Abstract
In the following subsections, we introduce the mathematical formalisms that underpin modelling, metamodelling and ontologies. Basic to a mathematical formalism are set theory and morphisms. First we consider, in Sect. 2.1, how set theory can represent modelling concepts. In Sect. 2.2, we look at mappings (functions) between pairs of sets–whether such a mapping is one-to-one, onto or both can be critical in understanding many of the ‘problems’ identified in the software engineering modelling literature
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Notes
- 1.
Although Halpin (2005) suggests that most of the examples in the literature showing modelling use of powertypes are not optimal and that powertype-based modelling is rarely needed.
- 2.
Some authors use the term ‘generalize’ to mean ignore details (see, for example, Alagar and Periyasamy 1998 p. 40). Here, the term generalize, and particularly ‘generalization’, is used in the object-oriented sense of a relationship between type and subtype (see later discussion).
- 3.
Thus confounding the abstraction and representation links depicted in the ‘meaning triangle’ (Fig. 1.3).
- 4.
Although we do not discuss notions here, we follow Kashek (2004) in assuming models to be implicitly systems of notions.
- 5.
Note that this is a very different meaning from the use of ‘reference model’ in software engineering standards published by ISO’s SC7 committee.
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Henderson-Sellers, B. (2012). Mathematics for Modelling. In: On the Mathematics of Modelling, Metamodelling, Ontologies and Modelling Languages. SpringerBriefs in Computer Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29825-7_2
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DOI: https://doi.org/10.1007/978-3-642-29825-7_2
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