#279 ⟨a, b, c | aaa=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. ba ⇒ ab [4]
2. bc ⇒ cb [3]
3. a3 ⇒ b [1]
4. c2a ⇒ ac2 [8]
5. c2b ⇒ 1 [5]
# abc:aaa=b,cbc=1 acb - -
ba=ab
bc=cb
aaa=b
cca=acc
ccb=1

Other submonoids of same group

12 unique, 27 total

Σ#PresentationPropertiesφ
7543⟨a, b, c | ba=ac, cb=a⟩Can Inf
7555⟨a, b, c | bb=ac, cb=a⟩Can Inf4
7558⟨a, b, c | bb=ac, cc=a⟩Can Inf2
7937⟨a, b, c | aa=b, cbc=a⟩Can Inf2
71042⟨a, b, c | ab=c, bcc=a⟩Can Inf2
83045⟨a, b, c | aba=c, abc=b⟩Can Inf2
83051⟨a, b, c | aba=c, bab=c⟩Can Inf
83071⟨a, b, c | abc=b, bca=c⟩Can Inf2
83595⟨a, b, c | ba=ac, cb=ac⟩Can Inf
85663⟨a, b, c | aa=b, aaa=cc⟩Can Inf1
86019⟨a, b, c | ab=c, aba=bc⟩Can Inf
86071⟨a, b, c | ab=c, bac=ca⟩Can Inf

