Based on the celebrated Birkhoff theorem, Doignon and Falmagne (1985) establish a one-to-one correspondence between certain binary relations on a domain of (dichotomous) items and the quasi-ordinal knowledge spaces defined on it. It is shown that this construction can be generalized to apply to polytomous response formats, where response values are partially ordered so that they form a lattice. Polytomous knowledge states are defined for the case where these response scales are identical across items, as well as for item-specific response scales, a generalization which emerges quite naturally within the developed theoretical framework. Certain collections of polytomous knowledge states are characterized through precedence relations that form particular quasi-orders on an extended set of (virtual) items, including for each item several dichotomous instances representing all the possible response values. These kinds of polytomous knowledge structures can be built from data through applying to the extended item set procedures commonly used in knowledge structure theory. Confining consideration to finite sets of items and response values, the presented theory generalizes previous approaches in this direction. Its application is illustrated for data from the 2011 Trends in International Mathematics and Science Study (TIMSS).

Doignon和Falmagne（1985）基于著名的Birkhoff定理，在（二分）项域上的某些二元关系与其上定义的准有序知识空间之间建立了一对一的对应关系。结果表明，这种构造可以推广到适用于多态响应格式，其中响应值是部分排序的，因此它们形成一个晶格。多项目知识状态是针对以下情况定义的：项目之间的这些反应量表是相同的，以及项目特定的反应量表，这种概化在已开发的理论框架中非常自然地出现。某些多态知识状态集合的特征在于优先级关系，这些优先级关系在扩展的（虚拟）项目集上形成特定的准顺序，包括针对每个项目的代表所有可能响应值的几个二分实例。通过将数据结构应用于知识结构理论中常用的扩展项目集过程，可以从数据中构建这些种类繁多的知识结构。限于对项目和响应值的有限集合的考虑，所提出的理论概括了该方向上的先前方法。2011年国际数学和科学研究趋势（TIMSS）中的数据对它的应用进行了说明。提出的理论在此方向上概括了先前的方法。2011年国际数学和科学研究趋势（TIMSS）中的数据对它的应用进行了说明。提出的理论在此方向上概括了先前的方法。2011年国际数学和科学研究趋势（TIMSS）中的数据对它的应用进行了说明。