#9739 ⟨a, b | aa=1, ababbbb=b⟩
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 34 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b8 = b and b ⋅ 1 = b, however b8 ≠ 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.
- 1 element center:
- 2 non-trivial idempotents:
- Order of generators:
- a: order 2
- b: index 1, period 8
- Histogram:
| order 2 | 1 element | a |
| index 1, period 1 | 2 elements | b8, ab8a |
| index 1, period 2 | 6 elements | b4, ab4, b4a, ab4a, ab8, ... |
| index 1, period 4 | 8 elements | b2, ab2, b2a, ab2a, b6, ... |
| index 1, period 8 | 16 elements | b, ab, ba, aba, b3, ... |
- 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:aa=1,ababbbb=b b/a - -
bbbbbbbbb=b
bab=abbbbbb
aa=1
1 unique, 1 total
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
2 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 9753 | ⟨a, b | aa=1, abbbabb=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 10051 | ⟨a, b | aa=1, babbbb=ab⟩ | φ(a) = a, 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 | 10059 | ⟨a, b | aa=1, bbabbb=ba⟩ | φ(a) = a, φ(b) = b |