#4691 ⟨a, b | aaab=b, babb=a⟩
Contents
- Properties
- Finite structure
- Right Cayley graph
- Left Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Anti-isomorphic instances
- Sum of relation sides is 10
- Finite non-commutative monoid with 28 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b9 = b and b ⋅ 1 = b, however b9 ≠ 1
Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.
- 4 element center:
- 1 non-trivial idempotent:
- Order of generators:
- a: index 1, period 3
- b: index 1, period 9
- Histogram:
| index 1, period 1 | 1 element | b9 |
| index 1, period 3 | 8 elements | a, a2, b3, b3a, b3a2, ... |
| index 1, period 9 | 18 elements | b, ba, b2, ba2, b2a, ... |
- Reduction order:
- Right-to-left recursive path with deg(b) = 0; deg(a) = 1
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:aaab=b,babb=a reversed:b/a - -
bbbbbbbbbb=b
bbbbbbbbba=a
ab=bbbba
aaa=bbbbbbbbb
2 unique, 5 total
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
2 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 4727 | ⟨a, b | aaba=b, babb=a⟩ | φ(a) = ba, φ(b) = b |
| 10 | 4730 | ⟨a, b | aaba=b, bbab=a⟩ | φ(a) = bbaa, φ(b) = b |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
1 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 4695 | ⟨a, b | aaab=b, bbab=a⟩ | φ(a) = a, φ(b) = b |