Cats algebra
2.9.0
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© 2002-2021 · LAMP/EPFL
© 2002-2021 · LAMP/EPFL
Cats algebra/algebra.lattice

algebra.lattice

package algebra.lattice

Members list

Concise view

Type members

Classlikes

trait Bool[A] extends Heyting[A] with GenBool[A]

Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).

Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).

This means that in addition to the laws Heyting algebras obey, boolean algebras also obey the following:

  • (a ∨ ¬a) = 1
  • ¬¬a = a

Boolean algebras generalize classical logic: one is equivalent to "true" and zero is equivalent to "false". Boolean algebras provide additional logical operators such as xor, nand, nor, and nxor which are commonly used.

Every boolean algebras has a dual algebra, which involves reversing true/false as well as and/or.

Attributes

Companion:
object
Source:
Bool.scala
Graph
Supertypes
trait GenBool[A]
trait Heyting[A]
trait Lattice[A]
class Any
Known subtypes
class DualBool[A]
Self type
Bool[A]
object Bool extends HeytingFunctions[Bool] with GenBoolFunctions[Bool]

Attributes

Companion:
trait
Source:
Bool.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
Bool.type
class BoolFromBoolRing[A](orig: BoolRing[A]) extends GenBoolFromBoolRng[A] with Bool[A]

Every Boolean ring gives rise to a Boolean algebra:

Every Boolean ring gives rise to a Boolean algebra:

  • 0 and 1 are preserved;
  • ring multiplication (times) corresponds to and;
  • ring addition (plus) corresponds to xor;
  • a or b is then defined as a xor b xor (a and b);
  • complement (¬a) is defined as a xor 1.

Attributes

Source:
Bool.scala
Graph
Supertypes
trait Bool[A]
trait Heyting[A]
trait GenBool[A]
trait Lattice[A]
class Object
trait Matchable
class Any
class BoolRingFromBool[A](orig: Bool[A]) extends BoolRngFromGenBool[A] with BoolRing[A]

Attributes

Source:
Bool.scala
Graph
Supertypes
trait BoolRing[A]
trait Ring[A]
trait Rig[A]
trait BoolRng[A]
trait Rng[A]
trait AdditiveGroup[A]
trait Semiring[A]
class Object
trait Matchable
class Any
class BoolRngFromGenBool[A](orig: GenBool[A]) extends BoolRng[A]

Attributes

Source:
GenBool.scala
Graph
Supertypes
trait BoolRng[A]
trait Rng[A]
trait AdditiveGroup[A]
trait Semiring[A]
class Object
trait Matchable
class Any
Known subtypes
trait BoundedDistributiveLattice[A] extends BoundedLattice[A] with DistributiveLattice[A]

A bounded distributive lattice is a lattice that both bounded and distributive

A bounded distributive lattice is a lattice that both bounded and distributive

Attributes

Companion:
object
Source:
BoundedDistributiveLattice.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
Self type
object BoundedDistributiveLattice extends BoundedMeetSemilatticeFunctions[BoundedDistributiveLattice] with BoundedJoinSemilatticeFunctions[BoundedDistributiveLattice]

Attributes

Companion:
trait
Source:
BoundedDistributiveLattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait BoundedJoinSemilattice[A] extends JoinSemilattice[A]

Attributes

Companion:
object
Source:
BoundedJoinSemilattice.scala
Graph
Supertypes
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
trait GenBool[A]
class SetLattice[A]
object BoundedJoinSemilattice extends JoinSemilatticeFunctions[BoundedJoinSemilattice] with BoundedJoinSemilatticeFunctions[BoundedJoinSemilattice]

Attributes

Companion:
trait
Source:
BoundedJoinSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait BoundedJoinSemilatticeFunctions[B <: (BoundedJoinSemilattice)] extends JoinSemilatticeFunctions[B]

Attributes

Source:
BoundedJoinSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object BoundedLattice.type
object DeMorgan.type
object Bool.type
object GenBool.type
object Heyting.type
trait BoundedLattice[A] extends Lattice[A] with BoundedMeetSemilattice[A] with BoundedJoinSemilattice[A]

A bounded lattice is a lattice that additionally has one element that is the bottom (zero, also written as ⊥), and one element that is the top (one, also written as ⊤).

