#1714 ⟨a, b, c | aab=bc, cba=1⟩

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Contents

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Other submonoids of anti-isomorphic group
  5. Isomorphic instances
  6. 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. db ⇒ 1 [19]
2. bd ⇒ 1 [24]
3. da2 ⇒ ab2ad3 [21]
4. bada ⇒ dab2ad [23]
5. bab2a ⇒ a2b3 [9]
6. d2ab2a ⇒ adab [26]
7. a2b2a ⇒ b [6]
8. baba2 ⇒ a2b5ad3 [31]
9. d2aba2 ⇒ adab3ad3 [34]
10. a2ba2 ⇒ b3ad3 [30]
11. c ⇒ ab2ad2 [15]
# abc:aab=bc,cba=1 db/a/c ac=d morph:2/0
db=1
bd=1
daa=abbaddd
bada=dabbad
babba=aabbb
ddabba=adab
aabba=b
babaa=aabbbbbaddd
ddabaa=adabbbaddd
aabaa=bbbaddd
c=abbadd

Other submonoids of same group

4 unique, 9 total

Σ#PresentationPropertiesφ
82988⟨a, b, c | aab=c, bcb=a⟩Can Inf2
83510⟨a, b, c | bb=ac, aba=c⟩Can Inf2
85585⟨a, b, c | ab=c, bcac=a⟩Can Inf
86037⟨a, b, c | ab=c, aca=bc⟩Can Inf1

Other submonoids of anti-isomorphic group

5 unique, 12 total

Σ#PresentationPropertiesφ
82984⟨a, b, c | aab=c, bbc=a⟩Can Inf5
83474⟨a, b, c | ba=ac, cbb=a⟩Can Inf
83524⟨a, b, c | bb=ac, baa=c⟩Can Inf2
83535⟨a, b, c | bb=ac, bca=c⟩Can Inf
85568⟨a, b, c | ab=c, bacc=a⟩Can Inf

Isomorphic instances

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

18 total

Σ#PresentationMapping
82513⟨a, b, c | aab=c, bacc=1⟩φ(a) = d, φ(b) = bba, φ(c) = a
82543⟨a, b, c | aab=c, cbac=1⟩φ(a) = d, φ(b) = bba, φ(c) = a
82553⟨a, b, c | aab=c, ccba=1⟩φ(a) = d, φ(b) = bba, φ(c) = a
83277⟨a, b, c | bb=ac, abca=1⟩φ(a) = b, φ(b) = a, φ(c) = daa
83297⟨a, b, c | bb=ac, bcaa=1⟩φ(a) = b, φ(b) = a, φ(c) = daa
83764⟨a, b, c | aab=1, bcacc=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83830⟨a, b, c | aab=1, cbcac=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83852⟨a, b, c | aab=1, ccbca=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83918⟨a, b, c | aba=1, accbc=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83952⟨a, b, c | aba=1, bcacc=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
85043⟨a, b, c | ab=c, acbca=1⟩φ(a) = b, φ(b) = da, φ(c) = a
85073⟨a, b, c | ab=c, bbacc=1⟩φ(a) = ad, φ(b) = b, φ(c) = a
85083⟨a, b, c | ab=c, bcaac=1⟩φ(a) = b, φ(b) = da, φ(c) = a
86432⟨a, b, c | ab=1, acbcca=1⟩φ(a) = b, φ(b) = d, φ(c) = a
86527⟨a, b, c | ab=1, bbcacc=1⟩φ(a) = d, φ(b) = b, φ(c) = a
86560⟨a, b, c | ab=1, bccaac=1⟩φ(a) = b, φ(b) = d, φ(c) = a
86603⟨a, b, c | ab=1, cbbcac=1⟩φ(a) = d, φ(b) = b, φ(c) = a
87052⟨a, b, c | ab=1, bcacc=a⟩φ(a) = d, φ(b) = b, φ(c) = a

Anti-isomorphic instances

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

20 total

Σ#PresentationMapping
81779⟨a, b, c | aab=cc, bac=1⟩φ(a) = d, φ(b) = aabb, φ(c) = a
81789⟨a, b, c | aab=cc, cba=1⟩φ(a) = d, φ(b) = aabb, φ(c) = a
82525⟨a, b, c | aab=c, bcac=1⟩φ(a) = d, φ(b) = abb, φ(c) = a
82547⟨a, b, c | aab=c, cbca=1⟩φ(a) = d, φ(b) = abb, φ(c) = a
83233⟨a, b, c | ba=ac, bcab=1⟩φ(a) = b, φ(b) = dab, φ(c) = a
83238⟨a, b, c | ba=ac, cabb=1⟩φ(a) = b, φ(b) = dab, φ(c) = a
83300⟨a, b, c | bb=ac, bcca=1⟩φ(a) = daa, φ(b) = a, φ(c) = b
83776⟨a, b, c | aab=1, bccac=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83834⟨a, b, c | aab=1, cbcca=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83914⟨a, b, c | aba=1, acbcc=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83957⟨a, b, c | aba=1, bccac=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
83968⟨a, b, c | aba=1, cacbc=1⟩φ(a) = d, φ(b) = bb, φ(c) = a
85047⟨a, b, c | ab=c, accba=1⟩φ(a) = b, φ(b) = ad, φ(c) = a
85057⟨a, b, c | ab=c, baacc=1⟩φ(a) = b, φ(b) = ad, φ(c) = a
85079⟨a, b, c | ab=c, bbcac=1⟩φ(a) = da, φ(b) = b, φ(c) = a
85102⟨a, b, c | ab=c, cbaac=1⟩φ(a) = b, φ(b) = ad, φ(c) = a
86445⟨a, b, c | ab=1, accbca=1⟩φ(a) = b, φ(b) = d, φ(c) = a
86533⟨a, b, c | ab=1, bbccac=1⟩φ(a) = d, φ(b) = b, φ(c) = a
86540⟨a, b, c | ab=1, bcaacc=1⟩φ(a) = b, φ(b) = d, φ(c) = a
87064⟨a, b, c | ab=1, bccac=a⟩φ(a) = d, φ(b) = b, φ(c) = a