#5 ⟨a, b, c | aa=b, ac=1⟩

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Contents

  1. Properties
  2. Rewriting system
  3. Isomorphic instances
  4. Anti-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. ac ⇒ 1 [2]
2. b ⇒ a2 [1]
# abc:aa=b,ac=1 ac/b - -
ac=1
b=aa

Isomorphic instances

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

129 total

Σ#PresentationMapping
510⟨a, b, c | ab=a, ca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
514⟨a, b, c | ab=c, ba=1⟩φ(a) = c, φ(b) = a, φ(c) = ca
523⟨a, b, c | ab=1, abc=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
638⟨a, b, c | ac=ab, ba=1⟩φ(a) = c, φ(b) = a, φ(c) = a
684⟨a, b, c | aa=a, abc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
6107⟨a, b, c | ab=a, bca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
6111⟨a, b, c | ab=a, cab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
6113⟨a, b, c | ab=a, cba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
6227⟨a, b, c | ab=1, aaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aaa
6228⟨a, b, c | ab=1, aab=c⟩φ(a) = a, φ(b) = c, φ(c) = a
6232⟨a, b, c | ab=1, aba=c⟩φ(a) = a, φ(b) = c, φ(c) = a
6233⟨a, b, c | ab=1, abc=a⟩φ(a) = a, φ(b) = c, φ(c) = a
6244⟨a, b, c | ab=1, baa=c⟩φ(a) = a, φ(b) = c, φ(c) = caa
7344⟨a, b, c | aba=a, abc=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
7348⟨a, b, c | aba=a, bac=1⟩φ(a) = c, φ(b) = a, φ(c) = 1
7394⟨a, b, c | abc=a, bca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7415⟨a, b, c | abc=b, bcc=1⟩φ(a) = a, φ(b) = aa, φ(c) = c
7685⟨a, b, c | abc=1, aabc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7695⟨a, b, c | abc=1, abca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7696⟨a, b, c | abc=1, abcb=1⟩φ(a) = a, φ(b) = 1, φ(c) = c
7715⟨a, b, c | abc=1, bcaa=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7725⟨a, b, c | aa=a, aabc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7726⟨a, b, c | aa=a, abac=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7728⟨a, b, c | aa=a, abca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
7801⟨a, b, c | ab=a, bbca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
7805⟨a, b, c | ab=a, bcab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
7807⟨a, b, c | ab=a, bcba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
7817⟨a, b, c | ab=a, cabb=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
7823⟨a, b, c | ab=a, cbab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
7825⟨a, b, c | ab=a, cbba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
71121⟨a, b, c | ab=1, aabbc=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
71135⟨a, b, c | ab=1, ababc=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
71145⟨a, b, c | ab=1, abcab=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
71275⟨a, b, c | ab=1, aaaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aaaa
71276⟨a, b, c | ab=1, aaab=c⟩φ(a) = a, φ(b) = c, φ(c) = aa
71280⟨a, b, c | ab=1, aaba=c⟩φ(a) = a, φ(b) = c, φ(c) = aa
71281⟨a, b, c | ab=1, aabb=c⟩φ(a) = a, φ(b) = c, φ(c) = 1
71294⟨a, b, c | ab=1, abaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aa
71295⟨a, b, c | ab=1, abab=c⟩φ(a) = a, φ(b) = c, φ(c) = 1
71299⟨a, b, c | ab=1, abba=c⟩φ(a) = a, φ(b) = c, φ(c) = ca
71301⟨a, b, c | ab=1, abbc=b⟩φ(a) = a, φ(b) = c, φ(c) = 1
71303⟨a, b, c | ab=1, abca=a⟩φ(a) = a, φ(b) = c, φ(c) = 1
71332⟨a, b, c | ab=1, baaa=c⟩φ(a) = a, φ(b) = c, φ(c) = caaa
71336⟨a, b, c | ab=1, baba=c⟩φ(a) = a, φ(b) = c, φ(c) = ca
71338⟨a, b, c | ab=1, babc=b⟩φ(a) = a, φ(b) = c, φ(c) = 1
71346⟨a, b, c | ab=1, bbaa=c⟩φ(a) = a, φ(b) = c, φ(c) = ccaa
71437⟨a, b, c | ab=1, aaa=ca⟩φ(a) = a, φ(b) = c, φ(c) = aa
71442⟨a, b, c | ab=1, aab=ca⟩φ(a) = a, φ(b) = c, φ(c) = 1
71456⟨a, b, c | ab=1, aba=ca⟩φ(a) = a, φ(b) = c, φ(c) = 1
71459⟨a, b, c | ab=1, abc=aa⟩φ(a) = a, φ(b) = c, φ(c) = aa
71460⟨a, b, c | ab=1, abc=ab⟩φ(a) = a, φ(b) = c, φ(c) = 1
71462⟨a, b, c | ab=1, abc=ba⟩φ(a) = a, φ(b) = c, φ(c) = ca
71463⟨a, b, c | ab=1, abc=bb⟩φ(a) = a, φ(b) = c, φ(c) = cc
71493⟨a, b, c | ab=1, baa=bc⟩φ(a) = a, φ(b) = c, φ(c) = aa
71494⟨a, b, c | ab=1, baa=ca⟩φ(a) = a, φ(b) = c, φ(c) = ca
71510⟨a, b, c | ab=1, bbc=bb⟩φ(a) = a, φ(b) = c, φ(c) = 1
71518⟨a, b, c | ab=1, bca=ba⟩φ(a) = a, φ(b) = c, φ(c) = 1
81545⟨a, b, c | aaa=aa, abc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
81601⟨a, b, c | aab=aa, bca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
81607⟨a, b, c | aab=aa, cba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
81614⟨a, b, c | aab=ab, abc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
