Multisets for Scala

Based on A New Look at Multisets by Norman J Wildberger.

The data structure provided by this library is MSet. An MSet[M,A] is a multiset of values of type A with multiplicities in M.

For example, an MSet[Natural,A] is a multiset where values of type A can have zero or more occurrences. An MSet[Boolean,A] is an ordinary set.

Using this library

To get all the functionality of this library, add the following imports:

import spire.syntax.all._
import mset._
import Realm._

Why this?

This type differs from most implementations of multisets in that the multiplicities can be integers (for negative occurrences), rational numbers (for fractional occurrences), intervals and probability distributions (for non-deterministic or "fuzzy" occurrences), or any other type of value. The algebra on that type will determine what operations are available on the MSet. In particular, MSet provides operations for any type of measure that forms a Realm, a type this library also provides:

trait Realm[A] {
  def meet(x: A, y: A): A
  def join(x: A, y: A): A
  def sum(x: A, y: A): A
  def zero: A
}

Notably, MSet[M,A] forms a Realm for any Realm[M].

A Realm[A] is a distributive lattice on A (through meet and join), and a commutative monoid (on sum and zero). Realms must obey the following laws (where ∧ is meet, ∨ is join, + is sum and 0 is zero):

Commutative laws

m + n ≡ n + m
m ∨ n ≡ n ∨ m
m ∧ n ≡ n ∧ m

Associative laws

k + (m + n) ≡ (k + m) + n
k ∨ (m ∨ n) ≡ (k ∨ m) ∨ n
k ∧ (m ∧ n) ≡ (k ∧ m) ∧ n

Distributive laws

k + (m ∨ n) ≡ (k + m) ∨ (k + n)
k + (m ∧ n) ≡ (k + m) ∧ (k + n)
k ∧ (m ∨ n) ≡ (k ∧ m) ∨ (k ∧ n)
k ∨ (m ∧ n) ≡ (k ∨ m) ∧ (k ∨ n)

Identity laws

0 + m ≡ m
0 ∨ m ≡ m
0 ∧ m ≡ 0 

Absorption laws

m ∨ (m ∧ n) ≡ m
m ∧ (m ∨ n) ≡ m

Idempotent laws

m ∨ m ≡ m
m ∧ m ≡ m

Summation law

(m ∨ n) + (m ∧ n) ≡ m + n

Partial order

Realm extends spire.algebra.Order because any join semilattice defines an order:

m ≤ n ≡ m ∨ n = n

Cancellative law

Some realms may additionally obey a cancellation law, and we call these cancellative realms:

(k + n = m + n) ⇒ (k = m)

Products and inverses

Two Realm subtypes with products are also provided which are RigRealm (realms with products) and RingRealm (realms with products and additive inverses).

A well-behaved realm with products requires products to distribute over the realm operations:

a * (b ∨ c) ≡ (a * b) ∨ (a * c)
a * (b ∧ c) ≡ (a * b) ∧ (a * c)
a * (b + c) ≡ (a * b) + (a * c)

A realm with inverses must obey a De Morgan law:

¬(a ∨ b) ≡ (¬a) ∧ (¬b)
¬(a ∧ b) ≡ (¬a) ∨ (¬b)

M-Realms and monus

An MRealm is a realm with a partial additive inverse called monus or ∸. Every RingRealm is an MRealm, where the monus happens to be exact. A monus operation is a kind of truncated or residuated subtraction.

The monus must be left adjoint to the addition:

a ∸ b ≤ c ≡ a ≤ b + c

In terms of the meet and join, that means:

a ∧ (b + (a ∸ b)) ≡ a
((b + c) ∸ b) ∨ c ≡ c

That is, monus(a,b) is the smallest object which when added to b is at least a.

Examples of monus operations include truncated subtraction on natural numbers, the quotient operation on integers, and the difference operation on sets and multisets.