algebra

lattice

package lattice

Visibility
  1. Public
  2. All

Type Members

  1. 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).

  2. class BoolFromBoolRing[A] extends GenBoolFromBoolRng[A] with Bool[A]

    Every Boolean ring gives rise to a Boolean algebra:

  3. trait BoundedDistributiveLattice[A] extends BoundedLattice[A] with DistributiveLattice[A]

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

  4. trait BoundedJoinSemilattice[A] extends JoinSemilattice[A]

  5. trait BoundedJoinSemilatticeFunctions[B[A] <: BoundedJoinSemilattice[A]] extends JoinSemilatticeFunctions[B]

  6. 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 ⊤).

  7. trait BoundedMeetSemilattice[A] extends MeetSemilattice[A]

  8. trait BoundedMeetSemilatticeFunctions[B[A] <: BoundedMeetSemilattice[A]] extends MeetSemilatticeFunctions[B]

  9. trait DistributiveLattice[A] extends Lattice[A]

    A distributive lattice a lattice where join and meet distribute:

  10. class DualBool[A] extends Bool[A]

  11. trait GenBool[A] extends DistributiveLattice[A] with BoundedJoinSemilattice[A]

    Generalized Boolean algebra, that is, a Boolean algebra without the top element.

  12. class GenBoolFromBoolRng[A] extends GenBool[A]

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

  13. trait GenBoolFunctions[G[A] <: GenBool[A]] extends BoundedJoinSemilatticeFunctions[G] with MeetSemilatticeFunctions[G]

  14. trait Heyting[A] extends BoundedDistributiveLattice[A]

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

  15. trait HeytingFunctions[H[A] <: Heyting[A]] extends BoundedMeetSemilatticeFunctions[H] with BoundedJoinSemilatticeFunctions[H]

  16. trait HeytingGenBoolOverlap[H[A] <: Heyting[A]] extends AnyRef

  17. 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.

  18. trait JoinSemilatticeFunctions[J[A] <: JoinSemilattice[A]] extends AnyRef

  19. trait Lattice[A] extends JoinSemilattice[A] with MeetSemilattice[A]

    A lattice is a set A together with two operations (meet and join).

  20. 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.

  21. trait MeetSemilatticeFunctions[M[A] <: MeetSemilattice[A]] extends AnyRef

  22. class MinMaxBoundedDistributiveLattice[A] extends MinMaxLattice[A] with BoundedDistributiveLattice[A]

  23. class MinMaxLattice[A] extends DistributiveLattice[A]

Ungrouped