81620⟨a, b, c | aab=ab, bca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
81722⟨a, b, c | aab=ca, abb=1⟩φ(a) = aa, φ(b) = c, φ(c) = a
81817⟨a, b, c | aba=ab, cab=1⟩φ(a) = 1, φ(b) = c, φ(c) = a
81819⟨a, b, c | aba=ab, cba=1⟩φ(a) = 1, φ(b) = c, φ(c) = a
81943⟨a, b, c | abc=ab, cab=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
81959⟨a, b, c | abc=ac, bca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
81962⟨a, b, c | abc=ac, cba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
82690⟨a, b, c | abc=a, abcb=1⟩φ(a) = a, φ(b) = c, φ(c) = a
82738⟨a, b, c | abc=a, cbab=1⟩φ(a) = a, φ(b) = c, φ(c) = a
82756⟨a, b, c | abc=b, aaab=1⟩φ(a) = a, φ(b) = ccc, φ(c) = c
83995⟨a, b, c | abc=1, aabcb=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
84010⟨a, b, c | abc=1, ababc=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
84022⟨a, b, c | abc=1, abcab=1⟩φ(a) = a, φ(b) = c, φ(c) = 1
84086⟨a, b, c | abc=1, bcbca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84607⟨a, b, c | aa=a, aaabc=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84608⟨a, b, c | aa=a, aabac=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84610⟨a, b, c | aa=a, aabca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84613⟨a, b, c | aa=a, abaac=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84615⟨a, b, c | aa=a, abaca=1⟩φ(a) = 1, φ(b) = a, φ(c) = c
84863⟨a, b, c | ab=a, bbbca=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84867⟨a, b, c | ab=a, bbcab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84869⟨a, b, c | ab=a, bbcba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84879⟨a, b, c | ab=a, bcabb=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84885⟨a, b, c | ab=a, bcbab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84887⟨a, b, c | ab=a, bcbba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84915⟨a, b, c | ab=a, cabbb=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84933⟨a, b, c | ab=a, cbabb=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84939⟨a, b, c | ab=a, cbbab=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
84941⟨a, b, c | ab=a, cbbba=1⟩φ(a) = c, φ(b) = 1, φ(c) = a
86780⟨a, b, c | ab=1, aaaaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aaaaa
86781⟨a, b, c | ab=1, aaaab=c⟩φ(a) = a, φ(b) = c, φ(c) = aaa
86785⟨a, b, c | ab=1, aaaba=c⟩φ(a) = a, φ(b) = c, φ(c) = aaa
86786⟨a, b, c | ab=1, aaabb=c⟩φ(a) = a, φ(b) = c, φ(c) = a
86799⟨a, b, c | ab=1, aabaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aaa
86800⟨a, b, c | ab=1, aabab=c⟩φ(a) = a, φ(b) = c, φ(c) = a
86804⟨a, b, c | ab=1, aabba=c⟩φ(a) = a, φ(b) = c, φ(c) = a
86805⟨a, b, c | ab=1, aabbc=a⟩φ(a) = a, φ(b) = c, φ(c) = a
86843⟨a, b, c | ab=1, abaaa=c⟩φ(a) = a, φ(b) = c, φ(c) = aaa
86844⟨a, b, c | ab=1, abaab=c⟩φ(a) = a, φ(b) = c, φ(c) = a
86848⟨a, b, c | ab=1, ababa=c⟩φ(a) = a, φ(b) = c, φ(c) = a
86849⟨a, b, c | ab=1, ababc=a⟩φ(a) = a, φ(b) = c, φ(c) = a
86861⟨a, b, c | ab=1, abbaa=c⟩φ(a) = a, φ(b) = c, φ(c) = caa
86865⟨a, b, c | ab=1, abbba=c⟩φ(a) = a, φ(b) = c, φ(c) = cca
86881⟨a, b, c | ab=1, abcab=a⟩φ(a) = a, φ(b) = c, φ(c) = a
86960⟨a, b, c | ab=1, baaaa=c⟩φ(a) = a, φ(b) = c, φ(c) = caaaa
86964⟨a, b, c | ab=1, baaba=c⟩φ(a) = a, φ(b) = c, φ(c) = caa
86974⟨a, b, c | ab=1, babaa=c⟩φ(a) = a, φ(b) = c, φ(c) = caa
87004⟨a, b, c | ab=1, bbaaa=c⟩φ(a) = a, φ(b) = c, φ(c) = ccaaa
87279⟨a, b, c | ab=1, aaaa=ca⟩φ(a) = a, φ(b) = c, φ(c) = aaa
87284⟨a, b, c | ab=1, aaab=ca⟩φ(a) = a, φ(b) = c, φ(c) = a
87298⟨a, b, c | ab=1, aaba=ca⟩φ(a) = a, φ(b) = c, φ(c) = a
87342⟨a, b, c | ab=1, abaa=ca⟩φ(a) = a, φ(b) = c, φ(c) = a
87358⟨a, b, c | ab=1, abba=bc⟩φ(a) = a, φ(b) = c, φ(c) = a
87365⟨a, b, c | ab=1, abbc=ba⟩φ(a) = a, φ(b) = c, φ(c) = a
87371⟨a, b, c | ab=1, abca=aa⟩φ(a) = a, φ(b) = c, φ(c) = a
87459⟨a, b, c | ab=1, baaa=bc⟩φ(a) = a, φ(b) = c, φ(c) = aaa
87460⟨a, b, c | ab=1, baaa=ca⟩φ(a) = a, φ(b) = c, φ(c) = caa
87473⟨a, b, c | ab=1, baba=bc⟩φ(a) = a, φ(b) = c, φ(c) = a
87478⟨a, b, c | ab=1, babc=ba⟩φ(a) = a, φ(b) = c, φ(c) = a
87503⟨a, b, c | ab=1, bbaa=bc⟩φ(a) = a, φ(b) = c, φ(c) = caa
87711⟨a, b, c | ab=1, abc=aaa⟩φ(a) = a, φ(b) = c, φ(c) = aaa
87712⟨a, b, c | ab=1, abc=aab⟩φ(a) = a, φ(b) = c, φ(c) = a
87714⟨a, b, c | ab=1, abc=aba⟩φ(a) = a, φ(b) = c, φ(c) = a
87737⟨a, b, c | ab=1, baa=abc⟩φ(a) = a, φ(b) = c, φ(c) = caa
87757⟨a, b, c | ab=1, bba=abc⟩φ(a) = a, φ(b) = c, φ(c) = cca
87762⟨a, b, c | ab=1, bbb=abc⟩φ(a) = a, φ(b) = c, φ(c) = ccc
87778⟨a, b, c | ab=1, bbc=bba⟩φ(a) = a, φ(b) = c, φ(c) = a
87788⟨a, b, c | ab=1, bca=baa⟩φ(a) = a, φ(b) = c, φ(c) = a

