#7 ⟨a, b, c | aa=b, cc=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. c2 ⇒ 1 [2]
2. b ⇒ a2 [1]
# abc:aa=b,cc=1 ac/b - -
cc=1
b=aa

Other submonoids of same group

2 unique, 235 total

Σ#PresentationPropertiesDescriptionφ
521⟨a, b, c | aa=1, bcb=1⟩Grp Infℤ2 ∗ ℤ225
653⟨a, b, c | bc=ac, cc=1⟩Can Infℤ2 ∗ ℕ8

Isomorphic instances

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

65 total

Σ#PresentationMapping
515⟨a, b, c | ab=c, bb=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
652⟨a, b, c | bb=ac, cc=1⟩φ(a) = aac, φ(b) = a, φ(c) = c
654⟨a, b, c | ca=bc, cc=1⟩φ(a) = a, φ(b) = cac, φ(c) = c
6212⟨a, b, c | aa=1, aba=c⟩φ(a) = c, φ(b) = a, φ(c) = cac
6213⟨a, b, c | aa=1, abb=c⟩φ(a) = c, φ(b) = a, φ(c) = caa
6217⟨a, b, c | aa=1, bab=c⟩φ(a) = c, φ(b) = a, φ(c) = aca
6220⟨a, b, c | aa=1, bbb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaa
71225⟨a, b, c | aa=1, aaab=c⟩φ(a) = c, φ(b) = a, φ(c) = ca
71226⟨a, b, c | aa=1, aaba=c⟩φ(a) = c, φ(b) = a, φ(c) = ac
71227⟨a, b, c | aa=1, aabb=c⟩φ(a) = c, φ(b) = a, φ(c) = aa
71231⟨a, b, c | aa=1, abab=c⟩φ(a) = c, φ(b) = a, φ(c) = caca
71235⟨a, b, c | aa=1, abba=c⟩φ(a) = c, φ(b) = a, φ(c) = caac
71236⟨a, b, c | aa=1, abbb=c⟩φ(a) = c, φ(b) = a, φ(c) = caaa
71248⟨a, b, c | aa=1, baab=c⟩φ(a) = c, φ(b) = a, φ(c) = aa
71251⟨a, b, c | aa=1, babb=c⟩φ(a) = c, φ(b) = a, φ(c) = acaa
71261⟨a, b, c | aa=1, bbbb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaaa
71386⟨a, b, c | aa=1, aab=ac⟩φ(a) = c, φ(b) = a, φ(c) = ca
71388⟨a, b, c | aa=1, aab=ca⟩φ(a) = c, φ(b) = a, φ(c) = ac
71390⟨a, b, c | aa=1, aab=cc⟩φ(a) = c, φ(b) = aa, φ(c) = a
71391⟨a, b, c | aa=1, aba=ac⟩φ(a) = c, φ(b) = a, φ(c) = ac
71393⟨a, b, c | aa=1, aba=cc⟩φ(a) = c, φ(b) = caac, φ(c) = a
71394⟨a, b, c | aa=1, abb=ac⟩φ(a) = c, φ(b) = a, φ(c) = aa
71396⟨a, b, c | aa=1, abb=ca⟩φ(a) = c, φ(b) = a, φ(c) = caac
71408⟨a, b, c | aa=1, bab=ac⟩φ(a) = c, φ(b) = a, φ(c) = caca
71417⟨a, b, c | aa=1, bbb=ac⟩φ(a) = c, φ(b) = a, φ(c) = caaa
86616⟨a, b, c | aa=1, aaaba=c⟩φ(a) = c, φ(b) = a, φ(c) = cac
86617⟨a, b, c | aa=1, aaabb=c⟩φ(a) = c, φ(b) = a, φ(c) = caa
86622⟨a, b, c | aa=1, aabab=c⟩φ(a) = c, φ(b) = a, φ(c) = aca
86626⟨a, b, c | aa=1, aabba=c⟩φ(a) = c, φ(b) = a, φ(c) = aac
