#1426 ⟨a, b | aababbaaab=1⟩

Up:Monoids with two generators and one relation
Prev:#1425 ⟨a, b | aabababbba=1⟩
Next:#1427 ⟨a, b | aababbaaba=1⟩

Quick links

  1. Properties
  2. Rewriting system
  3. Anti-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. bab2 ⇒ c [2]
2. babc ⇒ cab2 [4]
3. b2a3b ⇒ aca [22]
4. ba2ca ⇒ ca3b [7]
5. (ba)2ca ⇒ cba3b [25]
6. aba2c ⇒ ca3b [32]
7. b2a3c ⇒ aca2b2 [23]
8. (aba)2b ⇒ ca3 [35]
9. aca3b ⇒ ca3ba [19]
10. (aba)2c ⇒ ca4b2 [37]
11. ba(a2b)2 ⇒ a2ca2 [10]
12. ca3ba2 ⇒ 1 [28]
13. aca3c ⇒ b2 [18]
14. b2a4ca ⇒ acaba3b [24]
15. ca2ca3 ⇒ bab [29]
16. ba2c2a3 ⇒ ca3b2a(ab)2 [36]
17. (ba)2c2a3 ⇒ cba3b2a(ab)2 [40]
18. aba2bca3 ⇒ ca5bab [39]
19. aca4ca ⇒ ca3baba3b [30]
20. ba3bca3c ⇒ a2ca4cb2 [21]
21. ba3bca3ba ⇒ a2ca4c [17]
22. ba3baca3 ⇒ a2ca4bab [34]
23. b2a5ca2 ⇒ aca4ba2b [26]
24. (ba3)2ca ⇒ a2ca2ba3b [12]
25. ca3bca3c ⇒ ba2cb2 [20]
26. (ca3b)2a ⇒ ba2c [16]
27. ca3baca3 ⇒ ba(ab)2 [33]
28. ca5ca2 ⇒ aba2b [31]
29. b2a4c2a3 ⇒ acaba3b2a(ab)2 [44]
30. aba2ba3ca2 ⇒ ca6ba2b [38]
31. ba3ba4ca2 ⇒ a2ca5ba2b [15]
32. aca4c2a3 ⇒ ca3baba3b2a(ab)2 [46]
33. ba3(bca3)2 ⇒ a2ca4cba(ab)2 [47]
34. b2a5c2a3c ⇒ aca4ba2b2a2cb2 [54]
35. b2a5c2a3ba ⇒ aca4ba2b2a2c [48]
36. b2a5caca3 ⇒ aca4(ba2b)2ab [49]
37. (ba3)2c2a3 ⇒ a2ca2ba3b2a(ab)2 [50]
38. ca3(bca3)2 ⇒ ba2cba(ab)2 [43]
39. ca5c2a3c ⇒ aba2b2a2cb2 [45]
40. ca5c2a3ba ⇒ aba2b2a2c [41]
41. ca5caca3 ⇒ a(ba2b)2ab [42]
42. aba2ba3c2a3c ⇒ ca6ba2b2a2cb2 [55]
43. aba2ba3c2a3ba ⇒ ca6ba2b2a2c [51]
44. a(ba2)2(ac)2a3 ⇒ ca6(ba2b)2ab [52]
45. ba3ba4c2a3c ⇒ a2ca5ba2b2a2cb2 [60]
46. ba3ba4c2a3ba ⇒ a2ca5ba2b2a2c [56]
47. (ba3)2(ac)2a3 ⇒ a2ca5(ba2b)2ab [57]
48. b2a5c2a3bca3 ⇒ aca4ba2b2a2cba(ab)2 [58]
49. ca5c2a3bca3 ⇒ aba2b2a2cba(ab)2 [53]
50. aba2ba3c2a3bca3 ⇒ ca6ba2b2a2cba(ab)2 [59]
51. ba3ba4c2a3bca3 ⇒ a2ca5ba2b2a2cba(ab)2 [61]
# ab:aababbaaab=1 b/ca babb=c morph:4/2
babb=c
babc=cabb
bbaaab=aca
baaca=caaab
babaca=cbaaab
abaac=caaab
bbaaac=acaabb
abaabab=caaa
acaaab=caaaba
abaabac=caaaabb
baaabaab=aacaa
caaabaa=1
acaaac=bb
bbaaaaca=acabaaab
caacaaa=bab
baaccaaa=caaabbaabab
babaccaaa=cbaaabbaabab
abaabcaaa=caaaaabab
acaaaaca=caaababaaab
baaabcaaac=aacaaaacbb
baaabcaaaba=aacaaaac
baaabacaaa=aacaaaabab
bbaaaaacaa=acaaaabaab
baaabaaaca=aacaabaaab
caaabcaaac=baacbb
caaabcaaaba=baac
caaabacaaa=baabab
caaaaacaa=abaab
bbaaaaccaaa=acabaaabbaabab
abaabaaacaa=caaaaaabaab
baaabaaaacaa=aacaaaaabaab
acaaaaccaaa=caaababaaabbaabab
baaabcaaabcaaa=aacaaaacbaabab
bbaaaaaccaaac=acaaaabaabbaacbb
bbaaaaaccaaaba=acaaaabaabbaac
bbaaaaacacaaa=acaaaabaabbaabab
baaabaaaccaaa=aacaabaaabbaabab
caaabcaaabcaaa=baacbaabab
caaaaaccaaac=abaabbaacbb
caaaaaccaaaba=abaabbaac
caaaaacacaaa=abaabbaabab
abaabaaaccaaac=caaaaaabaabbaacbb
abaabaaaccaaaba=caaaaaabaabbaac
abaabaaacacaaa=caaaaaabaabbaabab
baaabaaaaccaaac=aacaaaaabaabbaacbb
baaabaaaaccaaaba=aacaaaaabaabbaac
baaabaaaacacaaa=aacaaaaabaabbaabab
bbaaaaaccaaabcaaa=acaaaabaabbaacbaabab
caaaaaccaaabcaaa=abaabbaacbaabab
abaabaaaccaaabcaaa=caaaaaabaabbaacbaabab
baaabaaaaccaaabcaaa=aacaaaaabaabbaacbaabab

Anti-isomorphic instances

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

2 total

Σ#PresentationMapping
101488⟨a, b | abaaabbaba=1⟩φ(a) = a, φ(b) = b
101521⟨a, b | ababbabbba=1⟩φ(a) = b, φ(b) = a