Safe Haskell	Safe-Inferred
Language	Haskell2010

Lattices.ResiduatedLattice

Synopsis

Documentation

A bounded lattice with additional laws and operations namely --> and tnorm

Commutativity

x tnorm y ≡ y tnorm x

Asociativity of tnorm

x tnorm (y tnorm z) ≡ (x tnorm y) tnorm z

Identity Element x tnorm 1 ≡ x

Adjointness x ≤ y --> z iff x tnorm y ≤ z

Minimal complete definition

Methods

(-->) :: l -> l -> l Source #

residuum

(<--) :: l -> l -> l Source #

residuum

(<-->) :: l -> l -> l Source #

biresiduum

tnorm :: l -> l -> l Source #

power :: l -> Nat -> l Source #

Instances details

ResiduatedLattice UIGodel Source #	Gödel structure of truth values
Instance details Defined in Lattices.UnitIntervalStructures.Godel Methods (-->) :: UIGodel -> UIGodel -> UIGodel Source # (<--) :: UIGodel -> UIGodel -> UIGodel Source # (<-->) :: UIGodel -> UIGodel -> UIGodel Source # tnorm :: UIGodel -> UIGodel -> UIGodel Source # negation :: UIGodel -> UIGodel Source # power :: UIGodel -> Nat -> UIGodel Source #
ResiduatedLattice UILukasiewicz Source #	Łukasiewicz structure of truth values
Instance details Defined in Lattices.UnitIntervalStructures.Lukasiewicz Methods (-->) :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # (<--) :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # (<-->) :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # tnorm :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # negation :: UILukasiewicz -> UILukasiewicz Source # power :: UILukasiewicz -> Nat -> UILukasiewicz Source #
ResiduatedLattice UIProduct Source #	Product (Goguen) structure of truth values
Instance details Defined in Lattices.UnitIntervalStructures.Product Methods (-->) :: UIProduct -> UIProduct -> UIProduct Source # (<--) :: UIProduct -> UIProduct -> UIProduct Source # (<-->) :: UIProduct -> UIProduct -> UIProduct Source # tnorm :: UIProduct -> UIProduct -> UIProduct Source # negation :: UIProduct -> UIProduct Source # power :: UIProduct -> Nat -> UIProduct Source #

class RealFrac l => BoundedLattice l where Source #

Lattice is an algebraic structure with join and meet operations. BoundedLattice has Top and Bottom elements Lattice satisfies following laws:

Associativity

x \/ (y \/ z) ≡ (x \/ y) \/ z
x /\ (y /\ z) ≡ (x /\ y) /\ z

Commutativity

x \/ y ≡ y \/ x
x /\ y ≡ y /\ x

Idempotency

x \/ x ≡ x
x /\ x ≡ x

Absorption

a \/ (a /\ b) ≡ a
a /\ (a \/ b) ≡ a

Methods

(/\) :: l -> l -> l Source #

meet

(\/) :: l -> l -> l Source #

join

also called upper bound

also called lower bound

mkLattice :: Double -> l Source #

constructor for lattice

Instances details

BoundedLattice UnitInterval Source #
Instance details Defined in Lattices.UnitInterval Methods (/\) :: UnitInterval -> UnitInterval -> UnitInterval Source # (\/) :: UnitInterval -> UnitInterval -> UnitInterval Source # top :: UnitInterval Source # bot :: UnitInterval Source # mkLattice :: Double -> UnitInterval Source #
BoundedLattice UIGodel Source #
Instance details Defined in Lattices.UnitIntervalStructures.Godel Methods (/\) :: UIGodel -> UIGodel -> UIGodel Source # (\/) :: UIGodel -> UIGodel -> UIGodel Source # top :: UIGodel Source # bot :: UIGodel Source # mkLattice :: Double -> UIGodel Source #
BoundedLattice UILukasiewicz Source #
Instance details Defined in Lattices.UnitIntervalStructures.Lukasiewicz Methods (/\) :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # (\/) :: UILukasiewicz -> UILukasiewicz -> UILukasiewicz Source # top :: UILukasiewicz Source # bot :: UILukasiewicz Source # mkLattice :: Double -> UILukasiewicz Source #
BoundedLattice UIProduct Source #
Instance details Defined in Lattices.UnitIntervalStructures.Product Methods (/\) :: UIProduct -> UIProduct -> UIProduct Source # (\/) :: UIProduct -> UIProduct -> UIProduct Source # top :: UIProduct Source # bot :: UIProduct Source # mkLattice :: Double -> UIProduct Source #

type Nat = Int Source #