#62 ⟨a, b | ab=a, ba=a⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 6
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ b = a and a ⋅ 1 = a, however b ≠ 1
- Gröbner basis: ⟨a, b | ab=a⟩
- Cancellative quotient is isomorphic to ℕ
- Enveloping group is isomorphic to ℤ
- Group of units is isomorphic to ℤ1
- 3 Archimedian components:
- Reduction order:
- Left-to-right shortlex with a < b
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ab=a,ba=a ab - -
ab=a
ba=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
34 total
| Σ | # | Presentation | Mapping |
|---|
| 7 | 269 | ⟨a, b | ab=a, bab=a⟩ | φ(a) = a, φ(b) = b |
| 7 | 279 | ⟨a, b | ab=a, ba=ab⟩ | φ(a) = a, φ(b) = b |
| 8 | 942 | ⟨a, b | ab=a, babb=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 1006 | ⟨a, b | ab=a, abb=ba⟩ | φ(a) = a, φ(b) = b |
| 8 | 1013 | ⟨a, b | ab=a, bab=ab⟩ | φ(a) = a, φ(b) = b |
| 9 | 1623 | ⟨a, b | aab=ab, aba=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2109 | ⟨a, b | aba=b, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2982 | ⟨a, b | ab=a, babbb=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 3102 | ⟨a, b | ab=a, abbb=ba⟩ | φ(a) = a, φ(b) = b |
| 9 | 3117 | ⟨a, b | ab=a, babb=ab⟩ | φ(a) = a, φ(b) = b |
| 9 | 3185 | ⟨a, b | ab=a, bab=abb⟩ | φ(a) = a, φ(b) = b |
| 10 | 5131 | ⟨a, b | aab=ab, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5251 | ⟨a, b | aba=ab, abaa=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 8806 | ⟨a, b | ab=a, babbbb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 9046 | ⟨a, b | ab=a, abbbb=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 9077 | ⟨a, b | ab=a, babbb=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9317 | ⟨a, b | ab=a, abbb=bab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9347 | ⟨a, b | ab=a, babb=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 12355 | ⟨a, b | aaba=ab, abaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14606 | ⟨a, b | aaba=b, aaaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15614 | ⟨a, b | aab=ab, aaaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15858 | ⟨a, b | aba=ab, abaaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19169 | ⟨a, b | aba=b, aaabaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19712 | ⟨a, b | aba=b, aaaba=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19720 | ⟨a, b | aba=b, aabaa=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20255 | ⟨a, b | aba=b, aaab=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20263 | ⟨a, b | aba=b, aaba=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20264 | ⟨a, b | aba=b, aaba=aba⟩ | φ(a) = b, φ(b) = a |
| 11 | 24522 | ⟨a, b | ab=a, babbbbb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24986 | ⟨a, b | ab=a, abbbbb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 25049 | ⟨a, b | ab=a, babbbb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25529 | ⟨a, b | ab=a, abbbb=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25591 | ⟨a, b | ab=a, babbb=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25854 | ⟨a, b | ab=a, babb=abbb⟩ | φ(a) = a, φ(b) = b |