Despite similarities between the Web Ontology Language (OWL) and schema languages traditionally used in relational databases, systems based on these languages exhibit quite different behavior in practice. The schema statements in relational databases are usually interpreted as integrity constraints and are used to check whether the data is structured according to the schema. OWL allows for axioms that resemble integrity constraints; however, these axioms are interpreted under the standard first-order semantics and not as checks. This often leads to confusion and is inappropriate in certain data-centric applications. To explain the source of this confusion, in this paper we compare OWL and relational databases w.r.t. their schema languages and basic computational problems. Based on this comparison, we extend OWL with integrity constraints that capture the intuition behind similar statements in relational databases. We show that, if the integrity constraints are satisfied, they need not be considered while answering a broad range of positive queries. Finally, we discuss several algorithms for checking integrity constraint satisfaction, each of which is suitable to different types of OWL knowledge bases.