#2584 ⟨a, b | aaa=1, abbb=ba⟩
Contents
- Properties
- Finite structure
- Right Cayley graph
- Left Cayley graph
- Rewriting system
- Isomorphic instances
- Anti-isomorphic instances
- Sum of relation sides is 9
- Finite non-commutative monoid with 81 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b26 = b and b ⋅ 1 = b, however b26 ≠ 1
Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.
- 3 element center:
- 1 non-trivial idempotent:
- Order of generators:
- a: order 3
- b: index 1, period 26
- Histogram:
| order 3 | 2 elements | a, a2 |
| index 1, period 1 | 1 element | b26 |
| index 1, period 2 | 1 element | b13 |
| index 1, period 3 | 26 elements | ab2, a2b2, ab4, a2b4, ab6, ... |
| index 1, period 6 | 26 elements | ab, a2b, ab3, a2b3, ab5, ... |
| index 1, period 13 | 12 elements | b2, b4, b6, b8, b10, ... |
| index 1, period 26 | 12 elements | b, b3, b5, b7, b9, ... |
- Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:aaa=1,abbb=ba b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbb
aaa=1
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
3 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 8045 | ⟨a, b | aaa=1, aaba=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 22263 | ⟨a, b | aaa=1, ababab=ba⟩ | φ(a) = a, φ(b) = aab |
| 11 | 22774 | ⟨a, b | aaa=1, aabbb=aba⟩ | φ(a) = a, φ(b) = b |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
3 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 7507 | ⟨a, b | aaa=1, aabbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 22776 | ⟨a, b | aaa=1, aabbb=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 22811 | ⟨a, b | aaa=1, abbba=aab⟩ | φ(a) = a, φ(b) = b |