#2183 ⟨a, b, c | aabb=1, bcac=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. a2b ⇒ cac [3]
2. ab2 ⇒ cbc [18]
3. bca ⇒ acb [6]
4. ba2 ⇒ cac [10]
5. b2a ⇒ cbc [15]
6. cacb ⇒ 1 [4]
7. acbc ⇒ 1 [7]
8. a(ac)2 ⇒ caca2 [11]
9. a3cb ⇒ cac2a [8]
10. a(bc)2 ⇒ (cb)2a [22]
11. abacb ⇒ cbc2a [21]
12. bc2ac ⇒ acbab [9]
13. bc2bc ⇒ acb3 [19]
14. b(ac)2 ⇒ (ca)2b [13]
15. b(bc)2 ⇒ cbcb2 [20]
16. (cac)2 ⇒ a2 [5]
17. cac2bc ⇒ ba [14]
18. a3c2ac ⇒ cac2a3 [12]
19. a3c2bc ⇒ cac2aba [16]
20. abac2ac ⇒ cbc2a3 [23]
21. abac2bc ⇒ cbc2aba [24]
# abc:aabb=1,bcac=1 cab - -
aab=cac
abb=cbc
bca=acb
baa=cac
bba=cbc
cacb=1
acbc=1
aacac=cacaa
aaacb=cacca
abcbc=cbcba
abacb=cbcca
bccac=acbab
bccbc=acbbb
bacac=cacab
bbcbc=cbcbb
caccac=aa
caccbc=ba
aaaccac=caccaaa
aaaccbc=caccaba
abaccac=cbccaaa
abaccbc=cbccaba

Other submonoids of same group

2 unique, 2 total

Σ#PresentationProperties
83495⟨a, b, c | bb=aa, cac=b⟩Can Inf
83599⟨a, b, c | bb=aa, cc=ab⟩Can Inf

Isomorphic instances

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

7 total

Σ#PresentationMapping
82269⟨a, b, c | abac=1, bccb=1⟩φ(a) = cbcac, φ(b) = bcacacbcacacb, φ(c) = b
82277⟨a, b, c | abac=1, cbbc=1⟩φ(a) = cbcac, φ(b) = bcacacbcacacb, φ(c) = b
82522⟨a, b, c | aab=c, bbcc=1⟩φ(a) = cb, φ(b) = cacacbcacacbb, φ(c) = b
82530⟨a, b, c | aab=c, bccb=1⟩φ(a) = cb, φ(b) = cacacbcacacbb, φ(c) = b
82546⟨a, b, c | aab=c, cbbc=1⟩φ(a) = cb, φ(b) = cacacbcacacbb, φ(c) = b
83256⟨a, b, c | bb=aa, accb=1⟩φ(a) = caccbcb, φ(b) = b, φ(c) = cacacb
83259⟨a, b, c | bb=aa, cabc=1⟩φ(a) = cbcbcac, φ(b) = b, φ(c) = cacacb