*Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by the means of basic binary relation theory.
*

## From Bijection to Function

We consider mappings from and to nodes of graphs from a domain lhs to a codomain rhs. The most trivial mapping is the 1-1 mapping, where each lhs node has exactly one rhs node, and vice versa. In mathematics this is said to be a bijection.

Such **a bijection can be generalised, by dropping some of its properties. **Therefore, we apply the four basic properties of binary relations*:

1. **lhs-total**, i.e. every node in lhs is mapped to rhs

2.** lhs-unique**, i.e. every node from rhs is mapped from max one node in lhs

3.** rhs-total**, i.e. every node in rhs is mapped from lhs

4.** rhs-unique**, i.e. every node from lhs maps to max one node in rhs

Thus, the most specialised concept is the bijection, having all four properties. **A more general concept is that of a function, where just properties 1 and 4 must always apply**, i.o.w. 2 and 3 MAY be dropped. Functions are a pretty useful concept in mathematics and its applications. However, as we will see next, for describing graph-to-graph abstractions, other combinations of the properties 1 to 4 are more suitable.

## From Bijection to Abstraction

**In a graph-to-graph mapping the absence of the four properties can be thought of in the following way:**

1. **Not lhs-total**: a ‘deflating’ mapping by picking certain nodes and omitting others. Examples: all leaf nodes of a tree; all business process steps with an external interface.

2. **Not lhs-unique**: a ‘deflating’ mapping by subsuming nodes of lhs in rhs. Examples: the concept of connected component in graph theory; structuring an object-oriented class model by building components.

3. **Not rhs-total**: an ‘inflating’ mapping, adding extra nodes to rhs, not mapped to from lhs. This can be thought of as adding a new context to the lhs. Examples: adding a ‘bottom node’ to a taxonomy s.t. it becomes a lattice; Adding dots to a drawing s.t. the shape becomes clear, a popular kind of puzzle.

4. **Not rhs-unique**: an ‘inflating’ mapping, that splits a node from lhs into two or more nodes in rhs. In some way this reveals the ‘internals’ of the node. Examples: two nodes from one (e.g. if it belongs to two cycles or 1 node incoming, 1 node outgoing edges); wall belonging to two rooms (which is also not lhs-unique)

Thus, in order to achieve the (strictly) reductional property of an abstraction, property 1 or 2 (or both) MUST be dropped. **We suggest to call a not lhs-total mapping subsuming, and a not lhs-unique one omitting abstraction.** Whereas dropping properties 3 and 4 each constitute a reverse abstraction, i.e. ‘inflating’ from lhs to rhs, and ‘deflating’ from rhs to lhs. Notice that subsuming functions may exist (props 1, not 2, 4).

## Thus …

**we can see that the concept of function is less important here. Central to abstraction, and thus to relational modelling, are the concepts of subsumption and omittance.** Look here for an illustration. In other words, not lhs-total and/or not rhs-unique is the necessary property to constitute an (non-trivial) abstraction.

So long

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* basic properties of binary relations: http://en.wikipedia.org/wiki/Binary_relation

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