Residuated Lattices

Origin. Ward, Dilworth (1939). Algebraic semantics. Substructural foundation. Residuation. Unifying algebraic structure.

Models. Lattices with monoid. Residuation property. Substructural algebraic semantics. Generalized implication.

Formalism.

Residuated lattice: ⟨L, ∧, ∨, ·, , /, e⟩ where: ⟨L, ∧, ∨⟩: lattice. ⟨L, ·, e⟩: monoid. , /: left/right residuals.

Residuation: a · b ≤ c iff a ≤ c / b iff b ≤ a \ c. Galois connection. Adjunction. Implication from fusion.

Right residual: c / b = max{a : a · b ≤ c}. "c divided by b." Right implication.

Left residual: a \ c = max{b : a · b ≤ c}. "a under c." Left implication.

Commutative case: If · commutative: a \ c = c / a. Single residual →. a · b ≤ c iff a ≤ b → c.

Logical interpretation: ·: fusion/tensor. , /: implications. ∧, ∨: meet, join. e: unit/truth.

Substructural correspondence: FL-algebras: full Lambek. Relevance algebras: ·, → with conditions. Heyting: a · b = a ∧ b. Boolean: classical.

Varieties: Integral: a ≤ e. Contractive: a ≤ a · a. Commutative: a · b = b · a. Different logics.

Symbols.

Metatheory. Algebraic semantics. Residuation. Substructural. Variety theory.

Applies to. Substructural logic. Fuzzy logic. Relevance logic. Lambek calculus.

Limitations. Abstract. Multiple conventions. Non-commutative complexity. Technical prerequisites.