A bounded lattice is a lattice that additionally has one element that is the bottom (zero, also written as ⊥), and one element that is the top (one, also written as ⊤).

This means that for any a in A:

join(zero, a) = a = meet(one, a)

Or written using traditional notation:

(0 ∨ a) = a = (1 ∧ a)

Attributes

Companion:
object
Source:
BoundedLattice.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
Self type
object BoundedLattice extends BoundedMeetSemilatticeFunctions[BoundedLattice] with BoundedJoinSemilatticeFunctions[BoundedLattice]

Attributes

Companion:
trait
Source:
BoundedLattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait BoundedMeetSemilattice[A] extends MeetSemilattice[A]

Attributes

Companion:
object
Source:
BoundedMeetSemilattice.scala
Graph
Supertypes
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
object BoundedMeetSemilattice extends BoundedMeetSemilatticeFunctions[BoundedMeetSemilattice]

Attributes

Companion:
trait
Source:
BoundedMeetSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait BoundedMeetSemilatticeFunctions[B <: (BoundedMeetSemilattice)] extends MeetSemilatticeFunctions[B]

Attributes

Source:
BoundedMeetSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object BoundedLattice.type
object DeMorgan.type
object Bool.type
object Heyting.type
trait DeMorgan[A] extends Logic[A]

De Morgan algebras are bounded lattices that are also equipped with a De Morgan involution.

De Morgan algebras are bounded lattices that are also equipped with a De Morgan involution.

De Morgan involution obeys the following laws:

  • ¬¬a = a
  • ¬(x∧y) = ¬x∨¬y

However, in De Morgan algebras this involution does not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1. De Morgan algebra do not not necessarily provide the law of non contradiction either. This means that there is no guarantee that (a ∧ ¬a) = 0.

De Morgan algebras are useful to model fuzzy logic. For a model of classical logic, see the boolean algebra type class implemented as Bool.

Attributes

Companion:
object
Source:
DeMorgan.scala
Graph
Supertypes
trait Logic[A]
trait Lattice[A]
class Any
Self type
object DeMorgan extends DeMorganFunctions[DeMorgan]

Attributes

Companion:
trait
Source:
DeMorgan.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait DeMorganFunctions[H <: (DeMorgan)] extends BoundedMeetSemilatticeFunctions[H] with BoundedJoinSemilatticeFunctions[H] with LogicFunctions[H]

Attributes

Source:
DeMorgan.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object DeMorgan.type
trait DistributiveLattice[A] extends Lattice[A]

A distributive lattice a lattice where join and meet distribute:

A distributive lattice a lattice where join and meet distribute:

  • a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
  • a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)

Attributes

Companion:
object
Source:
DistributiveLattice.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
trait GenBool[A]
class SetLattice[A]
class MinMaxLattice[A]
object DistributiveLattice extends JoinSemilatticeFunctions[DistributiveLattice] with MeetSemilatticeFunctions[DistributiveLattice]

Attributes

Companion:
trait
Source:
DistributiveLattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
class DualBool[A](orig: Bool[A]) extends Bool[A]

Attributes

Source:
Bool.scala
Graph
Supertypes
trait Bool[A]
trait GenBool[A]
trait Heyting[A]
trait Lattice[A]
class Object
trait Matchable
class Any
trait GenBool[A] extends DistributiveLattice[A] with BoundedJoinSemilattice[A]

Generalized Boolean algebra, that is, a Boolean algebra without the top element. Generalized Boolean algebras do not (in general) have (absolute) complements, but they have ''relative complements'' (see GenBool.without).