Anti-isomorphic instances

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

12 total

Σ#PresentationMapping
635⟨a, b, c | ab=aa, ca=1⟩φ(a) = a, φ(b) = a, φ(c) = c
6234⟨a, b, c | ab=1, abc=b⟩φ(a) = c, φ(b) = a, φ(c) = a
82760⟨a, b, c | abc=b, aabc=1⟩φ(a) = c, φ(b) = a, φ(c) = a
86806⟨a, b, c | ab=1, aabbc=b⟩φ(a) = c, φ(b) = a, φ(c) = a
86850⟨a, b, c | ab=1, ababc=b⟩φ(a) = c, φ(b) = a, φ(c) = a
87359⟨a, b, c | ab=1, abba=ca⟩φ(a) = c, φ(b) = a, φ(c) = a
87366⟨a, b, c | ab=1, abbc=bb⟩φ(a) = c, φ(b) = a, φ(c) = a
87374⟨a, b, c | ab=1, abca=ba⟩φ(a) = c, φ(b) = a, φ(c) = a
87479⟨a, b, c | ab=1, babc=bb⟩φ(a) = c, φ(b) = a, φ(c) = a
87715⟨a, b, c | ab=1, abc=abb⟩φ(a) = c, φ(b) = a, φ(c) = a
87742⟨a, b, c | ab=1, bab=abc⟩φ(a) = c, φ(b) = a, φ(c) = a
87779⟨a, b, c | ab=1, bbc=bbb⟩φ(a) = c, φ(b) = a, φ(c) = a