#243 ⟨a, b | aa=a, bab=a⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 7
- Infinite non-cancellative commutative monoid
- Element b has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
- Gröbner basis: ⟨a, b | a2=a, ab2=a⟩
- Cancellative quotient is isomorphic to ℤ2
- Enveloping group is isomorphic to ℤ2
- 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:aa=a,bab=a ab - -
aa=a
ba=ab
abb=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
100 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 894 | ⟨a, b | aa=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 966 | ⟨a, b | aa=a, bab=aa⟩ | φ(a) = a, φ(b) = b |
| 9 | 1628 | ⟨a, b | aab=ab, bab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 1640 | ⟨a, b | aab=ba, bab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 1647 | ⟨a, b | aab=bb, aba=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 1670 | ⟨a, b | aba=ab, bab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2111 | ⟨a, b | aba=b, aabb=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2113 | ⟨a, b | aba=b, abab=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2115 | ⟨a, b | aba=b, abba=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2119 | ⟨a, b | aba=b, baab=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2884 | ⟨a, b | aa=a, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 3026 | ⟨a, b | aa=a, baab=aa⟩ | φ(a) = a, φ(b) = b |
| 9 | 3144 | ⟨a, b | aa=a, bab=aaa⟩ | φ(a) = a, φ(b) = b |
| 10 | 5036 | ⟨a, b | aaa=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5144 | ⟨a, b | aab=ab, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5164 | ⟨a, b | aab=ba, aabb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5168 | ⟨a, b | aab=ba, abab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5170 | ⟨a, b | aab=ba, abba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5178 | ⟨a, b | aab=ba, baba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5182 | ⟨a, b | aab=ba, bbaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5260 | ⟨a, b | aba=ab, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5262 | ⟨a, b | aba=ab, baba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8620 | ⟨a, b | aa=a, baaaab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8892 | ⟨a, b | aa=a, baaab=aa⟩ | φ(a) = a, φ(b) = b |
| 10 | 9116 | ⟨a, b | aa=a, aaaa=bab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9164 | ⟨a, b | aa=a, baab=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 12273 | ⟨a, b | aaab=ba, aabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12277 | ⟨a, b | aaab=ba, abab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12279 | ⟨a, b | aaab=ba, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12353 | ⟨a, b | aaba=ab, aabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12357 | ⟨a, b | aaba=ab, abab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12359 | ⟨a, b | aaba=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12504 | ⟨a, b | abab=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12506 | ⟨a, b | abab=aa, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12599 | ⟨a, b | abba=bb, baba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12603 | ⟨a, b | abba=bb, bbaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14699 | ⟨a, b | aabb=a, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14763 | ⟨a, b | abab=a, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14765 | ⟨a, b | abab=a, baaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14769 | ⟨a, b | abab=a, babaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14821 | ⟨a, b | abba=a, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14848 | ⟨a, b | abba=b, aabbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14856 | ⟨a, b | abba=b, abbab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14858 | ⟨a, b | abba=b, abbba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14864 | ⟨a, b | abba=b, baabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15429 | ⟨a, b | aaa=aa, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15643 | ⟨a, b | aab=ab, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15679 | ⟨a, b | aab=ba, aaabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15685 | ⟨a, b | aab=ba, aabba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15691 | ⟨a, b | aab=ba, abaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15697 | ⟨a, b | aab=ba, abbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15707 | ⟨a, b | aab=ba, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15709 | ⟨a, b | aab=ba, baaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15721 | ⟨a, b | aab=ba, bbaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15875 | ⟨a, b | aba=ab, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15877 | ⟨a, b | aba=ab, baaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15881 | ⟨a, b | aba=ab, babaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 19167 | ⟨a, b | aba=b, aaaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19171 | ⟨a, b | aba=b, aaabab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19173 | ⟨a, b | aba=b, aaabba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19177 | ⟨a, b | aba=b, aabaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19179 | ⟨a, b | aba=b, aababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19183 | ⟨a, b | aba=b, aabbaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19191 | ⟨a, b | aba=b, abaaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19193 | ⟨a, b | aba=b, abaaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19213 | ⟨a, b | aba=b, baaaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19710 | ⟨a, b | aba=b, aaaab=bb⟩ | φ(a) = b, φ(b) = a |
| 11 | 19714 | ⟨a, b | aba=b, aaaba=bb⟩ | φ(a) = b, φ(b) = a |
| 11 | 19716 | ⟨a, b | aba=b, aaabb=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19717 | ⟨a, b | aba=b, aaabb=ba⟩ | φ(a) = b, φ(b) = a |
| 11 | 19721 | ⟨a, b | aba=b, aabaa=bb⟩ | φ(a) = b, φ(b) = a |
| 11 | 19723 | ⟨a, b | aba=b, aabab=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19724 | ⟨a, b | aba=b, aabab=ba⟩ | φ(a) = b, φ(b) = a |
| 11 | 19727 | ⟨a, b | aba=b, aabba=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19728 | ⟨a, b | aba=b, aabba=ba⟩ | φ(a) = b, φ(b) = a |
| 11 | 19735 | ⟨a, b | aba=b, abaab=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19736 | ⟨a, b | aba=b, abaab=ba⟩ | φ(a) = b, φ(b) = a |
| 11 | 19739 | ⟨a, b | aba=b, ababa=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19757 | ⟨a, b | aba=b, baaab=ab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20257 | ⟨a, b | aba=b, aaab=abb⟩ | φ(a) = b, φ(b) = a |
| 11 | 20259 | ⟨a, b | aba=b, aaab=bab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20260 | ⟨a, b | aba=b, aaab=bba⟩ | φ(a) = b, φ(b) = a |
| 11 | 20265 | ⟨a, b | aba=b, aaba=abb⟩ | φ(a) = b, φ(b) = a |
| 11 | 20267 | ⟨a, b | aba=b, aaba=bab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20268 | ⟨a, b | aba=b, aaba=bba⟩ | φ(a) = b, φ(b) = a |
| 11 | 20271 | ⟨a, b | aba=b, aabb=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20272 | ⟨a, b | aba=b, aabb=aba⟩ | φ(a) = b, φ(b) = a |
| 11 | 20274 | ⟨a, b | aba=b, aabb=baa⟩ | φ(a) = b, φ(b) = a |
| 11 | 20279 | ⟨a, b | aba=b, abab=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20280 | ⟨a, b | aba=b, abab=aba⟩ | φ(a) = b, φ(b) = a |
| 11 | 20282 | ⟨a, b | aba=b, abab=baa⟩ | φ(a) = b, φ(b) = a |
| 11 | 20287 | ⟨a, b | aba=b, abba=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20288 | ⟨a, b | aba=b, abba=aba⟩ | φ(a) = b, φ(b) = a |
| 11 | 20301 | ⟨a, b | aba=b, baab=aab⟩ | φ(a) = b, φ(b) = a |
| 11 | 20302 | ⟨a, b | aba=b, baab=aba⟩ | φ(a) = b, φ(b) = a |
| 11 | 24148 | ⟨a, b | aa=a, baaaaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24688 | ⟨a, b | aa=a, baaaab=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 25120 | ⟨a, b | aa=a, aaaaa=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25220 | ⟨a, b | aa=a, baaab=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 25690 | ⟨a, b | aa=a, baab=aaaa⟩ | φ(a) = a, φ(b) = b |