#63 ⟨a, b | ab=a, ba=b⟩

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Properties

Element profile

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. a2 ⇒ a
  2. ab ⇒ a
  3. ba ⇒ b
  4. b2 ⇒ b 
# ab:ab=a,ba=b ab
aa=a
ab=a
ba=b
bb=b

Cayley table

Idempotents are shown in bold.

1ab
11ab
aaaa
bbbb

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 2129 total

Σ#PresentationDescriptionRelated
56⟨a, b | aa=b, ab=1⟩Isomorphic to ℤ32029 iso
659⟨a, b | aa=b, ab=a⟩Isomorphic to ℕ(3 = 1)61 iso
660⟨a, b | aa=b, ab=b⟩Isomorphic to ℕ(3 = 2)23 iso
8891⟨a, b | aa=a, abba=b⟩Finite commutative monoid with 3 elements12 iso

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

61 total

Σ#PresentationMapping
7270⟨a, b | ab=a, bab=b⟩φ(a) = a, φ(b) = b
7272⟨a, b | ab=a, bba=b⟩φ(a) = a, φ(b) = b
8943⟨a, b | ab=a, babb=b⟩φ(a) = a, φ(b) = b
8947⟨a, b | ab=a, bbab=b⟩φ(a) = a, φ(b) = b
8949⟨a, b | ab=a, bbba=b⟩φ(a) = a, φ(b) = b
92039⟨a, b | aab=a, baab=b⟩φ(a) = a, φ(b) = b
92041⟨a, b | aab=a, baba=b⟩φ(a) = a, φ(b) = b
92043⟨a, b | aab=a, babb=b⟩φ(a) = a, φ(b) = b
92045⟨a, b | aab=a, bbaa=b⟩φ(a) = a, φ(b) = b
92047⟨a, b | aab=a, bbab=b⟩φ(a) = a, φ(b) = b
92049⟨a, b | aab=a, bbba=b⟩φ(a) = a, φ(b) = b
92983⟨a, b | ab=a, babbb=b⟩φ(a) = a, φ(b) = b
92991⟨a, b | ab=a, bbabb=b⟩φ(a) = a, φ(b) = b
92995⟨a, b | ab=a, bbbab=b⟩φ(a) = a, φ(b) = b
92997⟨a, b | ab=a, bbbba=b⟩φ(a) = a, φ(b) = b
108807⟨a, b | ab=a, babbbb=b⟩φ(a) = a, φ(b) = b
108823⟨a, b | ab=a, bbabbb=b⟩φ(a) = a, φ(b) = b
108831⟨a, b | ab=a, bbbabb=b⟩φ(a) = a, φ(b) = b
108835⟨a, b | ab=a, bbbbab=b⟩φ(a) = a, φ(b) = b
108837⟨a, b | ab=a, bbbbba=b⟩φ(a) = a, φ(b) = b
1114444⟨a, b | aaab=a, baaab=b⟩φ(a) = a, φ(b) = b
1114446⟨a, b | aaab=a, baaba=b⟩φ(a) = a, φ(b) = b
1114450⟨a, b | aaab=a, babaa=b⟩φ(a) = a, φ(b) = b
1114458⟨a, b | aaab=a, bbaaa=b⟩φ(a) = a, φ(b) = b
1114574⟨a, b | aaba=a, baaba=b⟩φ(a) = a, φ(b) = b
1114578⟨a, b | aaba=a, babaa=b⟩φ(a) = a, φ(b) = b
1114586⟨a, b | aaba=a, bbaaa=b⟩φ(a) = a, φ(b) = b
1114712⟨a, b | aabb=a, babbb=b⟩φ(a) = a, φ(b) = b
1114720⟨a, b | aabb=a, bbabb=b⟩φ(a) = a, φ(b) = b
1114724⟨a, b | aabb=a, bbbab=b⟩φ(a) = a, φ(b) = b
1114726⟨a, b | aabb=a, bbbba=b⟩φ(a) = a, φ(b) = b
1114776⟨a, b | abab=a, babbb=b⟩φ(a) = a, φ(b) = b
1114784⟨a, b | abab=a, bbabb=b⟩φ(a) = a, φ(b) = b
1114788⟨a, b | abab=a, bbbab=b⟩φ(a) = a, φ(b) = b
1114790⟨a, b | abab=a, bbbba=b⟩φ(a) = a, φ(b) = b
1118903⟨a, b | aab=a, baaabb=b⟩φ(a) = a, φ(b) = b
1118907⟨a, b | aab=a, baabab=b⟩φ(a) = a, φ(b) = b
1118909⟨a, b | aab=a, baabba=b⟩φ(a) = a, φ(b) = b
1118911⟨a, b | aab=a, baabbb=b⟩φ(a) = a, φ(b) = b
1118915⟨a, b | aab=a, babaab=b⟩φ(a) = a, φ(b) = b
1118917⟨a, b | aab=a, bababa=b⟩φ(a) = a, φ(b) = b
1118919⟨a, b | aab=a, bababb=b⟩φ(a) = a, φ(b) = b
1118921⟨a, b | aab=a, babbaa=b⟩φ(a) = a, φ(b) = b
1118923⟨a, b | aab=a, babbab=b⟩φ(a) = a, φ(b) = b
1118925⟨a, b | aab=a, babbba=b⟩φ(a) = a, φ(b) = b
1118931⟨a, b | aab=a, bbaaab=b⟩φ(a) = a, φ(b) = b
1118933⟨a, b | aab=a, bbaaba=b⟩φ(a) = a, φ(b) = b
1118935⟨a, b | aab=a, bbaabb=b⟩φ(a) = a, φ(b) = b
1118937⟨a, b | aab=a, bbabaa=b⟩φ(a) = a, φ(b) = b
1118939⟨a, b | aab=a, bbabab=b⟩φ(a) = a, φ(b) = b
1118941⟨a, b | aab=a, bbabba=b⟩φ(a) = a, φ(b) = b
1118945⟨a, b | aab=a, bbbaaa=b⟩φ(a) = a, φ(b) = b
1118947⟨a, b | aab=a, bbbaab=b⟩φ(a) = a, φ(b) = b
1118949⟨a, b | aab=a, bbbaba=b⟩φ(a) = a, φ(b) = b
1118953⟨a, b | aab=a, bbbbaa=b⟩φ(a) = a, φ(b) = b
1124523⟨a, b | ab=a, babbbbb=b⟩φ(a) = a, φ(b) = b
1124555⟨a, b | ab=a, bbabbbb=b⟩φ(a) = a, φ(b) = b
1124571⟨a, b | ab=a, bbbabbb=b⟩φ(a) = a, φ(b) = b
1124579⟨a, b | ab=a, bbbbabb=b⟩φ(a) = a, φ(b) = b
1124583⟨a, b | ab=a, bbbbbab=b⟩φ(a) = a, φ(b) = b
1124585⟨a, b | ab=a, bbbbbba=b⟩φ(a) = a, φ(b) = b

