#267 ⟨a, b | ab=a, baa=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, ab=a⟩
- Cancellative quotient is isomorphic to ℤ1
- Enveloping group is isomorphic to ℤ1
- 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,baa=a ab - -
aa=a
ab=a
ba=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
168 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 938 | ⟨a, b | ab=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 940 | ⟨a, b | ab=a, baba=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 1009 | ⟨a, b | ab=a, baa=ab⟩ | φ(a) = a, φ(b) = b |
| 9 | 1614 | ⟨a, b | aab=aa, baa=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 1634 | ⟨a, b | aab=ba, aba=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 1637 | ⟨a, b | aab=ba, abb=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 1638 | ⟨a, b | aab=ba, baa=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 1667 | ⟨a, b | aba=ab, abb=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 1668 | ⟨a, b | aba=ab, baa=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2036 | ⟨a, b | aab=a, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2038 | ⟨a, b | aab=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2040 | ⟨a, b | aab=a, baba=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2098 | ⟨a, b | aba=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2974 | ⟨a, b | ab=a, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2978 | ⟨a, b | ab=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2980 | ⟨a, b | ab=a, babba=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 3109 | ⟨a, b | ab=a, baab=ab⟩ | φ(a) = a, φ(b) = b |
| 9 | 3113 | ⟨a, b | ab=a, baba=ab⟩ | φ(a) = a, φ(b) = b |
| 9 | 3181 | ⟨a, b | ab=a, baa=abb⟩ | φ(a) = a, φ(b) = b |
| 10 | 5112 | ⟨a, b | aab=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5114 | ⟨a, b | aab=aa, baba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5139 | ⟨a, b | aab=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5160 | ⟨a, b | aab=ba, aaab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5162 | ⟨a, b | aab=ba, aaba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5166 | ⟨a, b | aab=ba, abaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5169 | ⟨a, b | aab=ba, abab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5171 | ⟨a, b | aab=ba, abba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5174 | ⟨a, b | aab=ba, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5236 | ⟨a, b | aba=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5253 | ⟨a, b | aba=ab, abab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5255 | ⟨a, b | aba=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 5258 | ⟨a, b | aba=ab, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8790 | ⟨a, b | ab=a, baabbb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8798 | ⟨a, b | ab=a, bababb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8802 | ⟨a, b | ab=a, babbab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8804 | ⟨a, b | ab=a, babbba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 9061 | ⟨a, b | ab=a, baabb=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9069 | ⟨a, b | ab=a, babab=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9073 | ⟨a, b | ab=a, babba=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 9316 | ⟨a, b | ab=a, abbb=baa⟩ | φ(a) = a, φ(b) = b |
| 10 | 9331 | ⟨a, b | ab=a, baab=abb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9339 | ⟨a, b | ab=a, baba=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 12226 | ⟨a, b | aaab=aa, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12228 | ⟨a, b | aaab=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12230 | ⟨a, b | aaab=aa, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12338 | ⟨a, b | aaba=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12458 | ⟨a, b | aabb=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12471 | ⟨a, b | aabb=ba, abab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12473 | ⟨a, b | aabb=ba, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12475 | ⟨a, b | aabb=ba, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12519 | ⟨a, b | abab=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12532 | ⟨a, b | abab=ba, abba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12533 | ⟨a, b | abab=ba, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12535 | ⟨a, b | abab=ba, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12536 | ⟨a, b | abab=ba, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12538 | ⟨a, b | abab=ba, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12540 | ⟨a, b | abab=ba, babb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12541 | ⟨a, b | abab=ba, bbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12543 | ⟨a, b | abab=ba, bbab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12569 | ⟨a, b | abba=ab, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12570 | ⟨a, b | abba=ab, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12572 | ⟨a, b | abba=ab, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12574 | ⟨a, b | abba=ab, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12589 | ⟨a, b | abba=ba, baba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12630 | ⟨a, b | baba=ab, bbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14441 | ⟨a, b | aaab=a, baaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14707 | ⟨a, b | aabb=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14767 | ⟨a, b | abab=a, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14771 | ⟨a, b | abab=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14773 | ⟨a, b | abab=a, babba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14825 | ⟨a, b | abba=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 14840 | ⟨a, b | abba=b, aaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14844 | ⟨a, b | abba=b, aabab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14846 | ⟨a, b | abba=b, aabba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14850 | ⟨a, b | abba=b, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14852 | ⟨a, b | abba=b, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14862 | ⟨a, b | abba=b, baaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15583 | ⟨a, b | aab=aa, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15589 | ⟨a, b | aab=aa, babba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15622 | ⟨a, b | aab=ab, aabba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15630 | ⟨a, b | aab=ab, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15675 | ⟨a, b | aab=ba, aaaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15677 | ⟨a, b | aab=ba, aaaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15680 | ⟨a, b | aab=ba, aaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15681 | ⟨a, b | aab=ba, aabaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15689 | ⟨a, b | aab=ba, abaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15692 | ⟨a, b | aab=ba, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15694 | ⟨a, b | aab=ba, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15698 | ⟨a, b | aab=ba, abbaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15705 | ⟨a, b | aab=ba, baaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15833 | ⟨a, b | aba=aa, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15860 | ⟨a, b | aba=ab, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15862 | ⟨a, b | aba=ab, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15866 | ⟨a, b | aba=ab, abbaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15873 | ⟨a, b | aba=ab, baaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18898 | ⟨a, b | aab=a, baaaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18900 | ⟨a, b | aab=a, baaaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18902 | ⟨a, b | aab=a, baaabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18904 | ⟨a, b | aab=a, baabaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18906 | ⟨a, b | aab=a, baabab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18908 | ⟨a, b | aab=a, baabba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18912 | ⟨a, b | aab=a, babaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18914 | ⟨a, b | aab=a, babaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18916 | ⟨a, b | aab=a, bababa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 18920 | ⟨a, b | aab=a, babbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 19140 | ⟨a, b | aba=a, baaaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 19144 | ⟨a, b | aba=a, baabab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 19390 | ⟨a, b | aab=a, aaabb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19398 | ⟨a, b | aab=a, aabab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19402 | ⟨a, b | aab=a, aabba=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19406 | ⟨a, b | aab=a, aabbb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19414 | ⟨a, b | aab=a, abaab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19418 | ⟨a, b | aab=a, ababa=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19422 | ⟨a, b | aab=a, ababb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19426 | ⟨a, b | aab=a, abbaa=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19430 | ⟨a, b | aab=a, abbab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19434 | ⟨a, b | aab=a, abbba=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19440 | ⟨a, b | aab=a, baaaa=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19444 | ⟨a, b | aab=a, baaab=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19445 | ⟨a, b | aab=a, baaab=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19448 | ⟨a, b | aab=a, baaba=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19449 | ⟨a, b | aab=a, baaba=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19453 | ⟨a, b | aab=a, baabb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19456 | ⟨a, b | aab=a, babaa=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19457 | ⟨a, b | aab=a, babaa=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19461 | ⟨a, b | aab=a, babab=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19465 | ⟨a, b | aab=a, babba=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19652 | ⟨a, b | aba=a, aabab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19664 | ⟨a, b | aba=a, abaab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19675 | ⟨a, b | aba=a, abbab=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19932 | ⟨a, b | aab=a, aaab=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19940 | ⟨a, b | aab=a, aaba=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19948 | ⟨a, b | aab=a, aabb=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19949 | ⟨a, b | aab=a, aabb=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19956 | ⟨a, b | aab=a, abaa=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19964 | ⟨a, b | aab=a, abab=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19965 | ⟨a, b | aab=a, abab=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19972 | ⟨a, b | aab=a, abba=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19973 | ⟨a, b | aab=a, abba=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19981 | ⟨a, b | aab=a, abbb=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19984 | ⟨a, b | aab=a, baaa=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 19985 | ⟨a, b | aab=a, baaa=aab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19986 | ⟨a, b | aab=a, baaa=aba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19993 | ⟨a, b | aab=a, baab=aab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19994 | ⟨a, b | aab=a, baab=aba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19995 | ⟨a, b | aab=a, baab=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 20001 | ⟨a, b | aab=a, baba=aab⟩ | φ(a) = a, φ(b) = b |
| 11 | 20002 | ⟨a, b | aab=a, baba=aba⟩ | φ(a) = a, φ(b) = b |
| 11 | 20003 | ⟨a, b | aab=a, baba=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 20011 | ⟨a, b | aab=a, babb=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 20210 | ⟨a, b | aba=a, abab=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 20218 | ⟨a, b | aba=a, abba=bab⟩ | φ(a) = a, φ(b) = b |
| 11 | 20230 | ⟨a, b | aba=a, baab=aba⟩ | φ(a) = a, φ(b) = b |
| 11 | 24490 | ⟨a, b | ab=a, baabbbb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24506 | ⟨a, b | ab=a, bababbb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24514 | ⟨a, b | ab=a, babbabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24518 | ⟨a, b | ab=a, babbbab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24520 | ⟨a, b | ab=a, babbbba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 25017 | ⟨a, b | ab=a, baabbb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25033 | ⟨a, b | ab=a, bababb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25041 | ⟨a, b | ab=a, babbab=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25045 | ⟨a, b | ab=a, babbba=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25528 | ⟨a, b | ab=a, abbbb=baa⟩ | φ(a) = a, φ(b) = b |
| 11 | 25559 | ⟨a, b | ab=a, baabb=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25575 | ⟨a, b | ab=a, babab=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25583 | ⟨a, b | ab=a, babba=abb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25835 | ⟨a, b | ab=a, baab=abbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25844 | ⟨a, b | ab=a, baba=abbb⟩ | φ(a) = a, φ(b) = b |