#67 ⟨a, b | aabbba=1⟩

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Quick links

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

Properties

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. ca ⇒ ac [5]
2. cb ⇒ bc [13]
3. cd ⇒ 1 [9]
4. da ⇒ ad [11]
5. db ⇒ bd [6]
6. dc ⇒ 1 [12]
7. a3 ⇒ c [2]
8. b3 ⇒ d [3]
# ab:aabbba=1 abcd aaa=c,bbb=d magic:0
ca=ac
cb=bc
cd=1
da=ad
db=bd
dc=1
aaa=c
bbb=d

Other submonoids of same group

5 unique, 6 total

Σ#PresentationDescriptionRelated
6124⟨a, b | bbb=aaa⟩Infinite cancellative non-commutative monoid
7229⟨a, b | ababa=bb⟩Infinite cancellative non-commutative monoid
8410⟨a, b | abbabba=b⟩Infinite cancellative non-commutative monoid1 iso
91115⟨a, b | abaaba=bab⟩Infinite cancellative non-commutative monoid
115304⟨a, b | abaaaba=baab⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

8 total

Σ#PresentationMapping
9635⟨a, b | aaabababa=1⟩φ(a) = a, φ(b) = aadb
9640⟨a, b | aaabbaaab=1⟩φ(a) = bbc, φ(b) = bbba
9658⟨a, b | aabaaabba=1⟩φ(a) = bbc, φ(b) = bbab
9668⟨a, b | aabababaa=1⟩φ(a) = a, φ(b) = aadb
9677⟨a, b | aabbaaaba=1⟩φ(a) = bbc, φ(b) = bbab
9697⟨a, b | abaaaabab=1⟩φ(a) = b, φ(b) = bbca
9710⟨a, b | ababaaaab=1⟩φ(a) = b, φ(b) = bbca
9729⟨a, b | abbbabbba=1⟩φ(a) = abaa, φ(b) = aad