Skills Algebra

Definition

A skills algebra is a formal structure $(\mathcal{S},\circ ,1,\mathcal{T},\varphi )$ where:

• $\mathcal{S}$ = set of skills
• $\circ :\mathcal{S}\times \mathcal{S}\to \mathcal{S}$ = composition operation
• $1\in \mathcal{S}$ = identity element
• $\mathcal{T}$ = set of tasks
• $\varphi :\mathcal{S}\times \mathcal{T}\to [0,1]$ = fitness function

Axioms

1. Associativity: $(S_1\circ S_2)\circ S_3=S_1\circ (S_2\circ S_3)$
2. Identity: $S\circ 1=1\circ S=S$
3. Fitness coherence: $\varphi (1,T)=0$ for all $T$

This makes $(\mathcal{S},\circ ,1)$ a monoid.

Induced Structures

Divisibility Pre-order

$S_1⊴S_2⟺\exists S^{\prime }:S_1\circ S^{\prime }=S_2\text{ or }{S}^{\prime }\circ S_1=S_2$

Complexity Filtration

$\lambda (S)=\min \{n:S=S_1\circ \cdots \circ S_n,S_i\in {\mathcal{S}}_0\}$

Induces strata: ${\mathcal{S}}_0\subseteq {\mathcal{S}}_{\le 1}\subseteq {\mathcal{S}}_{\le 2}\subseteq \cdots$

Task-Relative Ontology

${\mathcal{S}}_T^{ϵ}=\{S\in \mathcal{S}:\varphi (S,T)\ge ϵ\}$

Galois Connection

$(\text{skills},\text{tasks}):2^{\mathcal{T}}\leftrightarrows 2^{\mathcal{S}}$

Closed elements = natural kinds.

Full Skill Structure

Combining with mereology:

$(\mathcal{S},{⊑}_m,\circ ,1)$

A monoidal poset with coherence conditions relating mereological and algebraic structures.

Summary Table

Construct	Definition	Role
Skills monoid	$(\mathcal{S},\circ ,1)$	Generative engine
Divisibility	$⊴$	Algebraic order
Complexity	$\lambda (S)$	Stratification
Task-relative set	${\mathcal{S}}_T^{ϵ}$	Perspectival ontology
Galois connection	$(\text{skills},\text{tasks})$	Natural kinds
Mereological order	${⊑}_m$	Constitutional structure

Related Concepts

• skills — Elements of the algebra
• composition — The $\circ$ operation
• ontology — Induced from algebra
• mereology — Complementary structure
• ontological-expansion — Algebra growth

References

• A mereological reconciliation
• Clarifying “Algebra of Skills Ontology”