The notion of a quasi-order generated by a homomorphism from the semigroup of all words onto a finite ordered semigroup was introduced by Bucher et al. (Theor. Comput. Sci. 40, 131–148 1985). It naturally occurred in their studies of derivation relations associated with a given set of context-free rules, and they asked a crucial question, whether the resulting relation is a well quasi-order. We answer this question in the case of the quasi-order generated by a semigroup homomorphism. We show that the answer does not depend on the homomorphism, but it is a property of its image. Moreover, we give an algebraic characterization of those finite semigroups for which we get well quasi-orders. This characterization completes the structural characterization given by Kunc (Theor. Comput. Sci. 348, 277–293 2005) in the case of semigroups ordered by equality. Compared with Kunc’s characterization, the new one has no structural meaning, and we explain why that is so. In addition, we prove that the new condition is testable in polynomial time.

Bucher 等人引入了由所有单词的半群到有限有序半群的同态生成的拟序概念。 （理论。计算机科学。40，131–148 1985）。它自然地发生在他们对与给定的上下文无关规则集相关的派生关系的研究中，并且他们提出了一个关键问题，即所得到的关系是否是良好的准顺序。我们在半群同态生成拟序的情况下回答这个问题。我们证明答案并不取决于同态，而是其图像的属性。此外，我们给出了那些我们得到良好拟阶的有限半群的代数表征。该表征完成了 Kunc (Theor. Comput. Sci. 348 , 277–293 2005) 在按等式排序的半群情况下给出的结构表征。与昆克的表征相比，新的表征没有结构意义，我们解释为什么会这样。此外，我们证明新条件在多项式时间内是可测试的。