#366 ⟨a, b, c | aba=b, cbc=1⟩

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Contents

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

Properties

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. bc ⇒ cb [3]
2. c2b ⇒ 1 [5]
3. c2a ⇒ bac4 [11]
4. b2a ⇒ ab2 [4]
5. aba ⇒ b [1]
# abc:aba=b,cbc=1 cb/a - -
bc=cb
ccb=1
cca=bacccc
bba=abb
aba=b

Other submonoids of same group

10 unique, 24 total

Σ#PresentationPropertiesφ
7548⟨a, b, c | bb=aa, cc=a⟩Can Inf1
7938⟨a, b, c | aa=b, cbc=b⟩Can Inf5
82849⟨a, b, c | aaa=b, cac=b⟩Can Inf2
82851⟨a, b, c | aaa=b, cbc=a⟩Can Inf3
83041⟨a, b, c | aba=b, cac=b⟩Can Inf
83042⟨a, b, c | aba=b, cbc=a⟩Can Inf1
83472⟨a, b, c | ba=ac, cab=a⟩Can Inf
83501⟨a, b, c | bb=ac, aaa=c⟩Can Inf2
83594⟨a, b, c | ba=ac, cb=aa⟩Can Inf
86095⟨a, b, c | ab=c, bcc=ca⟩Can Inf

