Individual concept notes for mathematical structures and ideas. Several of these concepts serve as direct components of the semiotic universe hierarchy.
Algebraic structures
- heyting-algebra — a bounded lattice with relative pseudo-complementation; the algebraic semantics of intuitionistic logic and the base structure of the semiotic universe
- boolean-algebra — a complemented distributive lattice providing the algebraic semantics of classical logic; every Boolean algebra is a Heyting algebra, but not conversely
- residuated-lattice — a lattice with a multiplication-implication adjunction that generalizes Heyting algebras; the residuation pattern recurs at multiple levels of the relationality derivation
- lindenbaum-tarski-algebra — the quotient algebra of formulas modulo logical equivalence; connects syntactic theories to their algebraic counterparts
Order theory and fixed points
- closure-operator — a monotone, extensive, idempotent function whose fixed points form a complete lattice; the semiotic universe is built from three closure operators (semantic, syntactic, fusion)
- fixed-point — a value unchanged by a function; the semiotic universe’s initial semiotic structure is the least fixed point of the composite closure operator
Logic and foundations
- set-theory — the study of collections and membership; this vault builds on Heyting algebras and type theory rather than set-theoretic foundations
- curry-howard-correspondence — the structural isomorphism between proofs and programs, propositions and types
- proof-assistant — software for constructing and verifying formal proofs through the Curry-Howard correspondence
- Function
- Proof
- Relation
Curriculum
The closure operators and Heyting algebras concepts have associated learning skills in the vault’s curriculum graph. See the learn-closure-operators and learn-heyting-algebras skills for guided introductions.