#1312 ⟨a, b | aaaabaabba=1⟩

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Quick links

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Other submonoids of anti-isomorphic group
  5. Isomorphic instances
  6. Anti-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. dc ⇒ 1 [13]
2. cd ⇒ 1 [8]
3. ca ⇒ ac [5]
4. da ⇒ ad [12]
5. a5 ⇒ c [2]
6. b2c ⇒ a(a2b)2 [17]
7. ba2bd ⇒ a2db2 [14]
8. b2ac ⇒ a(a2b)2a [19]
9. ba2bad ⇒ a2db2a [15]
10. bcba2 ⇒ cba2b [20]
11. b2a2c ⇒ a3(ba2)2 [22]
12. (ba2)2d ⇒ a2db2a2 [23]
13. b2a3 ⇒ a3dbcb [26]
14. ba2ba3 ⇒ d(bc)2 [24]
15. ba2b2 ⇒ d [3]
# ab:aaaabaabba=1 reversed:cda/b aaaaa=c,baabb=d magic:0
dc=1
cd=1
ca=ac
da=ad
aaaaa=c
bbc=aaabaab
baabd=aadbb
bbac=aaabaaba
baabad=aadbba
bcbaa=cbaab
bbaac=aaabaabaa
baabaad=aadbbaa
bbaaa=aaadbcb
baabaaa=dbcbc
baabb=d

Other submonoids of same group

4 unique, 4 total

Σ#PresentationDescriptionRelated
8536⟨a, b | aabba=bbb⟩Infinite cancellative non-commutative monoid
9992⟨a, b | ababbba=bb⟩Infinite cancellative non-commutative monoid
101812⟨a, b | abbabbbba=b⟩Infinite cancellative non-commutative monoid
115698⟨a, b | aababa=babab⟩Infinite cancellative non-commutative monoid

Other submonoids of anti-isomorphic group

3 unique, 3 total

Σ#PresentationDescriptionRelated
8464⟨a, b | aabbba=bb⟩Infinite cancellative non-commutative monoid
9856⟨a, b | ababbbba=b⟩Infinite cancellative non-commutative monoid
114648⟨a, b | aabababa=bab⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

6 total

Σ#PresentationMapping
101348⟨a, b | aaabaabbaa=1⟩φ(a) = a, φ(b) = aaaadabaaaaad
101437⟨a, b | aabbaaaaab=1⟩φ(a) = a, φ(b) = aaaadaaaadaba
112871⟨a, b | aaaabbaaaba=1⟩φ(a) = aaaad, φ(b) = aaabaa
112957⟨a, b | aaabbaaabaa=1⟩φ(a) = aaaad, φ(b) = aabaaa
113010⟨a, b | aabaaaaabba=1⟩φ(a) = aaaad, φ(b) = aaabaa
113301⟨a, b | abbbabbbbba=1⟩φ(a) = aabaaa, φ(b) = aaaad

Anti-isomorphic instances

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

8 total

Σ#PresentationMapping
101324⟨a, b | aaaabbaaba=1⟩φ(a) = a, φ(b) = aaaadabaaaaad
101368⟨a, b | aaabbaabaa=1⟩φ(a) = a, φ(b) = aaaadabaaaaad
101478⟨a, b | abaaaaabba=1⟩φ(a) = a, φ(b) = aaaadabaaaaad
101540⟨a, b | abbabbbbba=1⟩φ(a) = aaaadabaaaaad, φ(b) = a
112845⟨a, b | aaaabaaabba=1⟩φ(a) = aaaad, φ(b) = aabaaa
112910⟨a, b | aaabaaabbaa=1⟩φ(a) = aaaad, φ(b) = aabaaa
112954⟨a, b | aaabbaaaaab=1⟩φ(a) = aaaad, φ(b) = aaaaba
113096⟨a, b | aabbaaaaaba=1⟩φ(a) = aaaad, φ(b) = aabaaa