86627⟨a, b, c | aa=1, aabbb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaa
86640⟨a, b, c | aa=1, abaab=c⟩φ(a) = c, φ(b) = a, φ(c) = caa
86644⟨a, b, c | aa=1, ababa=c⟩φ(a) = c, φ(b) = a, φ(c) = cacac
86645⟨a, b, c | aa=1, ababb=c⟩φ(a) = c, φ(b) = a, φ(c) = cacaa
86657⟨a, b, c | aa=1, abbab=c⟩φ(a) = c, φ(b) = a, φ(c) = caaca
86661⟨a, b, c | aa=1, abbba=c⟩φ(a) = c, φ(b) = a, φ(c) = caaac
86662⟨a, b, c | aa=1, abbbb=c⟩φ(a) = c, φ(b) = a, φ(c) = caaaa
86696⟨a, b, c | aa=1, baaab=c⟩φ(a) = c, φ(b) = a, φ(c) = aca
86699⟨a, b, c | aa=1, baabb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaa
86709⟨a, b, c | aa=1, babab=c⟩φ(a) = c, φ(b) = a, φ(c) = acaca
86713⟨a, b, c | aa=1, babbb=c⟩φ(a) = c, φ(b) = a, φ(c) = acaaa
86738⟨a, b, c | aa=1, bbabb=c⟩φ(a) = c, φ(b) = a, φ(c) = aacaa
86747⟨a, b, c | aa=1, bbbbb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaaaa
87120⟨a, b, c | aa=1, aaab=ca⟩φ(a) = c, φ(b) = a, φ(c) = cac
87122⟨a, b, c | aa=1, aaab=cc⟩φ(a) = c, φ(b) = caa, φ(c) = a
87123⟨a, b, c | aa=1, aaba=ac⟩φ(a) = c, φ(b) = a, φ(c) = cac
87127⟨a, b, c | aa=1, aaba=cc⟩φ(a) = c, φ(b) = aac, φ(c) = a
87128⟨a, b, c | aa=1, aabb=ac⟩φ(a) = c, φ(b) = a, φ(c) = caa
87130⟨a, b, c | aa=1, aabb=ca⟩φ(a) = c, φ(b) = a, φ(c) = aac
87142⟨a, b, c | aa=1, abab=ac⟩φ(a) = c, φ(b) = a, φ(c) = aca
87144⟨a, b, c | aa=1, abab=ca⟩φ(a) = c, φ(b) = a, φ(c) = cacac
87156⟨a, b, c | aa=1, abba=ac⟩φ(a) = c, φ(b) = a, φ(c) = aac
87159⟨a, b, c | aa=1, abbb=ac⟩φ(a) = c, φ(b) = a, φ(c) = aaa
87161⟨a, b, c | aa=1, abbb=ca⟩φ(a) = c, φ(b) = a, φ(c) = caaac
87197⟨a, b, c | aa=1, baab=ac⟩φ(a) = c, φ(b) = a, φ(c) = caa
87206⟨a, b, c | aa=1, babb=ac⟩φ(a) = c, φ(b) = a, φ(c) = cacaa
87208⟨a, b, c | aa=1, babb=ca⟩φ(a) = c, φ(b) = a, φ(c) = acaac
87238⟨a, b, c | aa=1, bbbb=ac⟩φ(a) = c, φ(b) = a, φ(c) = caaaa
87620⟨a, b, c | aa=1, aba=aac⟩φ(a) = c, φ(b) = a, φ(c) = cac
87621⟨a, b, c | aa=1, abb=aac⟩φ(a) = c, φ(b) = a, φ(c) = caa
87628⟨a, b, c | aa=1, aca=abb⟩φ(a) = c, φ(b) = a, φ(c) = aac
87636⟨a, b, c | aa=1, baa=acc⟩φ(a) = c, φ(b) = caa, φ(c) = a
87637⟨a, b, c | aa=1, bab=aac⟩φ(a) = c, φ(b) = a, φ(c) = aca
87639⟨a, b, c | aa=1, bab=aca⟩φ(a) = c, φ(b) = a, φ(c) = cacac
87654⟨a, b, c | aa=1, bbb=aac⟩φ(a) = c, φ(b) = a, φ(c) = aaa
87656⟨a, b, c | aa=1, bbb=aca⟩φ(a) = c, φ(b) = a, φ(c) = caaac