Isomorphic instances

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

146 total

Σ#PresentationMapping
7328⟨a, b, c | aab=c, bac=1⟩φ(a) = aacc, φ(b) = aaac, φ(c) = ac
7331⟨a, b, c | aab=c, bbc=1⟩φ(a) = c, φ(b) = cbac, φ(c) = cac
7333⟨a, b, c | aab=c, bcb=1⟩φ(a) = c, φ(b) = cbac, φ(c) = cac
7334⟨a, b, c | aab=c, bcc=1⟩φ(a) = cb, φ(b) = cca, φ(c) = a
7338⟨a, b, c | aab=c, cba=1⟩φ(a) = aacc, φ(b) = aaac, φ(c) = ac
7340⟨a, b, c | aab=c, cbc=1⟩φ(a) = cb, φ(b) = cca, φ(c) = a
7479⟨a, b, c | ba=ac, cab=1⟩φ(a) = a, φ(b) = aacaacc, φ(c) = ac
7611⟨a, b, c | aab=1, bcac=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = ac
7617⟨a, b, c | aab=1, bccc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = a
7633⟨a, b, c | aab=1, cbca=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = ac
7635⟨a, b, c | aab=1, cbcc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = a
7641⟨a, b, c | aab=1, ccbc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = a
7658⟨a, b, c | aba=1, acbc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = ac
7670⟨a, b, c | aba=1, bcac=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = ac
7673⟨a, b, c | aba=1, bccc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = a
7678⟨a, b, c | aba=1, cbcc=1⟩φ(a) = cb, φ(b) = cc, φ(c) = a
7755⟨a, b, c | aa=b, accb=1⟩φ(a) = a, φ(b) = aa, φ(c) = c
7770⟨a, b, c | aa=b, cabc=1⟩φ(a) = a, φ(b) = aa, φ(c) = c
7773⟨a, b, c | aa=b, cbcc=1⟩φ(a) = c, φ(b) = cc, φ(c) = a
7859⟨a, b, c | ab=c, acba=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
7865⟨a, b, c | ab=c, baac=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
7871⟨a, b, c | ab=c, bbac=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = ac
71131⟨a, b, c | ab=1, aacca=1⟩φ(a) = a, φ(b) = aacc, φ(c) = c
71163⟨a, b, c | ab=1, acbca=1⟩φ(a) = a, φ(b) = aacc, φ(c) = ac
71165⟨a, b, c | ab=1, accaa=1⟩φ(a) = a, φ(b) = aacc, φ(c) = c
71169⟨a, b, c | ab=1, accca=1⟩φ(a) = c, φ(b) = cb, φ(c) = a
71193⟨a, b, c | ab=1, bbbcc=1⟩φ(a) = aacc, φ(b) = a, φ(c) = c
71195⟨a, b, c | ab=1, bbcac=1⟩φ(a) = aacc, φ(b) = a, φ(c) = ac
71198⟨a, b, c | ab=1, bbccc=1⟩φ(a) = cb, φ(b) = c, φ(c) = a
71199⟨a, b, c | ab=1, bcaac=1⟩φ(a) = a, φ(b) = aacc, φ(c) = ac
71210⟨a, b, c | ab=1, caaac=1⟩φ(a) = a, φ(b) = aacc, φ(c) = c
71212⟨a, b, c | ab=1, caacc=1⟩φ(a) = c, φ(b) = cb, φ(c) = a
71220⟨a, b, c | ab=1, cbbcc=1⟩φ(a) = cb, φ(b) = c, φ(c) = a
71325⟨a, b, c | ab=1, acca=b⟩φ(a) = a, φ(b) = aacc, φ(c) = c
71356⟨a, b, c | ab=1, bbcc=a⟩φ(a) = aacc, φ(b) = a, φ(c) = c
71359⟨a, b, c | ab=1, bcac=a⟩φ(a) = aacc, φ(b) = a, φ(c) = ac
71367⟨a, b, c | ab=1, bccc=a⟩φ(a) = cb, φ(b) = c, φ(c) = a
71521⟨a, b, c | ab=1, bcc=aa⟩φ(a) = aacc, φ(b) = a, φ(c) = c
81570⟨a, b, c | aaa=bb, aca=1⟩φ(a) = a, φ(b) = cb, φ(c) = aaccaacc
81571⟨a, b, c | aaa=bb, acb=1⟩φ(a) = a, φ(b) = cb, φ(c) = aaccc
81575⟨a, b, c | aaa=bb, bcb=1⟩φ(a) = a, φ(b) = cb, φ(c) = cc
81778⟨a, b, c | aab=cc, bab=1⟩φ(a) = cbaacccb, φ(b) = caccacb, φ(c) = cb
81855⟨a, b, c | aba=bb, bcb=1⟩φ(a) = ac, φ(b) = aacc, φ(c) = aa
81878⟨a, b, c | aba=bc, cba=1⟩φ(a) = ac, φ(b) = cb, φ(c) = caac
81880⟨a, b, c | aba=bc, cbc=1⟩φ(a) = ac, φ(b) = cbaacccb, φ(c) = caac
81894⟨a, b, c | aba=cc, bab=1⟩φ(a) = cbaacccb, φ(b) = caac, φ(c) = cb
81898⟨a, b, c | aba=cc, bcb=1⟩φ(a) = cbaacccbaacccb, φ(b) = caac, φ(c) = cbaacccb
82258⟨a, b, c | abac=1, acbc=1⟩φ(a) = cbaaccaacc, φ(b) = aaac, φ(c) = ac
82267⟨a, b, c | abac=1, bcba=1⟩φ(a) = cbaacc, φ(b) = aac, φ(c) = c
82274⟨a, b, c | abac=1, cbab=1⟩φ(a) = cbaacc, φ(b) = aac, φ(c) = c
