#83 ⟨a, b, c | abc=1, cba=1⟩

Up:Monoid enumeration
Prev:#82 ⟨a, b, c | abc=1, bca=1⟩
Next:#85 ⟨a, b, c | aa=a, bac=1⟩

Contents

  1. Properties
  2. Commutative structure
  3. Rewriting system
  4. Other submonoids of same group
  5. Isomorphic instances

Properties

Commutative structure

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. ad ⇒ 1 [4]
2. da ⇒ 1 [7]
3. ba ⇒ ab [11]
4. bd ⇒ db [8]
5. bc ⇒ d [3]
6. ca ⇒ ac [12]
7. cd ⇒ dc [9]
8. cb ⇒ d [5]
# abc:abc=1,cba=1 adbc bc=d morph:2/1
ad=1
da=1
ba=ab
bd=db
bc=d
ca=ac
cd=dc
cb=d

Other submonoids of same group

3 unique, 10 total

Σ#PresentationPropertiesDescriptionφ
6159⟨a, b, c | ab=c, ba=c⟩Can Com Infℕ ⊕ ℕ5
7461⟨a, b, c | ba=ab, aca=1⟩Can Com Infℕ ⊕ ℤ2
83181⟨a, b, c | ba=ab, aaca=1⟩Can Com Infℕ ⊕ ℤ

Isomorphic instances

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

4 total

Σ#PresentationMapping
82221⟨a, b, c | aabc=1, caba=1⟩φ(a) = b, φ(b) = aac, φ(c) = c
82257⟨a, b, c | abac=1, acba=1⟩φ(a) = b, φ(b) = aac, φ(c) = c
82270⟨a, b, c | abac=1, caab=1⟩φ(a) = b, φ(b) = aac, φ(c) = c
84600⟨a, b, c | abc=1, cbaa=a⟩φ(a) = a, φ(b) = b, φ(c) = c