A relation consisting of tuples annotated by an element of a monoid K is called a K-relation. A K-database is a collection of K-relations. In this paper, we study entailment of inclusion dependencies over K-databases, where K is a positive commutative monoid. We establish a dichotomy regarding the axiomatisation of the entailment of inclusion dependencies over K-databases, based on whether the monoid K is weakly absorptive or weakly cancellative. We establish that, if the monoid is weakly cancellative then the standard axioms of inclusion dependencies are sound and complete for the implication problem. If the monoid is not weakly cancellative, it is weakly absorptive and the standard axioms of inclusion dependencies together with the weak symmetry axiom are sound and complete for the implication problem. In addition, we establish that the so-called balance axiom is further required, if one stipulates that the joint weights of each K-relation of a K-database need to be the same; this generalises the notion of a K-relation being a distribution. In conjunction with the balance axiom, weak symmetry axiom boils down to symmetry.

翻译：由幺半群K中元素标注的元组构成的关系称为K-关系。K-数据库是K-关系的集合。本文研究K-数据库上包含依赖的蕴涵问题，其中K是正交换幺半群。基于幺半群K是弱吸收还是弱可消的，我们建立了关于K-数据库上包含依赖蕴涵公理化的二分性结果。我们证明：若幺半群是弱可消的，则包含依赖的标准公理对于蕴涵问题是可靠且完备的；若幺半群不是弱可消的，则它是弱吸收的，此时包含依赖的标准公理与弱对称公理共同构成蕴涵问题的可靠且完备公理系统。此外，若要求K-数据库中每个K-关系的联合权重必须相同（这推广了K-关系作为分布的概念），则还需要引入所谓的平衡公理。在与平衡公理结合时，弱对称公理可简化为对称公理。