82278⟨a, b, c | abac=1, cbca=1⟩φ(a) = cbaaccaacc, φ(b) = aaac, φ(c) = ac
82649⟨a, b, c | aba=c, acbc=1⟩φ(a) = c, φ(b) = cba, φ(c) = ac
82664⟨a, b, c | aba=c, bcac=1⟩φ(a) = c, φ(b) = cba, φ(c) = ac
83239⟨a, b, c | ba=ac, cacb=1⟩φ(a) = cb, φ(b) = cbacc, φ(c) = ac
83669⟨a, b, c | aab=1, aacca=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83681⟨a, b, c | aab=1, abbcc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83689⟨a, b, c | aab=1, abccb=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83709⟨a, b, c | aab=1, accaa=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83715⟨a, b, c | aab=1, accca=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83727⟨a, b, c | aab=1, babcc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83735⟨a, b, c | aab=1, baccb=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83741⟨a, b, c | aab=1, bbacc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83753⟨a, b, c | aab=1, bbcca=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83773⟨a, b, c | aab=1, bcbcc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83775⟨a, b, c | aab=1, bccab=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83777⟨a, b, c | aab=1, bccba=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83779⟨a, b, c | aab=1, bccbc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83785⟨a, b, c | aab=1, caaac=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83791⟨a, b, c | aab=1, caacc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83797⟨a, b, c | aab=1, cabbc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83815⟨a, b, c | aab=1, cbabc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83821⟨a, b, c | aab=1, cbbac=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83833⟨a, b, c | aab=1, cbcbc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83835⟨a, b, c | aab=1, cbccb=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83837⟨a, b, c | aab=1, ccaaa=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83839⟨a, b, c | aab=1, ccaac=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83849⟨a, b, c | aab=1, ccbba=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83855⟨a, b, c | aab=1, cccaa=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83868⟨a, b, c | aba=1, aaacc=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83880⟨a, b, c | aba=1, aacca=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83882⟨a, b, c | aba=1, aaccc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83892⟨a, b, c | aba=1, abbcc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83899⟨a, b, c | aba=1, abccb=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83917⟨a, b, c | aba=1, accbb=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83919⟨a, b, c | aba=1, accca=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83929⟨a, b, c | aba=1, babcc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83934⟨a, b, c | aba=1, baccb=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83939⟨a, b, c | aba=1, bbacc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83956⟨a, b, c | aba=1, bcbcc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83958⟨a, b, c | aba=1, bccbc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83961⟨a, b, c | aba=1, caaac=1⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = c
83963⟨a, b, c | aba=1, caacc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = a
83965⟨a, b, c | aba=1, cabbc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83970⟨a, b, c | aba=1, cbabc=1⟩φ(a) = aacc, φ(b) = aa, φ(c) = c
