#2633 ⟨a, b | abbbba=babb⟩

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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. bcb2 ⇒ cbc [7]
2. bab2 ⇒ c [2]
3. babc ⇒ cab2 [4]
4. cb2a ⇒ bc [6]
5. cbca ⇒ b2c [8]
6. ab3c ⇒ cb2 [5]
7. cab4 ⇒ bacbc [10]
8. ab(bc)2 ⇒ cb4 [11]
9. (bc)3 ⇒ cbc2b2 [9]
10. cb4a ⇒ ab4c [16]
11. ab4a ⇒ c [3]
12. ab5c ⇒ cb3ca [12]
13. ab2c2bc ⇒ cb6 [15]
14. c2b3ca ⇒ bac(bc)2 [22]
15. cbc2b4 ⇒ b(cbc)2 [14]
16. abcbc2b2 ⇒ cb5c [20]
17. bacbc2b2 ⇒ c2b3c [13]
18. cbc2b3c ⇒ (bc2)2b2 [18]
19. ab4cbc ⇒ cb3cab2 [21]
20. cb6a ⇒ abcbc2 [26]
21. cb5ca ⇒ acbc2b2 [17]
22. cb8 ⇒ ab2c3bc [25]
23. cb3cab2a ⇒ ab6c [39]
24. cb2(bca)2 ⇒ ab7c [23]
25. ab(cbc)2bc ⇒ cb5c2b2 [35]
26. ba(cbc)2bc ⇒ c2b3c2b2 [19]
27. cb7ca ⇒ ab2c2b3c [27]
28. cb7cbc ⇒ ab2c3bc2b2 [28]
29. ba(cbc)3 ⇒ (c2b3)2b [29]
30. ab6acbc ⇒ cb3ca2b4 [24]
31. cb5c2b4 ⇒ ab(cbc)3 [36]
32. ab4(c2b)2b ⇒ (cb3c)2 [31]
33. ba(cbc2)2b2 ⇒ (c2b3)2c [42]
34. cb5c2b3c ⇒ ab(cbc2)2b2 [32]
35. c2b3c2b6 ⇒ bacb(c2bc)2 [30]
36. cb3ca2b6 ⇒ ab6ac2bc [47]
37. ab6(cbc)2 ⇒ cb3cabc2b4 [33]
38. (cb3ca)2a ⇒ ab6acb3c [48]
39. cb7c2b4 ⇒ ab2c2b2(cbc)2 [34]
40. ab6(c2b)2b ⇒ cb3cabc2b3c [37]
41. cb7c2b3c ⇒ a(b2c2)2bc2b2 [38]
42. ab4(c2b)2cbc ⇒ c(b3c2)2b2 [41]
43. ba(cbc2)2(bc)2 ⇒ c2(b3c2)2b2 [54]
44. cb3cabc2b6 ⇒ ab6cbc3bc [40]
45. ab6a(c2b)2b ⇒ (cb3ca)2bc [49]
46. c2b3c2b5cbc ⇒ bacbc(cbc2)2b2 [45]
47. ab4(c2b)3c ⇒ c(b3c2)2b4 [44]
48. (c2b3)3b ⇒ bacbc(c2b)3c [63]
49. ab4c(cbc2)2b2 ⇒ (cb3c)3 [52]
50. (c2b3)3c ⇒ ba(cbc2)3b2 [64]
51. ab6(c2b)2cbc ⇒ cb3ca(bc2b2)2 [43]
52. cb3cabc2b5cbc ⇒ ab6(cbc2)2b2 [56]
53. ab6a(c2b)2cbc ⇒ (cb3ca)2bc2b2 [55]
54. ab6acb2(cbc)2 ⇒ c(b3cac)2b4 [50]
55. cb3cab(c2b3)2b ⇒ ab6(c2b)3c [46]
56. ab6acb(bc2)2b2 ⇒ cb3c(acb3c)2 [51]
57. cb3cab(c2b3)2c ⇒ ab6c(cbc2)2b2 [53]
58. ab4c(cbc2)2(bc)2 ⇒ c(b3c2)3b2 [62]
59. (cb3ca)3 ⇒ ab6acb2c2(bc)2 [58]
60. c(b3cac)2b6 ⇒ ab6acb2cbc3bc [59]
61. (cb3ca)2bc2b4 ⇒ ab6a(c2b)3c [57]
62. (cb3ca)2bc2b3c ⇒ ab6ac(cbc2)2b2 [61]
63. ab6acb(bc2)2(bc)2 ⇒ cb3(cacb3)2c2b2 [66]
64. c(b3cac)2b5cbc ⇒ ab6acb2(cbc2)2b2 [65]
65. cb3(cacb3)2c2b4 ⇒ ab6acb2(c2b)3c [67]
66. (cb3ca)2(cb3c)2 ⇒ ab6acb2c(cbc2)2b2 [60]
