#1666 ⟨a, b | aabaaaaba=b⟩

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Quick links

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Other submonoids of anti-isomorphic group

Properties

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
#RuleProof
1. (a2b)2 ⇒ ba3ba [2]
2. a2ba4ba ⇒ b [1]
3. b(a3b)2 ⇒ a2b2a3ba [5]
4. a2ba4b2 ⇒ baba4ba [3]
5. ba5b2ab ⇒ a2b2(aba)2 [12]
6. a2ba6b2 ⇒ ba2ba5ba [6]
7. ba3ba5ba ⇒ a2b2 [4]
8. ba3ba5b2 ⇒ a2b2aba4ba [10]
9. a2b2a3ba4b ⇒ ba5b2a3ba [9]
10. ba2(a3b2)2 ⇒ a(ab2)2a4ba [13]
11. ba5b2a3ba2 ⇒ a2b2ab [17]
12. ba5b2a3bab ⇒ a2b(ba)3a3ba [18]
13. ba3ba7b2 ⇒ a2b2a2ba5ba [7]
14. b(a5b2)2 ⇒ a2b(ba)3a4ba [15]
15. a2b2a3ba6ba ⇒ ba5b2 [8]
16. (ba3ba)2a3b ⇒ (a2b2a3)2ba [16]
17. a2b2a3ba8b2 ⇒ ba5b2a2ba5ba [14]
18. (ba3ba)2a5ba ⇒ a2b2a5b2 [11]
19. (ba3ba)2a7b2 ⇒ a2b2a5b2a2ba5ba [19]
# ab:aabaaaaba=b reversed:ab
aabaab=baaaba
aabaaaaba=b
baaabaaab=aabbaaaba
aabaaaabb=babaaaaba
baaaaabbab=aabbabaaba
aabaaaaaabb=baabaaaaaba
baaabaaaaaba=aabb
baaabaaaaabb=aabbabaaaaba
aabbaaabaaaab=baaaaabbaaaba
baaaaabbaaabb=aabbabbaaaaba
baaaaabbaaabaa=aabbab
baaaaabbaaabab=aabbababaaaaba
baaabaaaaaaabb=aabbaabaaaaaba
baaaaabbaaaaabb=aabbababaaaaaba
aabbaaabaaaaaaba=baaaaabb
baaababaaabaaaab=aabbaaaaabbaaaba
aabbaaabaaaaaaaabb=baaaaabbaabaaaaaba
baaababaaabaaaaaaba=aabbaaaaabb
baaababaaabaaaaaaaabb=aabbaaaaabbaabaaaaaba

Other submonoids of same group

12 unique, 56 total

Σ#PresentationDescriptionRelated
6104⟨a, b | aaba=bb⟩Infinite cancellative non-commutative monoid
7145⟨a, b | aababba=1⟩Infinite non-Abelian group22 iso, 22 anti-iso
7193⟨a, b | ababba=b⟩Infinite cancellative non-commutative monoid
7252⟨a, b | aaba=bab⟩Infinite cancellative non-commutative monoid
9842⟨a, b | abaababa=b⟩Infinite cancellative non-commutative monoid
91126⟨a, b | ababba=bab⟩Infinite cancellative non-commutative monoid
91273⟨a, b | abbba=babb⟩Infinite cancellative non-commutative monoid
102633⟨a, b | abbbba=babb⟩Infinite cancellative non-commutative monoid
113716⟨a, b | abaaabaaba=b⟩Infinite cancellative non-commutative monoid
114876⟨a, b | abbabbba=bab⟩Infinite cancellative non-commutative monoid
115415⟨a, b | abbbbba=babb⟩Infinite cancellative non-commutative monoid
115898⟨a, b | ababba=babab⟩Infinite cancellative non-commutative monoid

Other submonoids of anti-isomorphic group

6 unique, 6 total

Σ#PresentationDescriptionRelated
687⟨a, b | aabba=b⟩Infinite cancellative non-commutative monoid
7179⟨a, b | aababa=b⟩Infinite cancellative non-commutative monoid
8374⟨a, b | aabaaba=b⟩Infinite cancellative non-commutative monoid
8586⟨a, b | baab=aaba⟩Infinite cancellative non-commutative monoid
9792⟨a, b | aabaaaba=b⟩Infinite cancellative non-commutative monoid
113536⟨a, b | aabaaaaaba=b⟩Infinite cancellative non-commutative monoid