## Reflections on Abstractions: Abstractive vs Functional Mappings

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 