Semantic Ambient

The semiotic universe begins with a semantic domain H that supports logical structure and temporal structure at once.

Complete Heyting algebra

H is a complete Heyting algebra. This provides:

• a partial order <=,
• meets and joins for any subset,
• an implication operator that encodes logical entailment.

Modality and trace

Two additional operators act on H:

• j is a modal closure operator (monotone, extensive, idempotent, join-preserving).
• G is a trace comonad that also preserves Heyting operations.

Together they describe a semantic system with modal stability (j) and temporal unfolding (G).