Generalized Boolean algebra, that is, a Boolean algebra without the top element. Generalized Boolean algebras do not (in general) have (absolute) complements, but they have ''relative complements'' (see GenBool.without).

Attributes

Companion:
object
Source:
GenBool.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
class SetLattice[A]
trait Bool[A]
class DualBool[A]
Self type
object GenBool extends GenBoolFunctions[GenBool]

Attributes

Companion:
trait
Source:
GenBool.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
GenBool.type
class GenBoolFromBoolRng[A](orig: BoolRng[A]) extends GenBool[A]

Every Boolean rng gives rise to a Boolean algebra without top:

Every Boolean rng gives rise to a Boolean algebra without top:

  • 0 is preserved;
  • ring multiplication (times) corresponds to and;
  • ring addition (plus) corresponds to xor;
  • a or b is then defined as a xor b xor (a and b);
  • relative complement a\b is defined as a xor (a and b).

BoolRng.asBool.asBoolRing gives back the original BoolRng.

Attributes

See also:
Source:
GenBool.scala
Graph
Supertypes
trait GenBool[A]
trait Lattice[A]
class Object
trait Matchable
class Any
Known subtypes
trait GenBoolFunctions[G <: (GenBool)] extends BoundedJoinSemilatticeFunctions[G] with MeetSemilatticeFunctions[G]

Attributes

Source:
GenBool.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object Bool.type
object GenBool.type
trait Heyting[A] extends BoundedDistributiveLattice[A]

Heyting algebras are bounded lattices that are also equipped with an additional binary operation imp (for implication, also written as →).

Heyting algebras are bounded lattices that are also equipped with an additional binary operation imp (for implication, also written as →).

Implication obeys the following laws:

  • a → a = 1
  • a ∧ (a → b) = a ∧ b
  • b ∧ (a → b) = b
  • a → (b ∧ c) = (a → b) ∧ (a → c)

In heyting algebras, and is equivalent to meet and or is equivalent to join; both methods are available.

Heyting algebra also define complement operation (sometimes written as ¬a). The complement of a is equivalent to (a → 0), and the following laws hold:

  • a ∧ ¬a = 0

However, in Heyting algebras this operation is only a pseudo-complement, since Heyting algebras do not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1.

Heyting algebras model intuitionistic logic. For a model of classical logic, see the boolean algebra type class implemented as Bool.

Attributes

Companion:
object
Source:
Heyting.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
trait Bool[A]
class DualBool[A]
Self type
object Heyting extends HeytingFunctions[Heyting] with HeytingGenBoolOverlap[Heyting]

Attributes

Companion:
trait
Source:
Heyting.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
Heyting.type
trait HeytingFunctions[H <: (Heyting)] extends BoundedMeetSemilatticeFunctions[H] with BoundedJoinSemilatticeFunctions[H]

Attributes

Source:
Heyting.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object Bool.type
object Heyting.type
trait HeytingGenBoolOverlap[H <: (Heyting)]

Attributes

Source:
Heyting.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object Heyting.type
trait JoinSemilattice[A] extends Serializable

A join-semilattice (or upper semilattice) is a semilattice whose operation is called "join", and which can be thought of as a least upper bound.

A join-semilattice (or upper semilattice) is a semilattice whose operation is called "join", and which can be thought of as a least upper bound.

Attributes

Companion:
object
Source:
JoinSemilattice.scala
Graph
Supertypes
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
trait GenBool[A]
class SetLattice[A]
trait Lattice[A]
class MinMaxLattice[A]
object JoinSemilattice extends JoinSemilatticeFunctions[JoinSemilattice]

Attributes

Companion:
trait
Source:
JoinSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait JoinSemilatticeFunctions[J <: (JoinSemilattice)]

Attributes

Source:
JoinSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object BoundedLattice.type
object DeMorgan.type
object Bool.type
object GenBool.type
object Heyting.type
object JoinSemilattice.type
object Lattice.type
trait Lattice[A] extends JoinSemilattice[A] with MeetSemilattice[A]

A lattice is a set A together with two operations (meet and join). Both operations individually constitute semilattices (join- and meet-semilattices respectively): each operation is commutative, associative, and idempotent.

