We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the hyper-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.

翻译：本文引入了抽象条件独立模型框架的代数概念，并定义了此类框架应满足封闭性的基本操作：复制与边缘化。此类框架的三个标准实例是（离散）概率条件独立结构、半图胚与结构半图胚。我们重点关注在集合论交运算下封闭的框架，因为对于这类框架，相应的条件独立模型族构成格结构。这使得我们可以运用格论与形式概念分析的理论成果，以条件独立语句间的蕴涵关系来描述这些模型族。本文的核心概念是在代数意义上定义的自黏合性，该概念是多拟阵与信息论领域中早期研究的自黏合性概念的组合反射。这一推广还引出了在条件独立框架超层次上定义的自黏合性算子。我们针对该方法的相关问题给出了部分解答，并提出了若干待解决的开放性问题。论文的核心在于计算过程。基于组合学的计算方法有望克服依赖多面体几何的软件包（尤其是采用SAT求解器时）存在的内存与空间限制。我们通过条件独立语句间的规范蕴涵关系，刻画了4变量条件下若干基本条件独立模型族的特征。最后，我们将该方法应用于信息论领域，完成了5变量条件下熵区域的边界划分任务。