#1431 ⟨a, b | aababbabba=1⟩

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Quick links

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Isomorphic instances

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. cd ⇒ 1 [20]
2. dc ⇒ 1 [14]
3. bc ⇒ cb [5]
4. bd ⇒ db [23]
5. b2 ⇒ c [2]
6. ca3 ⇒ a(ab)2 [28]
7. da2ba ⇒ a3db [25]
8. ba(ca)2 ⇒ ab(ac)2 [35]
9. (ba)2ca ⇒ (ca)3d [32]
10. cba3 ⇒ ba(ab)2 [29]
11. dba2ba ⇒ ba3db [27]
12. ba3ba ⇒ (ac)2a2d [34]
13. ba2baca ⇒ (ca)3ad [40]
14. a3baca ⇒ d [13]
15. da2(ca)3 ⇒ a4cac [43]
16. ba(ac)2a2 ⇒ (ca)3da2b [45]
17. dba2(ca)3 ⇒ ba4cac [44]
18. da3caca2 ⇒ (a3b)2 [46]
19. ba3caca2 ⇒ (ca)3ada2b [48]
20. ba3(ca)3 ⇒ (ac)2a2db(ac)2 [50]
21. c(ca)3ada2 ⇒ ba2(bac)2a2db [52]
22. a4caca2 ⇒ da2b [42]
# ab:aababbabba=1 cdb/a bb=c,abbaaaba=d magic:1
cd=1
dc=1
bc=cb
bd=db
bb=c
caaa=aabab
daaba=aaadb
bacaca=abacac
babaca=cacacad
cbaaa=baabab
dbaaba=baaadb
baaaba=acacaad
baabaca=cacacaad
aaabaca=d
daacacaca=aaaacac
baacacaa=cacacadaab
dbaacacaca=baaaacac
daaacacaa=aaabaaab
baaacacaa=cacacaadaab
baaacacaca=acacaadbacac
ccacacaadaa=baabacbacaadb
aaaacacaa=daab

Other submonoids of same group

2 unique, 2 total

Σ#PresentationDescriptionRelated
9932⟨a, b | aabaaba=bb⟩Infinite cancellative non-commutative monoid
114602⟨a, b | aabaaaba=bab⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

6 total

Σ#PresentationMapping
101432⟨a, b | aababbbaab=1⟩φ(a) = dbaabacaca, φ(b) = aabacacabab
101453⟨a, b | aabbabbaba=1⟩φ(a) = abaabacacab, φ(b) = adbaabacac
101462⟨a, b | aabbbabaab=1⟩φ(a) = aabacacab, φ(b) = a
101497⟨a, b | abaababbba=1⟩φ(a) = dbaabacaca, φ(b) = abaabacacab
101503⟨a, b | abaabbbaba=1⟩φ(a) = aabacacab, φ(b) = a
101538⟨a, b | abbabbaaab=1⟩φ(a) = a, φ(b) = aabacacab