#139 ⟨a, b | aaabbba=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 [6]
2. cb ⇒ bc [13]
3. cd ⇒ 1 [9]
4. da ⇒ ad [11]
5. db ⇒ bd [5]
6. dc ⇒ 1 [12]
7. b3 ⇒ d [3]
8. a4 ⇒ c [2]
# ab:aaabbba=1 abcd aaaa=c,bbb=d magic:0
ca=ac
cb=bc
cd=1
da=ad
db=bd
dc=1
bbb=d
aaaa=c

Other submonoids of same group

7 unique, 7 total

Σ#PresentationDescriptionRelated
7238⟨a, b | aaaa=bbb⟩Infinite cancellative non-commutative monoid
8550⟨a, b | ababa=bbb⟩Infinite cancellative non-commutative monoid
9988⟨a, b | abababa=bb⟩Infinite cancellative non-commutative monoid
9999⟨a, b | abbabba=bb⟩Infinite cancellative non-commutative monoid
101816⟨a, b | abbbabbba=b⟩Infinite cancellative non-commutative monoid
113820⟨a, b | abbabbabba=b⟩Infinite cancellative non-commutative monoid
115857⟨a, b | abaaba=babab⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

16 total

Σ#PresentationMapping
7148⟨a, b | aabbbaa=1⟩φ(a) = a, φ(b) = b
7150⟨a, b | aabbbba=1⟩φ(a) = b, φ(b) = a
101316⟨a, b | aaaabababa=1⟩φ(a) = a, φ(b) = aaadb
101354⟨a, b | aaabababaa=1⟩φ(a) = a, φ(b) = aaadb
101477⟨a, b | abaaaaabab=1⟩φ(a) = a, φ(b) = aaadb
101505⟨a, b | ababaaaaab=1⟩φ(a) = a, φ(b) = aaadb
112870⟨a, b | aaaabbaaaab=1⟩φ(a) = aaad, φ(b) = aaaab
112904⟨a, b | aaabaaaabba=1⟩φ(a) = aaad, φ(b) = aaaba
112936⟨a, b | aaababababa=1⟩φ(a) = b, φ(b) = bbca
112955⟨a, b | aaabbaaaaba=1⟩φ(a) = aaad, φ(b) = aaaba
113014⟨a, b | aabaaaabbaa=1⟩φ(a) = aaad, φ(b) = aabaa
113066⟨a, b | aababababaa=1⟩φ(a) = b, φ(b) = bbca
113179⟨a, b | abaaaababab=1⟩φ(a) = b, φ(b) = bbca
113228⟨a, b | ababaaaabab=1⟩φ(a) = b, φ(b) = bbca
113239⟨a, b | abababaaaab=1⟩φ(a) = b, φ(b) = bbca
113303⟨a, b | abbbbabbbba=1⟩φ(a) = abaaa, φ(b) = aaad