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This paper is concerned with the problem of ensuring the integrity of database systems that are accessed concurrently by a number of independent asychronously running transactions. It is assumed that the database system is partitioned into small units that are referred to as the database entities. The relation between the entities is represented by a directed acyclic graph in which the vertices correspond to the database entities and the arcs correspond to certain access rights. We develop a family of non-two-phase locking protocols for such systems that will be shown to ensure serializability and deadlock-freedom. This family is sufficientdy general to encompass all the previously developed non-two-phase lose locking protocols as well as a number of new protocols. One of these new protocols that seems to be particularly useful is also presented in this paper.