Analysis of Binary/Ternary Cardinality Combinations in Entity-Relationship Modeling

In this paper, we discuss the simultaneous existence, and relationships, between binary and ternary relationships in entity-relationship (ER) modeling. We define the various interpretations that can be applied to the simultaneous existence of ternary and binary relationships having the same participating entities. We have identified that only certain cardinalities are permitted to exist simultaneously in such ER structures. We demonstrate which binary relationship cardinalities are permitted within ternary relationships, during ER modeling. We develop an Implicit Binary Cardinality (IBC) rule, which states that, in any ternary relationship, the cardinality of any binary relationship embedded in the ternary, is many-to-many when there are no explicit constraints on the data instances. We then present an Explicit Binary Permission (EBP) rule, which explains and enumerates all permitted binary relationships for various cardinalities of ternary relationships. Finally we present an Implicit Binary Override (IBO) rule, which states that the implicit binary cardinalities can be constrained in a ternary relationship by an explicitly imposed binary relationship. We then use these rules to consider the further implicit dynamics of ternary relationships when multiple binary relationships are imposed. In discussing these findings, we consider the rules in the context of supporting functional dependency analysis. The relevance of the findings is presented in the context of ensuring that all functional dependencies associated with ternary relationships are correctly applied and identifying the potential for decomposing the ternary relationship into multiple binary relationships based on all explicit functional dependencies.