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Update algebra page #4881

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e65e693
Update jump_start_guide.md
Jul 4, 2026
bdb2130
Update algebra.md
Jul 5, 2026
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Fix the comment
Oct 12, 2026
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Merge branch 'main' into Update-algebra-page
Oct 25, 2026
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Update algebra.md
Oct 25, 2026
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144 changes: 120 additions & 24 deletions docs/algebra.md
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Expand Up @@ -39,6 +39,30 @@ The most basic structures can be found in the `algebra` package. They all implem
|CommutativeGroup | ✓| ✓| ✓| ✓| |
|BoundedSemilattice | ✓| ✓| ✓| | ✓|

Dependency diagram:

```mermaid
graph TB
Semigroup --> CommutativeSemigroup
Semigroup --> Monoid
Semigroup --> Band

CommutativeSemigroup --> Semilattice
CommutativeSemigroup --> CommutativeMonoid

Monoid --> Group
Monoid --> CommutativeMonoid

Band --> Semilattice

Semilattice --> BoundedSemilattice

CommutativeMonoid --> CommutativeGroup
CommutativeMonoid --> BoundedSemilattice

Group --> CommutativeGroup
```

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

## ring-like structures
Expand All @@ -47,18 +71,61 @@ The `algebra.ring` package contains more sophisticated structures which combine

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`?|Commutative `*`?|
|--------------------|-------------|--------|-----------------|----------------|
|Semiring | | | | |
|Rng | ✓| | | |
|Rig | | ✓| | |
|CommutativeRig | | ✓| | ✓|
|Ring | ✓| ✓| | |
|CommutativeRing | ✓| ✓| | ✓|
|Semifield | | ✓| ✓| |
|CommutativeSemifield| | ✓| ✓| ✓|
|Field | ✓| ✓| ✓| ✓|

| Name | Has `negate`? | Has `1`? | Has `reciprocal`? | Commutative `*`? | Has `gcd`/`lcm`? | Has `equot`/`emod`? | Idempotent `*`? |
|----------------------|----------------|-----------|--------------------|-------------------|-------------------|----------------------|------------------|
| Semiring | | | | | | | |
| Rng | ✓ | | | | | | |
| Rig | | ✓ | | | | | |
| CommutativeSemiring | | | | ✓ | | | |
| CommutativeRng | ✓ | | | ✓ | | | |
| CommutativeRig | | ✓ | | ✓ | | | |
| Ring | ✓ | ✓ | | | | | |
| Semifield | | ✓ | ✓ | | | | |
| CommutativeRing | ✓ | ✓ | | ✓ | | | |
| CommutativeSemifield | | ✓ | ✓ | ✓ | | | |
| DivisionRing | ✓ | ✓ | ✓ | | | | |
| BoolRng | ✓ | | | ✓ | | | ✓ |
| BoolRing | ✓ | ✓ | | ✓ | | | ✓ |
| GCDRing | ✓ | ✓ | | ✓ | ✓ | | |
| EuclideanRing | ✓ | ✓ | | ✓ | ✓ | ✓ | |
| Field | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Dependency diagram:

```mermaid
graph TB
Semiring --> Rng
Semiring --> Rig
Semiring --> CommutativeSemiring

Rng --> Ring
Rig --> Ring
Ring --> CommutativeRing

Rng --> CommutativeRng
CommutativeSemiring --> CommutativeRng
Rig --> CommutativeRig
CommutativeSemiring --> CommutativeRig
CommutativeRng --> CommutativeRing
CommutativeRig --> CommutativeRing

Rig --> Semifield
Semifield --> CommutativeSemifield
CommutativeRig --> CommutativeSemifield

Semifield --> DivisionRing
Ring --> DivisionRing
CommutativeRing --> GCDRing
GCDRing --> EuclideanRing

DivisionRing --> Field
EuclideanRing --> Field
CommutativeSemifield --> Field

CommutativeRng --> BoolRng
BoolRng --> BoolRing
CommutativeRing --> BoolRing
```

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

Expand Down Expand Up @@ -88,18 +155,47 @@ 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 | ✓| ✓| ✓| ✓| ✓| ✓| ✓|
| Name | Has `join`? | Has `meet`? | Has `zero`? | Has `one`? | Distributive | Has `imp`? | Has `without`? | Excludes middle? |
|----------------------------|-------------|-------------|-------------|------------|--------------|------------|----------------|------------------|
| JoinSemilattice | ✓ | | | | | | | |
| MeetSemilattice | | ✓ | | | | | | |
| BoundedJoinSemilattice | ✓ | | ✓ | | | | | |
| BoundedMeetSemilattice | | ✓ | | ✓ | | | | |
| Lattice | ✓ | ✓ | | | | | | |
| DistributiveLattice | ✓ | ✓ | | | ✓ | | | |
| BoundedLattice | ✓ | ✓ | ✓ | ✓ | | | | |
| BoundedDistributiveLattice | ✓ | ✓ | ✓ | ✓ | ✓ | | | |
| Heyting | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | | |
| GenBool | ✓ | ✓ | ✓ | | ✓ | | ✓ | |
| Bool | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

Dependency diagram:

```mermaid
graph TD
JoinSemilattice --> BoundedJoinSemilattice
MeetSemilattice --> BoundedMeetSemilattice

JoinSemilattice --> Lattice
MeetSemilattice --> Lattice

Lattice --> DistributiveLattice

Lattice --> BoundedLattice
BoundedJoinSemilattice --> BoundedLattice
BoundedMeetSemilattice --> BoundedLattice

DistributiveLattice --> BoundedDistributiveLattice
BoundedLattice --> BoundedDistributiveLattice

BoundedDistributiveLattice --> Heyting

DistributiveLattice --> GenBool
BoundedJoinSemilattice --> GenBool

Heyting --> Bool
GenBool --> Bool
```

Note that a `BoundedDistributiveLattice` gives you a `CommutativeRig`, but not the other way around: rigs aren't distributive with `a + (b * c) = (a + b) * (a + c)`.

Expand Down
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