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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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52 changes: 28 additions & 24 deletions docs/algebra.md
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Expand Up @@ -47,18 +47,21 @@ 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 `quot`/`mod`? |
|----------------------|----------------|-----------|--------------------|-------------------|-------------------|--------------------|
| Semiring | | | | | | |
| Rng | ✓ | | | | | |
| Rig | | ✓ | | | | |
| CommutativeSemiring | | | | ✓ | | |
| CommutativeRng | ✓ | | | ✓ | | |
| CommutativeRig | | ✓ | | ✓ | | |
| Ring | ✓ | ✓ | | | | |
| CommutativeRing | ✓ | ✓ | | ✓ | | |
| Semifield | | ✓ | ✓ | | | |
| CommutativeSemifield | | ✓ | ✓ | ✓ | | |
| GCDRing | ✓ | ✓ | | ✓ | ✓ | |
| EuclideanRing | ✓ | ✓ | | ✓ | ✓ | ✓ |
| Field | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

(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 +91,19 @@ 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`? | Excludes middle? |
|----------------------------|-------------|-------------|-------------|------------|--------------|------------|------------------|
| JoinSemilattice | ✓ | | | | | | |
| MeetSemilattice | | ✓ | | | | | |
| BoundedJoinSemilattice | ✓ | | ✓ | | | | |
| BoundedMeetSemilattice | | ✓ | | ✓ | | | |
| Lattice | ✓ | ✓ | | | | | |
| DistributiveLattice | ✓ | ✓ | | | ✓ | | |
| BoundedLattice | ✓ | ✓ | ✓ | ✓ | | | |
| BoundedDistributiveLattice | ✓ | ✓ | ✓ | ✓ | ✓ | | |
| Heyting | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
| 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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