Proceedings of the 3rd European Conference on Intellectual Capital

Eckhard Ammann
individualization is applied to give <strong>the</strong> result <strong>of</strong> <strong>the</strong> concatenated operators. With this notation, <strong>the</strong>
concept <strong>of</strong> inverse operators introduced in <strong>the</strong> previous sub-section can be stated as
x -> x -1 == id == x -1 -> x, where x, id: V -> V and id is <strong>the</strong> identity operator on <strong>the</strong> IC space.
3.2.1 Hierarchical concatenations
The first important class <strong>of</strong> concatenations <strong>of</strong> operators are hierarchical concatenations. They consist
<strong>of</strong> sequences <strong>of</strong> <strong>the</strong> same operator applied one after <strong>the</strong> o<strong>the</strong>r, i.e. <strong>the</strong>y come in <strong>the</strong> form
x -> x -> … -> x, where x: V -> V and x is one <strong>of</strong> <strong>the</strong> set { Ind, Res, Ref, Gen, Enl, Coa }.
This is important for <strong>the</strong> downward operator refinement and restriction, where a meaningful procedure
for IC advancement is <strong>the</strong> repeated refinement from high level components to elements and variables
or <strong>the</strong> repeated restriction from <strong>the</strong> company-wide viewpoint to parts <strong>of</strong> <strong>the</strong> company. The case <strong>of</strong> <strong>the</strong>
individualization operator is not important here, because a single application <strong>of</strong> it already leads to <strong>the</strong>
full detail <strong>of</strong> individuals and single items or processes.
Using hierarchical concatenations with upward operators describe <strong>the</strong> repeated consolidation <strong>of</strong>
detailed statements into higher level viewpoints with growing overall perspective in <strong>the</strong> company.
3.2.2 Symmetric concatenations
Symmetric concatenations are defined as concatenations <strong>of</strong> <strong>the</strong> form
-1 -1 -1
x1 -> x2 -> … -> xn -> xn -> … -> x2 -> x1 == id, where xi: V -> V and i = 1,2, …,n.
They always effect in identity in <strong>the</strong> IC space. By far <strong>the</strong> most important ones in this class, and we
want to restrict our definition to <strong>the</strong>se cases, are those, where <strong>the</strong> x1, …,xn are all downward
operators. With <strong>the</strong> help <strong>of</strong> <strong>the</strong>m, you break down your IC viewpoint with <strong>the</strong> help <strong>of</strong> several (possibly
different) downward operators to <strong>the</strong> point <strong>of</strong> detail in <strong>the</strong> IC space you want <strong>the</strong> see and develop with
knowledge development measures, and <strong>the</strong>n proceed upwards on <strong>the</strong> same path in reverse order,
until <strong>the</strong> starting point <strong>of</strong> <strong>the</strong> procedure is reached again. An example scenario with a symmetric
concatenation path is visually depicted in Figure 3 in <strong>the</strong> next section 4 on model application
scenarios.
Important subclasses <strong>of</strong> symmetric concatenations are those, where <strong>the</strong> xi’s in <strong>the</strong> previous definition
are all ei<strong>the</strong>r refinements or restrictions. Then we have homogeneous refinement or restriction
followed by homogeneous coarsing or enlargement, respectively.
3.2.3 Roundtrip concatenations
Roundtrip concatenations are defined as concatenations <strong>of</strong> <strong>the</strong> form
x1 -> x2 -> … -> xn == id, where xi: V -> V and i = 1,2, …,n.
They always effect in identity in <strong>the</strong> IC space. By far <strong>the</strong> most important ones in this class, and we
want to restrict our definition to <strong>the</strong>se cases, are those, where <strong>the</strong> first x1, …,xi are all downward
operators and <strong>the</strong> last xi+1, …,xn are all upward operators (1