Residuated Lattice

Synonyms

Structure of truth values

Definition

The residuated lattice is a basic algebraic structure accepted as a structure of truth values for fuzzy logic and fuzzy set theory. In general, it is an algebra

$$ \left\langle L,\vee, \wedge, \otimes, \to, 0,1\right\rangle $$

where L is a support, ∨ and ∧ are binary lattice operations of join and meet, 0 is the smallest and 1 is the greatest element. The ⊗ is additional binary operation of product that is associative and commutative, and a ⊗ 1 = a holds for every a ∈ L. The → is a binary residuation operation that is adjoined with ⊗ as follows:

$$ a\otimes b\le c\ \mathrm{if}\ \mathrm{and}\ \mathrm{only}\ \mathrm{if}\ a\le b\to c $$

for arbitrary elements a, b, c ∈ L. The residuation operation is a generalization of the classical implication.

Key Points

The residuated lattice is naturally ordered by the classical lattice ordering relation defined by

$$ a\le b\ \ \ \ \mathrm{if}\ \ \mathrm{and}\ \ \mathrm{only}\ \ \mathrm{if}\ \ \ \ a\wedge...