#257 ⟨a, b | aa=b, bbb=a⟩

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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. a6 ⇒ a
  2. b ⇒ a2 
# ab:aa=b,bbb=a a/b
aaaaaa=a
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5
11aa2a3a4a5
aaa2a3a4a5a
a2a2a3a4a5aa2
a3a3a4a5aa2a3
a4a4a5aa2a3a4
a5a5aa2a3a4a5

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 1864 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-iso
7245⟨a, b | aa=a, bbb=a⟩Isomorphic to ℕ(6 = 3)49 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91427⟨a, b | abba=b, baba=1⟩Finite non-Abelian group with 6 elements66 iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Other isomorphic instances

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

61 total

Σ#PresentationMapping
8906⟨a, b | aa=b, aabb=a⟩φ(a) = bbb, φ(b) = b
8908⟨a, b | aa=b, abab=a⟩φ(a) = bbb, φ(b) = b
8910⟨a, b | aa=b, abba=a⟩φ(a) = bbb, φ(b) = b
8914⟨a, b | aa=b, baab=a⟩φ(a) = bbb, φ(b) = b
92002⟨a, b | aaa=b, aaab=a⟩φ(a) = bb, φ(b) = b
92004⟨a, b | aaa=b, aaba=a⟩φ(a) = bb, φ(b) = b
92021⟨a, b | aab=a, aaaa=b⟩φ(a) = bbbb, φ(b) = b
92085⟨a, b | aba=a, aaaa=b⟩φ(a) = bbbb, φ(b) = b
92898⟨a, b | aa=b, aaaab=a⟩φ(a) = bbb, φ(b) = b
92900⟨a, b | aa=b, aaaba=a⟩φ(a) = bbb, φ(b) = b
92904⟨a, b | aa=b, aabaa=a⟩φ(a) = bbb, φ(b) = b
92937⟨a, b | ab=a, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
108640⟨a, b | aa=b, aaaaaa=a⟩φ(a) = bbb, φ(b) = b
108715⟨a, b | ab=a, aaaaab=b⟩φ(a) = b, φ(b) = bbbbb
108717⟨a, b | ab=a, aaaaba=b⟩φ(a) = b, φ(b) = bbbbb
108721⟨a, b | ab=a, aaabaa=b⟩φ(a) = b, φ(b) = bbbbb
108729⟨a, b | ab=a, aabaaa=b⟩φ(a) = b, φ(b) = bbbbb
108745⟨a, b | ab=a, abaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1114440⟨a, b | aaab=a, abbbb=b⟩φ(a) = bb, φ(b) = b
1114672⟨a, b | aabb=a, aaabb=b⟩φ(a) = bbb, φ(b) = b
1114678⟨a, b | aabb=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114684⟨a, b | aabb=a, abaab=b⟩φ(a) = bbb, φ(b) = b
1114686⟨a, b | aabb=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114690⟨a, b | aabb=a, abbaa=b⟩φ(a) = bbb, φ(b) = b
1114736⟨a, b | abab=a, aaabb=b⟩φ(a) = bbb, φ(b) = b
1114740⟨a, b | abab=a, aabab=b⟩φ(a) = bbb, φ(b) = b
1114742⟨a, b | abab=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114748⟨a, b | abab=a, abaab=b⟩φ(a) = bbb, φ(b) = b
1114750⟨a, b | abab=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114754⟨a, b | abab=a, abbaa=b⟩φ(a) = bbb, φ(b) = b
1114806⟨a, b | abba=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114812⟨a, b | abba=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114847⟨a, b | abba=b, aabbb=a⟩φ(a) = bb, φ(b) = b
1114853⟨a, b | abba=b, ababb=a⟩φ(a) = bb, φ(b) = b
1114855⟨a, b | abba=b, abbab=a⟩φ(a) = bb, φ(b) = b
1114857⟨a, b | abba=b, abbba=a⟩φ(a) = bb, φ(b) = b
1114863⟨a, b | abba=b, baabb=a⟩φ(a) = bb, φ(b) = b
1114865⟨a, b | abba=b, babab=a⟩φ(a) = bb, φ(b) = b
1118760⟨a, b | aaa=b, aaaaaa=a⟩φ(a) = bb, φ(b) = b
1118835⟨a, b | aab=a, aaaaab=b⟩φ(a) = bbbb, φ(b) = b
1118837⟨a, b | aab=a, aaaaba=b⟩φ(a) = bbbb, φ(b) = b
1118841⟨a, b | aab=a, aaabaa=b⟩φ(a) = bbbb, φ(b) = b
1118849⟨a, b | aab=a, aabaaa=b⟩φ(a) = bbbb, φ(b) = b
1118865⟨a, b | aab=a, abaaaa=b⟩φ(a) = bbbb, φ(b) = b
1119093⟨a, b | aba=a, aaaaba=b⟩φ(a) = bbbb, φ(b) = b
1119097⟨a, b | aba=a, aaabaa=b⟩φ(a) = bbbb, φ(b) = b
1124339⟨a, b | ab=a, aaaaabb=b⟩φ(a) = b, φ(b) = bbbbb
1124343⟨a, b | ab=a, aaaabab=b⟩φ(a) = b, φ(b) = bbbbb
1124345⟨a, b | ab=a, aaaabba=b⟩φ(a) = b, φ(b) = bbbbb
1124351⟨a, b | ab=a, aaabaab=b⟩φ(a) = b, φ(b) = bbbbb
1124353⟨a, b | ab=a, aaababa=b⟩φ(a) = b, φ(b) = bbbbb
1124357⟨a, b | ab=a, aaabbaa=b⟩φ(a) = b, φ(b) = bbbbb
1124367⟨a, b | ab=a, aabaaab=b⟩φ(a) = b, φ(b) = bbbbb
1124369⟨a, b | ab=a, aabaaba=b⟩φ(a) = b, φ(b) = bbbbb
1124373⟨a, b | ab=a, aababaa=b⟩φ(a) = b, φ(b) = bbbbb
1124381⟨a, b | ab=a, aabbaaa=b⟩φ(a) = b, φ(b) = bbbbb
1124399⟨a, b | ab=a, abaaaab=b⟩φ(a) = b, φ(b) = bbbbb
1124401⟨a, b | ab=a, abaaaba=b⟩φ(a) = b, φ(b) = bbbbb
1124405⟨a, b | ab=a, abaabaa=b⟩φ(a) = b, φ(b) = bbbbb
1124413⟨a, b | ab=a, ababaaa=b⟩φ(a) = b, φ(b) = bbbbb
1124429⟨a, b | ab=a, abbaaaa=b⟩φ(a) = b, φ(b) = bbbbb