Isomorphic instances

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

93 total

Σ#PresentationMapping
7486⟨a, b, c | bb=aa, cac=1⟩φ(a) = cbcb, φ(b) = cbcba, φ(c) = c
7608⟨a, b, c | aab=1, bbcc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7616⟨a, b, c | aab=1, bccb=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7632⟨a, b, c | aab=1, cbbc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7669⟨a, b, c | aba=1, bbcc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7672⟨a, b, c | aba=1, bccb=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7677⟨a, b, c | aba=1, cbbc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = cbcba
7767⟨a, b, c | aa=b, bccb=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
7772⟨a, b, c | aa=b, cbbc=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
81729⟨a, b, c | aab=ca, bac=1⟩φ(a) = c, φ(b) = cbcbac, φ(c) = cccbcba
81785⟨a, b, c | aab=cc, bcc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcbacc, φ(c) = c
81791⟨a, b, c | aab=cc, cbc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcbacc, φ(c) = c
82078⟨a, b, c | aaab=1, bcac=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82099⟨a, b, c | aaab=1, cbca=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82123⟨a, b, c | aaba=1, acbc=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82137⟨a, b, c | aaba=1, bcac=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82149⟨a, b, c | aaba=1, cacb=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82155⟨a, b, c | aaba=1, cbca=1⟩φ(a) = cb, φ(b) = ccc, φ(c) = ccbcba
82328⟨a, b, c | aaa=b, accb=1⟩φ(a) = c, φ(b) = ccc, φ(c) = cbcba
82343⟨a, b, c | aaa=b, cabc=1⟩φ(a) = c, φ(b) = ccc, φ(c) = cbcba
82499⟨a, b, c | aab=c, acba=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = cccbcba
82507⟨a, b, c | aab=c, baac=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = cccbcba
82519⟨a, b, c | aab=c, bbbc=1⟩φ(a) = cbcba, φ(b) = baccccccbcba, φ(c) = cbcbaccbcba
82521⟨a, b, c | aab=c, bbcb=1⟩φ(a) = cbcba, φ(b) = baccccccbcba, φ(c) = cbcbaccbcba
82527⟨a, b, c | aab=c, bcbb=1⟩φ(a) = cbcba, φ(b) = baccccccbcba, φ(c) = cbcbaccbcba
82531⟨a, b, c | aab=c, bccc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcbac, φ(c) = c
82541⟨a, b, c | aab=c, cbaa=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = cccbcba
82549⟨a, b, c | aab=c, cbcc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcbac, φ(c) = c
82555⟨a, b, c | aab=c, ccbc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcbac, φ(c) = c
82607⟨a, b, c | aba=b, acbc=1⟩φ(a) = cccbcba, φ(b) = cbcba, φ(c) = bacccccb
82624⟨a, b, c | aba=b, cabc=1⟩φ(a) = cccbcba, φ(b) = bacccc, φ(c) = cb
83212⟨a, b, c | ba=ac, abca=1⟩φ(a) = c, φ(b) = cccbcbacb, φ(c) = ccbcba
83237⟨a, b, c | ba=ac, caab=1⟩φ(a) = c, φ(b) = cccbcbacb, φ(c) = ccbcba
83706⟨a, b, c | aab=1, acbca=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83758⟨a, b, c | aab=1, bcaac=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83782⟨a, b, c | aab=1, bcccc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83828⟨a, b, c | aab=1, cbcaa=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83836⟨a, b, c | aab=1, cbccc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83854⟨a, b, c | aab=1, ccbcc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83860⟨a, b, c | aab=1, cccbc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83879⟨a, b, c | aba=1, aacbc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83912⟨a, b, c | aba=1, acbca=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83949⟨a, b, c | aba=1, bcaac=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cccbcba
83960⟨a, b, c | aba=1, bcccc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83975⟨a, b, c | aba=1, cbccc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
83977⟨a, b, c | aba=1, ccbcc=1⟩φ(a) = bacccc, φ(b) = cbcbacbcba, φ(c) = c
84212⟨a, b, c | aab=1, acca=b⟩φ(a) = c, φ(b) = cbcb, φ(c) = cbcba
84264⟨a, b, c | aab=1, bcac=a⟩φ(a) = cb, φ(b) = cc, φ(c) = ccbcba
84340⟨a, b, c | aab=1, ccaa=b⟩φ(a) = c, φ(b) = cbcb, φ(c) = cbcba
84383⟨a, b, c | aba=1, aacc=b⟩φ(a) = c, φ(b) = cbcb, φ(c) = cbcba
84416⟨a, b, c | aba=1, acca=b⟩φ(a) = c, φ(b) = cbcb, φ(c) = cbcba
84451⟨a, b, c | aba=1, bcac=a⟩φ(a) = cb, φ(b) = cc, φ(c) = ccbcba
84479⟨a, b, c | aba=1, cccc=b⟩φ(a) = cbcba, φ(b) = baccccbacccc, φ(c) = c
84552⟨a, b, c | abc=1, baaa=c⟩φ(a) = c, φ(b) = cbcba, φ(c) = bacccccb
84575⟨a, b, c | abc=1, bbba=c⟩φ(a) = ccbcbacb, φ(b) = c, φ(c) = bacccccb
84674⟨a, b, c | aa=b, aaccb=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
84691⟨a, b, c | aa=b, abcca=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
84708⟨a, b, c | aa=b, accab=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
84755⟨a, b, c | aa=b, caabc=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
84757⟨a, b, c | aa=b, cabac=1⟩φ(a) = c, φ(b) = cc, φ(c) = cbcba
84768⟨a, b, c | aa=b, cbccc=1⟩φ(a) = cbcba, φ(b) = cbcbacbcba, φ(c) = c
84770⟨a, b, c | aa=b, ccbcc=1⟩φ(a) = cbcba, φ(b) = cbcbacbcba, φ(c) = c
85003⟨a, b, c | ab=c, aacba=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = ccbcba
85039⟨a, b, c | ab=c, acbaa=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = ccbcba
85053⟨a, b, c | ab=c, baaac=1⟩φ(a) = c, φ(b) = cbcba, φ(c) = ccbcba
85074⟨a, b, c | ab=c, bbbac=1⟩φ(a) = ccbcbacb, φ(b) = c, φ(c) = ccbcba
86289⟨a, b, c | ab=1, aaacca=1⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
86324⟨a, b, c | ab=1, aacbca=1⟩φ(a) = c, φ(b) = cb, φ(c) = ccbcba
86327⟨a, b, c | ab=1, aaccaa=1⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
86426⟨a, b, c | ab=1, acbbca=1⟩φ(a) = cb, φ(b) = c, φ(c) = cccbcba
86428⟨a, b, c | ab=1, acbcaa=1⟩φ(a) = c, φ(b) = cb, φ(c) = ccbcba
86435⟨a, b, c | ab=1, accaaa=1⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
86451⟨a, b, c | ab=1, acccca=1⟩φ(a) = cbcba, φ(b) = bacccc, φ(c) = c
86518⟨a, b, c | ab=1, bbbbcc=1⟩φ(a) = cb, φ(b) = c, φ(c) = cbcba
86520⟨a, b, c | ab=1, bbbcac=1⟩φ(a) = cb, φ(b) = c, φ(c) = ccbcba
86524⟨a, b, c | ab=1, bbcaac=1⟩φ(a) = cb, φ(b) = c, φ(c) = cccbcba
86536⟨a, b, c | ab=1, bbcccc=1⟩φ(a) = bacccc, φ(b) = cbcba, φ(c) = c
86537⟨a, b, c | ab=1, bcaaac=1⟩φ(a) = c, φ(b) = cb, φ(c) = ccbcba
86570⟨a, b, c | ab=1, caaaac=1⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
86578⟨a, b, c | ab=1, caaccc=1⟩φ(a) = cbcba, φ(b) = bacccc, φ(c) = c
86604⟨a, b, c | ab=1, cbbccc=1⟩φ(a) = bacccc, φ(b) = cbcba, φ(c) = c
86609⟨a, b, c | ab=1, ccaacc=1⟩φ(a) = cbcba, φ(b) = bacccc, φ(c) = c
86835⟨a, b, c | ab=1, aacca=b⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
86935⟨a, b, c | ab=1, acbca=b⟩φ(a) = c, φ(b) = cb, φ(c) = ccbcba
86941⟨a, b, c | ab=1, accaa=b⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
87026⟨a, b, c | ab=1, bbbcc=a⟩φ(a) = cb, φ(b) = c, φ(c) = cbcba
87031⟨a, b, c | ab=1, bbcac=a⟩φ(a) = cb, φ(b) = c, φ(c) = ccbcba
87043⟨a, b, c | ab=1, bcaac=a⟩φ(a) = cb, φ(b) = c, φ(c) = cccbcba
87072⟨a, b, c | ab=1, bcccc=a⟩φ(a) = bacccc, φ(b) = cbcba, φ(c) = c
87438⟨a, b, c | ab=1, acca=bb⟩φ(a) = c, φ(b) = cb, φ(c) = cbcba
87532⟨a, b, c | ab=1, bbcc=aa⟩φ(a) = cb, φ(b) = c, φ(c) = cbcba
87541⟨a, b, c | ab=1, bcac=aa⟩φ(a) = cb, φ(b) = c, φ(c) = ccbcba
87797⟨a, b, c | ab=1, bcc=aaa⟩φ(a) = cb, φ(b) = c, φ(c) = cbcba