On decomposition theory: Duality

Decomposition theory is concerned with the structures that arise in the decomposition of systems. It states from the premise that any method of system decomposition is based, either explicitly or implicitly, on some concept of dependence. The formal setting of decomposition theory is the dependence, an ordered-triple (
E
,
M
,
D
), where
E
is a nonempty set,
M
is a collection of subsets of
E
, and
D
is a relation from nonempty subsets of
M
to subsets of
M
. If (
A
,
B
)∈
D
, it is said that `
A
depends on
B
'. Duality is considered here. In particular, given a dependence (
E
,
M
,
D
), its dual is a dependence (
E
,
M
,
D
'). Duality plays a role here similar to duality in other formal systems such as graphs, matroids, lattices, circuits, control systems, and so forth. It deepens our understanding of dependence by pairing seemingly different concepts.