Together is the meet (intersection) operator on relations. It yields the shared refinement of two recognitions --- what is recognized under both.
Formal Signature
Together : (Rel, Rel) → Rel
Definition
Together(r, s) = r ∩ s. The result is a relation that holds exactly where both r and s hold. Together is commutative, associative, and idempotent.
Together participates in the central residuation law that links it to Implies and Entails:
Entails(Together(a, b), c) ⟺ Entails(a, Implies(b, c))
This adjunction is what makes the relational algebra a Heyting algebra rather than a mere lattice.
Derivational context
Together arises in Movement I: Logical Origination from the requirement that multiple distinctions be combinable. If you can distinguish A and distinguish B, coherence demands a way to recognize what is both A and B — the shared refinement. Together formalizes this combining, and its interaction with Implies through the residuation law gives the logical core its constructive character. In mathematical terms, Together is the meet of a Heyting algebra, but relationally it is the act of combining recognitions.