| Up: | Monoid enumeration |
|---|---|
| Prev: | #62 ⟨a, b | ab=a, ba=a⟩ |
| Next: | #64 ⟨a, b | aa=1, aaaa=1⟩ |
| index 1, period 1 | 2 elements | a, b |
| 1 | a | b | |
|---|---|---|---|
| 1 | 1 | a | b |
| a | a | a | a |
| b | b | b | b |
Idempotents are shown in bold.
Idempotents are shown in bold.
| # | Rule | Proof |
|---|---|---|
| 1. | a2 ⇒ a | [3] |
| 2. | ab ⇒ a | [1] |
| 3. | ba ⇒ b | [2] |
| 4. | b2 ⇒ b | [4] |
# ab:ab=a,ba=b ab - - aa=a ab=a ba=b bb=b
| Σ | # | Presentation | Properties | Description | φ |
|---|---|---|---|---|---|
| 5 | 6 | ⟨a, b | aa=b, ab=1⟩ | Grp Com 3 | ℤ3 | 2029 |
| 6 | 59 | ⟨a, b | aa=b, ab=a⟩ | Noc Com 3 | ℕ(3 = 1) | 61 |
| 6 | 60 | ⟨a, b | aa=b, ab=b⟩ | Noc Com 3 | ℕ(3 = 2) | 23 |
| 8 | 891 | ⟨a, b | aa=a, abba=b⟩ | Noc Com 3 | 12 |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
61 total
| Σ | # | Presentation | Mapping |
|---|---|---|---|
| 7 | 270 | ⟨a, b | ab=a, bab=b⟩ | φ(a) = a, φ(b) = b |
| 7 | 272 | ⟨a, b | ab=a, bba=b⟩ | φ(a) = a, φ(b) = b |
| 8 | 943 | ⟨a, b | ab=a, babb=b⟩ | φ(a) = a, φ(b) = b |
| 8 | 947 | ⟨a, b | ab=a, bbab=b⟩ | φ(a) = a, φ(b) = b |
| 8 | 949 | ⟨a, b | ab=a, bbba=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2039 | ⟨a, b | aab=a, baab=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2041 | ⟨a, b | aab=a, baba=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2043 | ⟨a, b | aab=a, babb=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2045 | ⟨a, b | aab=a, bbaa=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2047 | ⟨a, b | aab=a, bbab=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2049 | ⟨a, b | aab=a, bbba=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2983 | ⟨a, b | ab=a, babbb=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2991 | ⟨a, b | ab=a, bbabb=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2995 | ⟨a, b | ab=a, bbbab=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2997 | ⟨a, b | ab=a, bbbba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8807 | ⟨a, b | ab=a, babbbb=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8823 | ⟨a, b | ab=a, bbabbb=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8831 | ⟨a, b | ab=a, bbbabb=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8835 | ⟨a, b | ab=a, bbbbab=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8837 | ⟨a, b | ab=a, bbbbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14444 | ⟨a, b | aaab=a, baaab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14446 | ⟨a, b | aaab=a, baaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14450 | ⟨a, b | aaab=a, babaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14458 | ⟨a, b | aaab=a, bbaaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14574 | ⟨a, b | aaba=a, baaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14578 | ⟨a, b | aaba=a, babaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14586 | ⟨a, b | aaba=a, bbaaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14712 | ⟨a, b | aabb=a, babbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14720 | ⟨a, b | aabb=a, bbabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14724 | ⟨a, b | aabb=a, bbbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14726 | ⟨a, b | aabb=a, bbbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14776 | ⟨a, b | abab=a, babbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14784 | ⟨a, b | abab=a, bbabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14788 | ⟨a, b | abab=a, bbbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 14790 | ⟨a, b | abab=a, bbbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18903 | ⟨a, b | aab=a, baaabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18907 | ⟨a, b | aab=a, baabab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18909 | ⟨a, b | aab=a, baabba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18911 | ⟨a, b | aab=a, baabbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18915 | ⟨a, b | aab=a, babaab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18917 | ⟨a, b | aab=a, bababa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18919 | ⟨a, b | aab=a, bababb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18921 | ⟨a, b | aab=a, babbaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18923 | ⟨a, b | aab=a, babbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18925 | ⟨a, b | aab=a, babbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18931 | ⟨a, b | aab=a, bbaaab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18933 | ⟨a, b | aab=a, bbaaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18935 | ⟨a, b | aab=a, bbaabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18937 | ⟨a, b | aab=a, bbabaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18939 | ⟨a, b | aab=a, bbabab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18941 | ⟨a, b | aab=a, bbabba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18945 | ⟨a, b | aab=a, bbbaaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18947 | ⟨a, b | aab=a, bbbaab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18949 | ⟨a, b | aab=a, bbbaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18953 | ⟨a, b | aab=a, bbbbaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24523 | ⟨a, b | ab=a, babbbbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24555 | ⟨a, b | ab=a, bbabbbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24571 | ⟨a, b | ab=a, bbbabbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24579 | ⟨a, b | ab=a, bbbbabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24583 | ⟨a, b | ab=a, bbbbbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24585 | ⟨a, b | ab=a, bbbbbba=b⟩ | φ(a) = a, φ(b) = b |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
17 total
| Σ | # | Presentation | Mapping |
|---|---|---|---|
| 9 | 2091 | ⟨a, b | aba=a, aabb=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2093 | ⟨a, b | aba=a, abab=b⟩ | φ(a) = b, φ(b) = a |
| 9 | 2097 | ⟨a, b | aba=a, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14544 | ⟨a, b | aaba=a, aaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14548 | ⟨a, b | aaba=a, aabab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14556 | ⟨a, b | aaba=a, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14820 | ⟨a, b | abba=a, abbbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19103 | ⟨a, b | aba=a, aaabbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19109 | ⟨a, b | aba=a, aababb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19113 | ⟨a, b | aba=a, aabbab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19117 | ⟨a, b | aba=a, aabbbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19123 | ⟨a, b | aba=a, abaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19125 | ⟨a, b | aba=a, ababab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19129 | ⟨a, b | aba=a, ababbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19131 | ⟨a, b | aba=a, abbaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19133 | ⟨a, b | aba=a, abbabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 19135 | ⟨a, b | aba=a, abbbab=b⟩ | φ(a) = b, φ(b) = a |