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We introduce a mathematical framework where a formal semantics for object identity can be built irrespectively to computer related things like object identifiers, memory allocations etc. Then, on this base, we build formal semantics for a few major constructs of conceptual modeling (CM) such as association, aggregation, generalization, isA- and isPartOf-relationships. We also give a formal meaning to the two fundamental dichotomies of CM: objects vs. values and entities vs. relationships.On the syntactical side, the language we use for specifying our formal semantic constructs is graph-based and brief: specifications are directed graphs consisting only of three kinds of items--nodes, arrows and marked diagrams. The latter are configurations of nodes and arrows closed in some technical sense and marked with predicate labels taken from a predefined signature. We show that this format does provide a universal abstract syntax for the entire CM-field. Then any particular CM-notation appears as a particular visualization superstructure (concrete syntax) over the same basic specification format as above.