algebra

the vision

This repo represents an attempt to unify the basic algebraic type classes from Spire and Algebird. By targeting just these type classes, we will distribute an algebra package with no dependencies that works with Scala 2.10 and 2.11, and which can be shared by all Scala libraries interested in abstract algebra.

getting algebra

Algebra supports Scala 2.10 and 2.11, and is available from Sonatype (and Maven Central). To use algebra in your own projects, include this snippet in your build.sbt file:

resolvers += Resolver.sonatypeRepo("releases")

libraryDependencies += "org.spire-math" %% "algebra" % "0.4.0"

As of 0.3.1 algebra also supports scala.js!

what we have so far

This repo has been seeded with most of the contents of spire.algebra, and has been modified to fit an initially-conservative and shared vision for algebraic type classes. Discussions over removals and additions should take place on the issue tracker or on relevant pull requests.

participants

Anyone who wants to participate should feel free to send pull requests to this repo. The following people have push access:

• Oscar Boykin
• Avi Bryant
• Lars Hupel
• Erik Osheim
• Tom Switzer

development process

Please make a pull request against the master branch. For those that have merge access to the repo, we follow these rules:

1. Do not merge your own PR unless N people have signed-off on the PR (e.g. given a thumbs-up, +1, shipit, etc) and Travis is green.

2. If you are not the author, and you see N-1 sign-offs and Travis is green, just click merge and delete the branch.

3. Currently, N = 1.

algebra overview

Algebra uses type classes to represent algebraic structures. You can use these type classes to represent the abstract capabilities (and requirements) you want generic parameters to possess.

This section will explain the structures available.

algebraic properties and terminology

We will be talking about properties like associativity and commutativity. Here is a quick explanation of what those properties mean:

Name	Description
Associative	If ⊕ is associative, then a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c.
Commutative	If ⊕ is commutative, then a ⊕ b = b ⊕ a.
Identity	If id is an identity for ⊕, then a ⊕ id = id ⊕ a = a.
Inverse	If ¬ is an inverse for ⊕ and id, then a ⊕ ¬a = ¬a ⊕ a = id.
Distributive	If ⊕ and ⊙ distribute, then a ⊙ (b ⊕ c) = (a ⊙ b) ⊕ (a ⊙ c) and (a ⊕ b) ⊙ c = (a ⊙ c) ⊕ (b ⊙ c).
Idempotent	If ⊕ is idempotent, then a ⊕ a = a

Though these properties are illustrated with symbolic operators, they work equally-well with functions. When you see a ⊕ b that is equivalent to f(a, b): ⊕ is an infix representation of the binary function f, and a and b are values (of some type A).

Similarly, when you see ¬a that is equivalent to g(a): ¬ is a prefix representation of the unary function g, and a is a value (of some type A).

basic algebraic structures

The most basic structures can be found in the algebra package. They all implement a method called combine, which is associative. The identity element (if present) will be called empty, and the inverse method (if present) will be called inverse.

Name	Associative?	Commutative?	Identity?	Inverse?	Idempotent?
Semigroup	✓
CommutativeSemigroup	✓	✓
Monoid	✓		✓
Band	✓				✓
Semilattice	✓	✓			✓
Group	✓		✓	✓
CommutativeMonoid	✓	✓	✓
CommutativeGroup	✓	✓	✓	✓
BoundedSemilattice	✓	✓	✓		✓

(For a description of what each column means, see §algebraic properties and terminology.)

ring-like structures

The algebra.ring contains more sophisticated structures which combine an additive operation (called plus) and a multiplicative operation (called times). Additive identity and inverses will be called zero and negate (respectively); multiplicative identity and inverses will be called one and reciprocal (respectively).

Additionally some structures support a quotient operation, called quot.

All ring-like structures are associative for both + and *, have commutative +, and have a zero element (an identity for +).

Name	Has negate?	Has 1?	Has reciprocal?	Has quot?	Commutative *?
Semiring
Rng	✓
Rig		✓
CommutativeRig		✓			✓
Ring	✓	✓
CommutativeRing	✓	✓			✓
EuclideanRing	✓	✓		✓	✓
Field	✓	✓	✓	✓	✓

With the exception of CommutativeRig and Rng, every lower structure is also an instance of the structures above it. For example, every Ring is a Rig, every Field is a CommutativeRing, and so on.

(For a description of what the terminology in each column means, see §algebraic properties and terminology.)

lattice-like structures

The algebra.lattice contains more structures that can be somewhat ring-like. Rather than plus and times we have meet and join both of which are always associative, commutative and idempotent, and as such each can be viewed as a semilattice. Meet can be thought of as the greatest lower bound of two items while join can be thought of as the least upper bound between two items.

When zero is present, join(a, zero) = a. When one is present meet(a, one) = a.

When meet and join are both present, they obey the absorption law:

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

Sometimes meet and join distribute, we say it is distributive in this case:

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

Sometimes an additional binary operation imp (for impliciation, also written as →, meet written as ∧) is present. Implication obeys the following laws:

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

The law of the excluded middle can be expressed as:

• (a ∨ (a → 0)) = 1

Name	Has join?	Has meet?	Has zero?	Has one?	Distributive	Has imp?	Excludes middle?
JoinSemilattice	✓
MeetSemilattice		✓
BoundedJoinSemilattice	✓		✓
BoundedMeetSemilattice		✓		✓
Lattice	✓	✓
DistributiveLattice	✓	✓			✓
BoundedLattice	✓	✓	✓	✓
BoundedDistributiveLattice	✓	✓	✓	✓	✓
Heyting	✓	✓	✓	✓	✓	✓
Bool	✓	✓	✓	✓	✓	✓	✓

Note that a BoundedDistributiveLattice gives you a CommutativeRig, but not the other way around (since rigs aren't distributive with a + (b * c) = (a + b) * (a + c)). Also, a Bool gives rise to a BoolRing, since each element can be defined as its own negation. Note, Bool's asBoolRing is not an extension of the asCommutativeRig as the plus operations are defined differently.