#18727 ⟨a, b | aaa=a, ababba=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. a3 ⇒ a
  2. a2b ⇒ b
  3. ba2 ⇒ b
  4. bab2 ⇒ aba
  5. b3a ⇒ (ab)3
  6. b(ba)2 ⇒ ab2ab
  7. (ba)3 ⇒ ab3
  8. b5 ⇒ b 
# ab:aaa=a,ababba=b reversed:a/b
aaa=a
aab=b
baa=b
babb=aba
bbba=ababab
bbaba=abbab
bababa=abbb
bbbbb=b

Cayley table

Idempotents are shown in bold.

1aba2abbab2abaab2babb2ab3(ab)2ab2aab3(ba)2b2abb4a(ba)2ab2abab4b(ab)2(ab)3
11aba2abbab2abaab2babb2ab3(ab)2ab2aab3(ba)2b2abb4a(ba)2ab2abab4b(ab)2(ab)3
aaa2abababaab2bab2(ab)2ab2aab3babb2ab3a(ba)2ab2abab4(ba)2b2abb4(ab)3b(ab)2
bbbab2bbabb2ab3(ba)2abab2ab(ab)3b4b(ab)2ab(ab)2ab2abab2abab3ab2baa(ba)2ab4
a2a2aba2abbab2abaab2babb2ab3(ab)2ab2aab3(ba)2b2abb4a(ba)2ab2abab4b(ab)2(ab)3
abababaab2ab(ab)2ab2aab3a(ba)2baab2abb(ab)2ab4(ab)3bbabb2abb2aabb3b2aba(ba)2b4
bababbabbab2(ba)2abab2ab3b(ab)2ab(ab)2b2ab(ab)3b4ab3ab2baab2abab2abab4a(ba)2
b2b2b2ab3b2b2ab(ab)3b4ab2ab(ba)2ab2aab4ba(ba)2babb(ab)2ab2abb2(ab)2abab2aab3ba
abaabaab(ab)2abaab2a(ba)2baab2aab3(ab)3bbabab2abb(ab)2ab4b3b2abab2abb2aabb4(ba)2
ab2ab2ab2aab3ab2ab2abb(ab)2ab4b2aba(ba)2b2ab4ab(ba)2(ab)2(ab)3b2bab2babbaab2ab3aba
babbab(ba)2abababb(ab)2ab(ab)2ab3b2aab2a(ba)2baab4b2b2abab2a(ab)3babb4b3(ba)2ab2abb
b2ab2ab2b2abb2ab3ab2ab(ba)2(ab)3b4a(ba)2babb(ab)2ab2aab4b(ab)2abab2aab2abb2baab3
b3b3(ab)3b4b3ab2aab4bab2ab2ababbab2ab3b2aba(ba)2abababb3b(ab)2(ba)2(ab)3(ab)2b2a
(ab)2(ab)2a(ba)2ba(ab)2(ab)3bbabb3ab2ab2(ba)2abab4ab2ab2abb2ab(ab)2(ab)2ab4ab3a(ba)2b2abab
ab2aab2aab2ab2abab2aab3b2aba(ba)2b(ab)2ab4(ba)2(ab)2(ab)3b2ab4abbabbaab2ab2bab2abab3
ab3ab3b(ab)2ab4ab3b2ab4abb2b2abbabaab2b3ab2ab(ba)2ba(ab)2ab3(ab)3a(ba)2b(ab)2babab2a
(ba)2(ba)2babb(ab)2(ba)2abaab3b2aab(ab)2ab4b2b2abab2a(ba)2bab4b3(ba)2ab2a(ab)3babbab2ab
b2abb2abab2ab(ba)2b2aba(ba)2babb(ab)2(ab)2(ab)3abaab3b2abab3ab2aabab4b2abbb4ab2abab2b2
b4b4ab4bb4abbab2abaab2babb2ab3(ab)2ab2aab3(ba)2b2abb4a(ba)2ab2abab4b(ab)2(ab)3
a(ba)2a(ba)2(ab)2(ab)3a(ba)2bab3ab2abbabb4ab2ab2abb2(ba)2abaab4ab3a(ba)2b2ab(ab)2(ab)2abb2ab
ab2abab2abb2aba(ba)2ab2ab(ba)2(ab)2(ab)3babb(ab)2bab3ab2aabaab3b2abb4ab2ababab4b2abb2ab2
ab4ab4b4abab4babaab2bab2(ab)2ab2aab3babb2ab3a(ba)2ab2abab4(ba)2b2abb4(ab)3b(ab)2
b(ab)2b(ab)2ab3b2ab(ab)2ab4b2b2abb4abb3ab2ab(ba)2babaab2(ab)3a(ba)2b(ab)2ba(ab)2ab3ab2abab
(ab)3(ab)3b3ab2a(ab)3b4ab2ab2abab4bab3b2aba(ba)2abbab2b(ab)2(ba)2(ab)3abababb3b2a(ab)2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

1 unique, 1 total

Σ#PresentationDescriptionRelated
1120300⟨a, b | aba=b, baab=aaa⟩Finite non-commutative monoid with 23 elements