83974⟨a, b, c | aba=1, cbcbc=1⟩φ(a) = c, φ(b) = cbcb, φ(c) = cac
83991⟨a, b, c | abc=1, aabba=1⟩φ(a) = a, φ(b) = c, φ(c) = cbaacc
84003⟨a, b, c | abc=1, aacca=1⟩φ(a) = cbac, φ(b) = cbaacc, φ(c) = c
84013⟨a, b, c | abc=1, abbaa=1⟩φ(a) = a, φ(b) = c, φ(c) = cbaacc
84016⟨a, b, c | abc=1, abbba=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = cbaacc
84042⟨a, b, c | abc=1, accaa=1⟩φ(a) = cbac, φ(b) = cbaacc, φ(c) = c
84046⟨a, b, c | abc=1, accca=1⟩φ(a) = c, φ(b) = cbaacc, φ(c) = a
84049⟨a, b, c | abc=1, baaab=1⟩φ(a) = a, φ(b) = c, φ(c) = cbaacc
84051⟨a, b, c | abc=1, baabb=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = cbaacc
84065⟨a, b, c | abc=1, bbaaa=1⟩φ(a) = a, φ(b) = c, φ(c) = cbaacc
84066⟨a, b, c | abc=1, bbaab=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = cbaacc
84070⟨a, b, c | abc=1, bbbaa=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = cbaacc
84103⟨a, b, c | abc=1, ccaaa=1⟩φ(a) = cbac, φ(b) = cbaacc, φ(c) = c
84327⟨a, b, c | aab=1, cbbc=a⟩φ(a) = aacc, φ(b) = aa, φ(c) = aaccc
84463⟨a, b, c | aba=1, caac=a⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = ac
84467⟨a, b, c | aba=1, cabc=b⟩φ(a) = a, φ(b) = aaccaacc, φ(c) = ac
84472⟨a, b, c | aba=1, cbbc=a⟩φ(a) = aacc, φ(b) = aa, φ(c) = aaccc
84588⟨a, b, c | abc=1, bcba=c⟩φ(a) = a, φ(b) = ac, φ(c) = cbaaccaacc
84673⟨a, b, c | aa=b, aacca=1⟩φ(a) = a, φ(b) = aa, φ(c) = c
84704⟨a, b, c | aa=b, acbbc=1⟩φ(a) = a, φ(b) = aa, φ(c) = aaccc
84712⟨a, b, c | aa=b, accca=1⟩φ(a) = c, φ(b) = cc, φ(c) = a
84744⟨a, b, c | aa=b, bcacb=1⟩φ(a) = a, φ(b) = aa, φ(c) = aaccc
84751⟨a, b, c | aa=b, bccbc=1⟩φ(a) = cb, φ(b) = cbcb, φ(c) = cac
84754⟨a, b, c | aa=b, caaac=1⟩φ(a) = a, φ(b) = aa, φ(c) = c
84756⟨a, b, c | aa=b, caacc=1⟩φ(a) = c, φ(b) = cc, φ(c) = a
84767⟨a, b, c | aa=b, cbcbc=1⟩φ(a) = cb, φ(b) = cbcb, φ(c) = cac
84788⟨a, b, c | ab=a, aacca=1⟩φ(a) = a, φ(b) = 1, φ(c) = c
84828⟨a, b, c | ab=a, accaa=1⟩φ(a) = a, φ(b) = 1, φ(c) = c
84834⟨a, b, c | ab=a, accca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84904⟨a, b, c | ab=a, caaac=1⟩φ(a) = a, φ(b) = 1, φ(c) = c
84910⟨a, b, c | ab=a, caacc=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84958⟨a, b, c | ab=a, ccaac=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84995⟨a, b, c | ab=c, aabba=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
85006⟨a, b, c | ab=c, aacca=1⟩φ(a) = a, φ(b) = aaccc, φ(c) = c
85017⟨a, b, c | ab=c, abbaa=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
85019⟨a, b, c | ab=c, abbba=1⟩φ(a) = acaacc, φ(b) = a, φ(c) = ac
85040⟨a, b, c | ab=c, acbac=1⟩φ(a) = cb, φ(b) = cac, φ(c) = ac
85045⟨a, b, c | ab=c, accaa=1⟩φ(a) = a, φ(b) = aaccc, φ(c) = c
85049⟨a, b, c | ab=c, accca=1⟩φ(a) = c, φ(b) = cba, φ(c) = a
85064⟨a, b, c | ab=c, bacac=1⟩φ(a) = cb, φ(b) = cac, φ(c) = ac
85066⟨a, b, c | ab=c, bacbc=1⟩φ(a) = acc, φ(b) = cb, φ(c) = ac
85077⟨a, b, c | ab=c, bbbcc=1⟩φ(a) = caacc, φ(b) = a, φ(c) = c
85082⟨a, b, c | ab=c, bbccc=1⟩φ(a) = acb, φ(b) = c, φ(c) = a
85087⟨a, b, c | ab=c, bcbac=1⟩φ(a) = acc, φ(b) = cb, φ(c) = ac
85094⟨a, b, c | ab=c, caaac=1⟩φ(a) = a, φ(b) = aaccc, φ(c) = c
85096⟨a, b, c | ab=c, caacc=1⟩φ(a) = c, φ(b) = cba, φ(c) = a
85104⟨a, b, c | ab=c, cbbcc=1⟩φ(a) = acb, φ(b) = c, φ(c) = a
86429⟨a, b, c | ab=1, acbcac=1⟩φ(a) = cb, φ(b) = c, φ(c) = ac
86545⟨a, b, c | ab=1, bcacac=1⟩φ(a) = cb, φ(b) = c, φ(c) = ac
86546⟨a, b, c | ab=1, bcacbc=1⟩φ(a) = c, φ(b) = cb, φ(c) = ac
86556⟨a, b, c | ab=1, bcbcac=1⟩φ(a) = c, φ(b) = cb, φ(c) = ac
87107⟨a, b, c | ab=1, cbcac=a⟩φ(a) = c, φ(b) = cb, φ(c) = ac