# ab:abbbba=babb reversed:bca babb=c magic:0
bcbb=cbc
babb=c
babc=cabb
cbba=bc
cbca=bbc
abbbc=cbb
cabbbb=bacbc
abbcbc=cbbbb
bcbcbc=cbccbb
cbbbba=abbbbc
abbbba=c
abbbbbc=cbbbca
abbccbc=cbbbbbb
ccbbbca=bacbcbc
cbccbbbb=bcbccbc
abcbccbb=cbbbbbc
bacbccbb=ccbbbc
cbccbbbc=bccbccbb
abbbbcbc=cbbbcabb
cbbbbbba=abcbcc
cbbbbbca=acbccbb
cbbbbbbbb=abbcccbc
cbbbcabba=abbbbbbc
cbbbcabca=abbbbbbbc
abcbccbcbc=cbbbbbccbb
bacbccbcbc=ccbbbccbb
cbbbbbbbca=abbccbbbc
cbbbbbbbcbc=abbcccbccbb
bacbccbccbc=ccbbbccbbbb
abbbbbbacbc=cbbbcaabbbb
cbbbbbccbbbb=abcbccbccbc
abbbbccbccbb=cbbbccbbbc
bacbcccbccbb=ccbbbccbbbc
cbbbbbccbbbc=abcbcccbccbb
ccbbbccbbbbbb=bacbccbcccbc
cbbbcaabbbbbb=abbbbbbaccbc
abbbbbbcbccbc=cbbbcabccbbbb
cbbbcacbbbcaa=abbbbbbacbbbc
cbbbbbbbccbbbb=abbccbbcbccbc
abbbbbbccbccbb=cbbbcabccbbbc
cbbbbbbbccbbbc=abbccbbccbccbb
abbbbccbccbcbc=cbbbccbbbccbb
bacbcccbccbcbc=ccbbbccbbbccbb
cbbbcabccbbbbbb=abbbbbbcbcccbc
abbbbbbaccbccbb=cbbbcacbbbcabc
ccbbbccbbbbbcbc=bacbccbcccbccbb
abbbbccbccbccbc=cbbbccbbbccbbbb
ccbbbccbbbccbbbb=bacbcccbccbccbc
abbbbccbcccbccbb=cbbbccbbbccbbbc
ccbbbccbbbccbbbc=bacbcccbcccbccbb
abbbbbbccbccbcbc=cbbbcabccbbbccbb
cbbbcabccbbbbbcbc=abbbbbbcbcccbccbb
abbbbbbaccbccbcbc=cbbbcacbbbcabccbb
abbbbbbacbbcbccbc=cbbbcacbbbcacbbbb
cbbbcabccbbbccbbbb=abbbbbbccbccbccbc
abbbbbbacbbccbccbb=cbbbcacbbbcacbbbc
cbbbcabccbbbccbbbc=abbbbbbccbcccbccbb
abbbbccbcccbccbcbc=cbbbccbbbccbbbccbb
cbbbcacbbbcacbbbca=abbbbbbacbbccbcbc
cbbbcacbbbcacbbbbbb=abbbbbbacbbcbcccbc
cbbbcacbbbcabccbbbb=abbbbbbaccbccbccbc
cbbbcacbbbcabccbbbc=abbbbbbaccbcccbccbb
abbbbbbacbbccbccbcbc=cbbbcacbbbcacbbbccbb
cbbbcacbbbcacbbbbbcbc=abbbbbbacbbcbcccbccbb
cbbbcacbbbcacbbbccbbbb=abbbbbbacbbccbccbccbc
cbbbcacbbbcacbbbccbbbc=abbbbbbacbbccbcccbccbb

Other submonoids of same group

11 unique, 55 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
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

7 unique, 7 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
101666⟨a, b | aabaaaaba=b⟩Infinite cancellative non-commutative monoid
113536⟨a, b | aabaaaaaba=b⟩Infinite cancellative non-commutative monoid