#461 ⟨a, b, c | ba=ab, aca=1⟩

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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. ca ⇒ ac [3]
2. ba ⇒ ab [1]
3. bc ⇒ cb [5]
4. a2c ⇒ 1 [6]
# abc:ba=ab,aca=1 acb - -
ca=ac
ba=ab
bc=cb
aac=1

Other submonoids of same group

3 unique, 12 total

Σ#PresentationPropertiesDescriptionφ
683⟨a, b, c | abc=1, cba=1⟩Grp Com Infℤ ⊕ ℤ4
6159⟨a, b, c | ab=c, ba=c⟩Can Com Infℕ ⊕ ℕ5
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.

2 total

Σ#PresentationMapping
84396⟨a, b, c | aba=1, abca=c⟩φ(a) = a, φ(b) = c, φ(c) = b
84405⟨a, b, c | aba=1, acab=c⟩φ(a) = a, φ(b) = c, φ(c) = b