The semijoin operation is a fundamental operation of relational algebra that has been extensively used in query processing. Furthermore, semijoins have been used to formulate desirable properties of acyclic schemas; in particular, a schema is acyclic if and only if it has a full reducer, i.e., a sequence of semijoins that converts a given collection of relations to a globally consistent collection of relations. In recent years, the study of acyclicity has been extended to annotated relations, where the annotations are values from some positive commutative monoid. So far, however, it has not been known if the characterization of acyclicity in terms of full reducers extends to annotated relations. Here, we develop a theory of semijoins of annotated relations. To this effect, we first introduce the notion of a semijoin function on a monoid and then characterize the positive commutative monoids for which a semijoin function exists. After this, we introduce the notion of a full reducer for a schema on a monoid and show that the following is true for every positive commutative monoid that has the inner consistency property: a schema is acyclic if and only if it has a full reducer on that monoid.

翻译：半连接操作是关系代数中的基本运算，已在查询处理中得到广泛应用。此外，半连接被用于表述无环模式的可取性质；具体而言，一个模式是无环的当且仅当其存在完全归约器，即通过一系列半连接将给定关系集合转换为全局一致的关系集合。近年来，无环性研究已扩展至带注释的关系，其中注释值取自某个正交换幺半群。然而迄今为止，尚不清楚基于完全归约器的无环性特征是否适用于带注释的关系。本文发展了带注释关系的半连接理论。为此，我们首先引入幺半群上半函数的概念，进而刻画存在半函数的正交换幺半群的特征。在此基础上，我们提出幺半群上模式的完全归约器概念，并证明以下结论对所有具有内一致性的正交换幺半群均成立：一个模式是无环的当且仅当在该幺半群上存在完全归约器。