Concept
Residuated lattice
Overview
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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There are nevertheless, in mathematics, some examples of so-called Induction, in which the conclusion does bear the appearance of a generalization grounded on some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the _law_ of the series, does not hesitate to fill up any number…
John Stuart Mill, A System of Logic
States what Residuated lattice is.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Residuated lattice is.
Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid--corpus solidum--and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
Friedrich Engels, Landmarks of Scientific Socialism
States what Residuated lattice is.
Related ideas
- Formal language
- Logical conjunction
- Mathematics
- Total order
- Boolean algebra (structure)
- Linear logic
- MV-algebra
Place in the tradition
Part of Fuzzy logic.