A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $\mathbb{E}$, we give a counting method for the number of quasi-hereditary structures. In the case of a hereditary incidence algebra, we present a necessary and sufficient condition for its poset of quasi-hereditary structures to be a lattice.

准遗传代数是一个 Artin 代数，在其满足特定条件的简单模的同构类集合上具有偏序。在本文中，我们研究了在给定代数上产生准遗传结构的所有可能选择，特别是我们介绍和研究了我们称之为准遗传结构的偏序集。我们的技术涉及某些颤动分解和幂等减少。对于 Dynkin 类型的路径代数$\mathbb{一种}$，我们提供了其准遗传结构的完整分类。对于类型$\mathbb{D}$ 和 $\mathbb{乙}$，我们给出了准遗传结构数量的计数方法。在遗传关联代数的情况下，我们提出了其准遗传结构的偏序是格子的充分必要条件。