By refining this result we provide a polynomial time procedure to determine the semantical equality of two terms.

中文翻译：

---

序数加法乘以$$ \ omega $$ω的序数的加法结构中的等式的测试集

我们考虑通过向右乘以\（\ omega \）来增加序数的可加结构。我们证明，当且仅当代数中的两个项在包含0和每个有限度的至少一个序数的集合的所有元素上取相同的值时，它们在语义上相等。在这些（所谓的）测试集中，最自然的测试集是通过考虑0以及所有小于\（\ omega ^ {\ omega} \）的\（\ omega \）-次幂而获得的。通过完善此结果，我们提供了多项式时间过程来确定两个术语的语义相等性。