Relation: Dominance

Module: Rock_Paper_Scissors
Relation Type: Cyclic Triadic Dominance
ID: dominance

1. Formal Definition#

Let E = {Rock, Paper, Scissors} and ≻ denote "dominates."

Rock     ≻ Scissors   (crushes)
Scissors ≻ Paper      (cuts)
Paper    ≻ Rock       (covers)

1.1 Algebraic Properties#

Property	Holds?	Notes
Irreflexivity	✅	No entity dominates itself
Asymmetry	✅	If A ≻ B then ¬(B ≻ A)
Transitivity	❌	Rock ≻ Scissors ≻ Paper does not imply Rock ≻ Paper
Cyclicity	✅	Length-3 cycle; group symmetry Z₃

1.2 Cycle Diagram#

     Rock
    ↗    ↘
Paper ←— Scissors

(Arrow = "is beaten by")

2. Triadic Interpretation#

Move	Interpretation
Firstness ≻ Thirdness (Rock ≻ Scissors)	Brute immediacy destroys mechanism of mediation
Thirdness ≻ Secondness (Scissors ≻ Paper)	Differentiation severs relational coverage
Secondness ≻ Firstness (Paper ≻ Rock)	Context supersedes pure presence

No mode is universally supreme — encoding the irreducibility of all three Peircean categories.

3. Adjacency Matrix#

Rows = winner, Columns = loser. 1 = row dominates column.

	Rock	Paper	Scissors
Rock	0	0	1
Paper	1	0	0
Scissors	0	1	0

4. JSON Encoding#

{
  "relation_id": "dominance",
  "type": "cyclic_binary",
  "edges": [
    {"from": "rock",     "to": "scissors", "label": "crushes"},
    {"from": "scissors", "to": "paper",    "label": "cuts"},
    {"from": "paper",    "to": "rock",     "label": "covers"}
  ],
  "properties": {
    "irreflexive": true, "asymmetric": true,
    "transitive": false, "cyclic": true,
    "cycle_length": 3,   "group_symmetry": "Z3"
  }
}