A lattice is a set A together with two operations (meet and join). Both operations individually constitute semilattices (join- and meet-semilattices respectively): each operation is commutative, associative, and idempotent.

Join can be thought of as finding a least upper bound (supremum), and meet can be thought of as finding a greatest lower bound (infimum).

The join and meet operations are also linked by absorption laws:

meet(a, join(a, b)) = join(a, meet(a, b)) = a

Attributes

Companion:
object
Source:
Lattice.scala
Graph
Supertypes
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
trait GenBool[A]
class SetLattice[A]
class MinMaxLattice[A]
Self type
object Lattice extends JoinSemilatticeFunctions[Lattice] with MeetSemilatticeFunctions[Lattice]

Attributes

Companion:
trait
Source:
Lattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
Lattice.type
trait Logic[A] extends BoundedDistributiveLattice[A]

Logic models a logic generally. It is a bounded distributive lattice with an extra negation operator.

Logic models a logic generally. It is a bounded distributive lattice with an extra negation operator.

The negation operator obeys the weak De Morgan laws:

  • ¬(x∨y) = ¬x∧¬y
  • ¬(x∧y) = ¬¬(¬x∨¬y)

For intuitionistic logic see Heyting For fuzzy logic see DeMorgan

Attributes

Companion:
object
Source:
Logic.scala
Graph
Supertypes
trait Lattice[A]
class Any
Known subtypes
trait DeMorgan[A]
Self type
object Logic extends LogicFunctions[Logic]

Attributes

Companion:
trait
Source:
Logic.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
Logic.type
trait LogicFunctions[H <: (Logic)]

Attributes

Source:
Logic.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object DeMorgan.type
object Logic.type
trait MeetSemilattice[A] extends Serializable

A meet-semilattice (or lower semilattice) is a semilattice whose operation is called "meet", and which can be thought of as a greatest lower bound.

A meet-semilattice (or lower semilattice) is a semilattice whose operation is called "meet", and which can be thought of as a greatest lower bound.

Attributes

Companion:
object
Source:
MeetSemilattice.scala
Graph
Supertypes
class Any
Known subtypes
trait Heyting[A]
trait Bool[A]
class DualBool[A]
trait Logic[A]
trait DeMorgan[A]
trait Lattice[A]
trait GenBool[A]
class SetLattice[A]
class MinMaxLattice[A]
object MeetSemilattice extends MeetSemilatticeFunctions[MeetSemilattice]

Attributes

Companion:
trait
Source:
MeetSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Self type
trait MeetSemilatticeFunctions[M <: (MeetSemilattice)]

Attributes

Source:
MeetSemilattice.scala
Graph
Supertypes
class Object
trait Matchable
class Any
Known subtypes
object BoundedLattice.type
object DeMorgan.type
object Bool.type
object Heyting.type
object GenBool.type
object Lattice.type
object MeetSemilattice.type
class MinMaxBoundedDistributiveLattice[A](min: A, max: A)(implicit o: Order[A]) extends MinMaxLattice[A] with BoundedDistributiveLattice[A]

Attributes

Source:
BoundedDistributiveLattice.scala
Graph
Supertypes
class MinMaxLattice[A]
trait Lattice[A]
class Object
trait Matchable
class Any
class MinMaxLattice[A](implicit order: Order[A]) extends DistributiveLattice[A]

Attributes

Source:
DistributiveLattice.scala
Graph
Supertypes
trait Lattice[A]
class Object
trait Matchable
class Any
Known subtypes
Generated with
© 2002-2021 · LAMP/EPFL
© 2002-2021 · LAMP/EPFL