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：我们引入了抽象条件独立（CI）模型框架的代数概念，以及该框架应封闭的基本操作：复制和边缘化。此类框架的三个标准例子是（离散）概率CI结构、半图结构和结构半图结构。我们重点关注那些在集合论交运算下封闭的框架，因为对于这些框架，相应的CI模型族构成格结构。这使得我们可以应用格论和形式概念分析的结果，通过CI语句之间的蕴涵关系来描述这些族。本文的核心概念是代数意义上的自粘性，它是早期在多拟阵和信息论背景下研究的自粘性概念的组合反映。这一推广还导致在CI框架的元层次上定义的自粘性算子。我们回答了与此方法相关的一些问题，并提出了其他开放性问题。本文的核心在于计算。基于组合方法的计算可能克服基于多面体几何的软件包在内存和空间上的限制，尤其是在利用SAT求解器时。我们通过CI语句之间的典范蕴涵关系刻画了4个变量上的一些基本CI族。我们将该方法应用于信息论背景下5个变量上的熵区域划分任务。