[6, 3841] finitely based
[6, 3841] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 6 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 2 |
| 3 | 1 | 1 | 1 | 1 | 1 | 3 |
| 4 | 1 | 1 | 2 | 1 | 4 | 1 |
| 5 | 1 | 1 | 3 | 1 | 5 | 1 |
| 6 | 1 | 2 | 1 | 4 | 1 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal; grey: the zero.
Structure
- Smallsemi
- SmallSemigroup(6, 3841)
- Idempotents
- 1, 5, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 4, generated by {3, 4, 5, 6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 6 identities reduced
- x² ≈ x³
- x²yx ≈ xyx
- x²y² ≈ y²x²
- xyxzx ≈ xzxyx
- xyzxtyux ≈ yzxtyux
- xyztxuz ≈ xyztxuzx
Reduced as far as the search went: some identity may still follow from the others. Obtained from a basis certified in Lean by removing identities that follow from the others, with the prover Vampire; the equivalence rests on Vampire's proofs, not on Lean. Shortest known, not known to be minimal.
Basis certified in Lean
The Lean theorem certifies a longer basis, of 22 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6SporadicSection18.C7.S6_3841.opposite_basis
BasisFor CoRoots.Order6SporadicSection18.C7.S6_3841.oppositeTable.semigroup (reversedBasis CoRoots.Order6SporadicSection18.basis)
- Table
- The theorem is about the opposite semigroup, with the transposed table; read backwards, its basis is the basis of this table shown above.
- Method
- family proof. One basis is proved once to derive every identity of every semigroup of a family; on each table only the identities of the basis are checked, by computation.
- Size
- Checking this class alone compiles 70 Lean files with 17,264 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 8,911 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_3841 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_3841 is the table, with the elements numbered 0 to 5.
Variety
[6, 3841] generates the variety V[3841]; no other semigroup of order six generates it, and 2,031 semigroups of order six lie in it. [6, 3841] lies in 5 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[3841] in the inclusion graph, with the varieties above and below it.
Nearby tables
Semigroups whose Smallsemi table differs from this one in one or two entries (row, column), as the tables are listed, not up to renumbering.