#577 ⟨a, b | aaa=b, abb=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. a7 ⇒ a
  2. b ⇒ a3 
# ab:aaa=b,abb=a a/b
aaaaaaa=a
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6
11aa2a3a4a5a6
aaa2a3a4a5a6a
a2a2a3a4a5a6aa2
a3a3a4a5a6aa2a3
a4a4a5a6aa2a3a4
a5a5a6aa2a3a4a5
a6a6aa2a3a4a5a6

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1164 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92074⟨a, b | aab=b, babb=a⟩Finite commutative monoid with 7 elements20 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
104162⟨a, b | abb=aab, bbb=a⟩Isomorphic to ℕ(7 = 5)42 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
108987⟨a, b | ab=a, aaaaa=bb⟩Finite commutative monoid with 7 elements5 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Other isomorphic instances

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

59 total

Σ#PresentationMapping
8579⟨a, b | aaa=b, bab=a⟩φ(a) = a, φ(b) = aaa
8912⟨a, b | aa=b, abbb=a⟩φ(a) = a, φ(b) = aa
8916⟨a, b | aa=b, babb=a⟩φ(a) = a, φ(b) = aa
92902⟨a, b | aa=b, aaabb=a⟩φ(a) = a, φ(b) = aa
92906⟨a, b | aa=b, aabab=a⟩φ(a) = a, φ(b) = aa
92908⟨a, b | aa=b, aabba=a⟩φ(a) = a, φ(b) = aa
92912⟨a, b | aa=b, abaab=a⟩φ(a) = a, φ(b) = aa
92914⟨a, b | aa=b, ababa=a⟩φ(a) = a, φ(b) = aa
92924⟨a, b | aa=b, baaab=a⟩φ(a) = a, φ(b) = aa
104632⟨a, b | aaaa=b, aaab=a⟩φ(a) = a, φ(b) = aaaa
104634⟨a, b | aaaa=b, aaba=a⟩φ(a) = a, φ(b) = aaaa
104677⟨a, b | aaab=b, aabb=a⟩φ(a) = aaaa, φ(b) = a
104681⟨a, b | aaab=b, abab=a⟩φ(a) = aaaa, φ(b) = a
104716⟨a, b | aaba=b, aabb=a⟩φ(a) = aaaa, φ(b) = a
104719⟨a, b | aaba=b, abab=a⟩φ(a) = aaaa, φ(b) = a
104721⟨a, b | aaba=b, abba=a⟩φ(a) = aaaa, φ(b) = a
104723⟨a, b | aaba=b, baab=a⟩φ(a) = aaaa, φ(b) = a
104725⟨a, b | aaba=b, baba=a⟩φ(a) = aaaa, φ(b) = a
104728⟨a, b | aaba=b, bbaa=a⟩φ(a) = aaaa, φ(b) = a
106280⟨a, b | aaa=b, aaaab=a⟩φ(a) = a, φ(b) = aaa
106282⟨a, b | aaa=b, aaaba=a⟩φ(a) = a, φ(b) = aaa
106286⟨a, b | aaa=b, aabaa=a⟩φ(a) = a, φ(b) = aaa
106319⟨a, b | aab=a, aaaaa=b⟩φ(a) = aaaaa, φ(b) = a
106396⟨a, b | aab=b, aabbb=a⟩φ(a) = aaa, φ(b) = a
106404⟨a, b | aab=b, ababb=a⟩φ(a) = aaa, φ(b) = a
106408⟨a, b | aab=b, abbab=a⟩φ(a) = aaa, φ(b) = a
106420⟨a, b | aab=b, baabb=a⟩φ(a) = aaa, φ(b) = a
106432⟨a, b | aab=b, bbaab=a⟩φ(a) = aaa, φ(b) = a
106447⟨a, b | aba=a, aaaaa=b⟩φ(a) = aaaaa, φ(b) = a
106500⟨a, b | aba=b, aabbb=a⟩φ(a) = aaa, φ(b) = a
106506⟨a, b | aba=b, ababb=a⟩φ(a) = aaa, φ(b) = a
106508⟨a, b | aba=b, abbab=a⟩φ(a) = aaa, φ(b) = a
106510⟨a, b | aba=b, abbba=a⟩φ(a) = aaa, φ(b) = a
106516⟨a, b | aba=b, baabb=a⟩φ(a) = aaa, φ(b) = a
106518⟨a, b | aba=b, babab=a⟩φ(a) = aaa, φ(b) = a
108642⟨a, b | aa=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
108644⟨a, b | aa=b, aaaaba=a⟩φ(a) = a, φ(b) = aa
108648⟨a, b | aa=b, aaabaa=a⟩φ(a) = a, φ(b) = aa
108713⟨a, b | ab=a, aaaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1114479⟨a, b | aaab=b, aaabb=a⟩φ(a) = aa, φ(b) = a
1114491⟨a, b | aaab=b, abaab=a⟩φ(a) = aa, φ(b) = a
1114507⟨a, b | aaab=b, baaab=a⟩φ(a) = aa, φ(b) = a
1114607⟨a, b | aaba=b, aaabb=a⟩φ(a) = aa, φ(b) = a
1114611⟨a, b | aaba=b, aabab=a⟩φ(a) = aa, φ(b) = a
1114613⟨a, b | aaba=b, aabba=a⟩φ(a) = aa, φ(b) = a
1114619⟨a, b | aaba=b, abaab=a⟩φ(a) = aa, φ(b) = a
1114621⟨a, b | aaba=b, ababa=a⟩φ(a) = aa, φ(b) = a
1114625⟨a, b | aaba=b, abbaa=a⟩φ(a) = aa, φ(b) = a
1114635⟨a, b | aaba=b, baaab=a⟩φ(a) = aa, φ(b) = a
1114637⟨a, b | aaba=b, baaba=a⟩φ(a) = aa, φ(b) = a
1114641⟨a, b | aaba=b, babaa=a⟩φ(a) = aa, φ(b) = a
1114649⟨a, b | aaba=b, bbaaa=a⟩φ(a) = aa, φ(b) = a
1124188⟨a, b | aa=b, aaaaaaa=a⟩φ(a) = a, φ(b) = aa
1124335⟨a, b | ab=a, aaaaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1124337⟨a, b | ab=a, aaaaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1124341⟨a, b | ab=a, aaaabaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124349⟨a, b | ab=a, aaabaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124365⟨a, b | ab=a, aabaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124397⟨a, b | ab=a, abaaaaa=b⟩φ(a) = a, φ(b) = aaaaaa