# Combinatorial Monoid Extension
The extension of UOR principles to combinatorial monoids and incidence algebras, where connected components serve as prime elements.
## Definition
Key examples include:
- Combinatorial Monoids: In incidence algebras, connected components decompose uniquely, with φ capturing component counts.
## Mathematical Formulation
$
\text{For incidence algebras over a poset } P,
$
$
\text{connected components } C_i \text{ decompose uniquely}
$
$
\phi_{Inc(P)}(f) = (v_{C_1}(f), v_{C_2}(f), \ldots, v_{C_n}(f))
$
$
\text{where } v_{C_i}(f) \text{ captures the count of component } C_i \text{ in } f
$
## Related Concepts
- [[uor-c-031|general-uor-existence]]
- [[uor-c-002|prime-decomposition]]
## Metadata
- **ID:** urn:uor:concept:combinatorial-monoid-extension
- **Code:** UOR-C-033