Other anti-isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

17 total

Σ#PresentationMapping
92091⟨a, b | aba=a, aabb=b⟩φ(a) = a, φ(b) = b
92093⟨a, b | aba=a, abab=b⟩φ(a) = a, φ(b) = b
92097⟨a, b | aba=a, abbb=b⟩φ(a) = a, φ(b) = b
1114544⟨a, b | aaba=a, aaabb=b⟩φ(a) = a, φ(b) = b
1114548⟨a, b | aaba=a, aabab=b⟩φ(a) = a, φ(b) = b
1114556⟨a, b | aaba=a, abaab=b⟩φ(a) = a, φ(b) = b
1114820⟨a, b | abba=a, abbbb=b⟩φ(a) = a, φ(b) = b
1119103⟨a, b | aba=a, aaabbb=b⟩φ(a) = a, φ(b) = b
1119109⟨a, b | aba=a, aababb=b⟩φ(a) = a, φ(b) = b
1119113⟨a, b | aba=a, aabbab=b⟩φ(a) = a, φ(b) = b
1119117⟨a, b | aba=a, aabbbb=b⟩φ(a) = a, φ(b) = b
1119123⟨a, b | aba=a, abaabb=b⟩φ(a) = a, φ(b) = b
1119125⟨a, b | aba=a, ababab=b⟩φ(a) = a, φ(b) = b
1119129⟨a, b | aba=a, ababbb=b⟩φ(a) = a, φ(b) = b
1119131⟨a, b | aba=a, abbaab=b⟩φ(a) = a, φ(b) = b
1119133⟨a, b | aba=a, abbabb=b⟩φ(a) = a, φ(b) = b
1119135⟨a, b | aba=a, abbbab=b⟩φ(a) = a, φ(b) = b