We consider implicit representation of an arbitrary family of relations on finite sets. We derive upper and lower bounds for the general cases and for a number of restricted subfamilies, in particular for sparse and symmetric relations, and for relations first-order definable from families for which labeling schemes are already known. Our work extends existing work on implicit representation of graphs in two ways: (i) the known upper and lower bounds for many standard families of graphs are special cases of the results we derive; (ii) we allow families of relations to relate elements on both distinct sets and on multiple copies of the same set, and for different relations in the same family to have different arities, and to be defined on distinct or overlapping sets. The present paper is the first to study bounds on the size of labeling schemes for relations (including graphs) defined from existing relations using basic operations such as first-order logic. The techniques used to prove new results in this setting may be of independent interest.

我们考虑有限集上任意关系族的隐式表示。我们推导出一般情况和许多受限子族的上限和下界，特别是稀疏和对称关系，以及可从标签方案已知的族中一阶定义的关系。我们的工作以两种方式扩展了图隐式表示方面的现有工作：（i）许多标准图族的已知上限和下限是我们得出的结果的特殊情况；(ii) 我们允许关系族将不同集合和同一集合的多个副本上的元素关联起来，并且同一族中的不同关系具有不同的元数，并在不同或重叠的集合上定义。本文首次研究了使用一阶逻辑等基本运算从现有关系定义的关系（包括图）的标签方案大小的界限。在这种情况下用于证明新结果的技术可能具有独立的意义。