#500 ⟨a, b, c | bb=ac, bca=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. b2a2 ⇒ (ab)2 [24]
2. b2ab ⇒ a [23]
3. daba2 ⇒ bab [22]
4. d(ab)2 ⇒ 1 [25]
5. ad ⇒ db6a [14]
6. bd ⇒ db4 [12]
7. c ⇒ dab [7]
# abc:bb=ac,bca=1 ab/d/c cc=d morph:2/2
bbaa=abab
bbab=a
dabaa=bab
dabab=1
ad=dbbbbbba
bd=dbbbb
c=dab

Other submonoids of same group

2 unique, 7 total

Σ#PresentationPropertiesφ
7916⟨a, b, c | aa=b, acb=c⟩Can Inf2
71024⟨a, b, c | ab=c, aca=b⟩Can Inf3

Isomorphic instances

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

40 total

Σ#PresentationMapping
7869⟨a, b, c | ab=c, bacc=1⟩φ(a) = bc, φ(b) = ab, φ(c) = b
7875⟨a, b, c | ab=c, bcac=1⟩φ(a) = bab, φ(b) = c, φ(c) = b
7882⟨a, b, c | ab=c, cbac=1⟩φ(a) = bc, φ(b) = ab, φ(c) = b
71202⟨a, b, c | ab=1, bcacc=1⟩φ(a) = c, φ(b) = ab, φ(c) = b
71206⟨a, b, c | ab=1, bccac=1⟩φ(a) = ab, φ(b) = c, φ(c) = b
71221⟨a, b, c | ab=1, cbcac=1⟩φ(a) = c, φ(b) = ab, φ(c) = b
82492⟨a, b, c | aab=c, abbc=1⟩φ(a) = dabac, φ(b) = abbabbc, φ(c) = c
82510⟨a, b, c | aab=c, babc=1⟩φ(a) = ab, φ(b) = cb, φ(c) = abb
82524⟨a, b, c | aab=c, bcab=1⟩φ(a) = dabac, φ(b) = abbabbc, φ(c) = c
82526⟨a, b, c | aab=c, bcba=1⟩φ(a) = ab, φ(b) = cb, φ(c) = abb
82642⟨a, b, c | aba=c, abbc=1⟩φ(a) = ab, φ(b) = bc, φ(c) = abb
82657⟨a, b, c | aba=c, bacb=1⟩φ(a) = dabac, φ(b) = abbb, φ(c) = c
84336⟨a, b, c | aab=1, cbcc=a⟩φ(a) = dabac, φ(b) = abbabb, φ(c) = c
84469⟨a, b, c | aba=1, cacc=a⟩φ(a) = c, φ(b) = abab, φ(c) = b
84470⟨a, b, c | aba=1, cacc=b⟩φ(a) = c, φ(b) = abab, φ(c) = abb
84475⟨a, b, c | aba=1, cbcc=a⟩φ(a) = dabac, φ(b) = abbabb, φ(c) = c
84518⟨a, b, c | abc=1, abba=c⟩φ(a) = bc, φ(b) = ab, φ(c) = daba
84555⟨a, b, c | abc=1, baab=c⟩φ(a) = bc, φ(b) = ab, φ(c) = daba
84558⟨a, b, c | abc=1, baba=c⟩φ(a) = bab, φ(b) = c, φ(c) = daba
84606⟨a, b, c | abc=1, ccaa=b⟩φ(a) = cb, φ(b) = daba, φ(c) = ab
84711⟨a, b, c | aa=b, accbc=1⟩φ(a) = c, φ(b) = cc, φ(c) = abb
84761⟨a, b, c | aa=b, cacbc=1⟩φ(a) = abb, φ(b) = abbabb, φ(c) = c
85018⟨a, b, c | ab=c, abbac=1⟩φ(a) = bc, φ(b) = ab, φ(c) = b
85021⟨a, b, c | ab=c, abbca=1⟩φ(a) = bab, φ(b) = c, φ(c) = b
85027⟨a, b, c | ab=c, abcba=1⟩φ(a) = bc, φ(b) = ab, φ(c) = b
85055⟨a, b, c | ab=c, baabc=1⟩φ(a) = bc, φ(b) = ab, φ(c) = b
85059⟨a, b, c | ab=c, babac=1⟩φ(a) = bab, φ(b) = c, φ(c) = b
85065⟨a, b, c | ab=c, bacba=1⟩φ(a) = bab, φ(b) = c, φ(c) = b
86412⟨a, b, c | ab=1, acacca=1⟩φ(a) = c, φ(b) = ab, φ(c) = abb
86436⟨a, b, c | ab=1, accaac=1⟩φ(a) = c, φ(b) = ab, φ(c) = abb
86439⟨a, b, c | ab=1, accaca=1⟩φ(a) = abb, φ(b) = dabac, φ(c) = c
86531⟨a, b, c | ab=1, bbcbcc=1⟩φ(a) = ab, φ(b) = c, φ(c) = abb
86534⟨a, b, c | ab=1, bbccbc=1⟩φ(a) = dabac, φ(b) = abb, φ(c) = c
86555⟨a, b, c | ab=1, bcbbcc=1⟩φ(a) = dabac, φ(b) = abb, φ(c) = c
86564⟨a, b, c | ab=1, bccbbc=1⟩φ(a) = ab, φ(b) = c, φ(c) = abb
86576⟨a, b, c | ab=1, caacac=1⟩φ(a) = c, φ(b) = ab, φ(c) = abb
86587⟨a, b, c | ab=1, cacaac=1⟩φ(a) = abb, φ(b) = dabac, φ(c) = c
87061⟨a, b, c | ab=1, bcbcc=a⟩φ(a) = ab, φ(b) = c, φ(c) = abb
87067⟨a, b, c | ab=1, bccbc=a⟩φ(a) = dabac, φ(b) = abb, φ(c) = c
87104⟨a, b, c | ab=1, cbbcc=a⟩φ(a) = dabac, φ(b) = abb, φ(c) = c