#30 ⟨a, b | aabba=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. b2 ⇒ d [3]
2. ca ⇒ ac [6]
3. cb ⇒ bc [13]
4. cd ⇒ 1 [9]
5. da ⇒ ad [11]
6. db ⇒ bd [5]
7. dc ⇒ 1 [12]
8. a3 ⇒ c [2]
# ab:aabba=1 abcd aaa=c,bb=d magic:0
bb=d
ca=ac
cb=bc
cd=1
da=ad
db=bd
dc=1
aaa=c

Other submonoids of same group

12 unique, 17 total

Σ#PresentationDescriptionRelated
543⟨a, b | abba=b⟩Infinite cancellative non-commutative monoid
546⟨a, b | aaa=bb⟩Infinite cancellative non-commutative monoid
553⟨a, b | aba=bb⟩Infinite cancellative non-commutative monoid
691⟨a, b | ababa=b⟩Infinite cancellative non-commutative monoid5 iso
6123⟨a, b | bab=aba⟩Infinite cancellative non-commutative monoid
7267⟨a, b | abba=bab⟩Infinite cancellative non-commutative monoid
8555⟨a, b | abbba=bab⟩Infinite cancellative non-commutative monoid
91137⟨a, b | abbbba=bab⟩Infinite cancellative non-commutative monoid
91262⟨a, b | ababa=baab⟩Infinite cancellative non-commutative monoid
102361⟨a, b | abbbbba=bab⟩Infinite cancellative non-commutative monoid
114887⟨a, b | abbbbbba=bab⟩Infinite cancellative non-commutative monoid
115855⟨a, b | abaaba=baaab⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

40 total

Σ#PresentationMapping
533⟨a, b | abbba=1⟩φ(a) = b, φ(b) = a
7135⟨a, b | aaababa=1⟩φ(a) = a, φ(b) = aadb
7143⟨a, b | aababaa=1⟩φ(a) = a, φ(b) = aadb
7146⟨a, b | aabbaab=1⟩φ(a) = bc, φ(b) = bba
7151⟨a, b | abaaaab=1⟩φ(a) = a, φ(b) = aadb
7154⟨a, b | abaabba=1⟩φ(a) = bc, φ(b) = bab
7159⟨a, b | abbabba=1⟩φ(a) = bab, φ(b) = bc
8306⟨a, b | aabababa=1⟩φ(a) = b, φ(b) = bca
8323⟨a, b | abaaabab=1⟩φ(a) = b, φ(b) = bca
8329⟨a, b | ababaaab=1⟩φ(a) = b, φ(b) = bca
9613⟨a, b | aaaabaaba=1⟩φ(a) = a, φ(b) = aadbaad
9629⟨a, b | aaabaabaa=1⟩φ(a) = a, φ(b) = aadbaad
9653⟨a, b | aabaaaaab=1⟩φ(a) = a, φ(b) = aadaadb
9696⟨a, b | abaaaaaba=1⟩φ(a) = a, φ(b) = aadbaad
9707⟨a, b | abaabbaab=1⟩φ(a) = bcaadbc, φ(b) = babba
9712⟨a, b | ababaabba=1⟩φ(a) = aadbcbc, φ(b) = babba
9726⟨a, b | abbababba=1⟩φ(a) = babab, φ(b) = bcaadbc
101351⟨a, b | aaababaaab=1⟩φ(a) = bc, φ(b) = bbba
101398⟨a, b | aabaaababa=1⟩φ(a) = bc, φ(b) = bbab
101416⟨a, b | aababaaaba=1⟩φ(a) = bc, φ(b) = bbab
101483⟨a, b | abaaabaaab=1⟩φ(a) = bc, φ(b) = bbab
112811⟨a, b | aaaaabaaaba=1⟩φ(a) = a, φ(b) = aadaadbaad
112843⟨a, b | aaaabaaabaa=1⟩φ(a) = a, φ(b) = aadbaadaad
112899⟨a, b | aaabaaaaaab=1⟩φ(a) = a, φ(b) = aadaadaadb
112906⟨a, b | aaabaaabaaa=1⟩φ(a) = a, φ(b) = aadbaadaad
112914⟨a, b | aaabaabaaba=1⟩φ(a) = b, φ(b) = bcabc
113006⟨a, b | aabaaaaaaba=1⟩φ(a) = a, φ(b) = aadaadbaad
113011⟨a, b | aabaaaabaab=1⟩φ(a) = b, φ(b) = bcbca
113029⟨a, b | aabaabaaaab=1⟩φ(a) = b, φ(b) = bcbca
113032⟨a, b | aabaabaabaa=1⟩φ(a) = b, φ(b) = bcabc
113055⟨a, b | aababaaabab=1⟩φ(a) = bba, φ(b) = aadbcbcbc
113058⟨a, b | aababaababa=1⟩φ(a) = bba, φ(b) = bcaadbcbc
113178⟨a, b | abaaaabaaba=1⟩φ(a) = b, φ(b) = bcabc
113189⟨a, b | abaaababaab=1⟩φ(a) = bab, φ(b) = bcaadbcbc
113207⟨a, b | abaababaaab=1⟩φ(a) = bab, φ(b) = bcaadbcbc
113215⟨a, b | abaabbaabab=1⟩φ(a) = bcaadbcaadbc, φ(b) = bababba
113231⟨a, b | ababaaababa=1⟩φ(a) = bba, φ(b) = bcaadbcbc
113236⟨a, b | ababaabbaab=1⟩φ(a) = aadbcbcaadbc, φ(b) = babbaba
113241⟨a, b | abababaabba=1⟩φ(a) = aadbcaadbcbc, φ(b) = babbaba
113288⟨a, b | abbabababba=1⟩φ(a) = bababab, φ(b) = bcaadbcaadbc