#1427 ⟨a, b | abba=b, baba=1⟩

Up:Monoids with two generators and two relations
Prev:#1423 ⟨a, b | abba=a, bbbb=1⟩
Next:#1434 ⟨a, b | baab=a, bbbb=1⟩

Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Cayley table

1ababb2ab2
11ababb2ab2
aa1abbab2b2
bbab2b2a1ab
ababb2ab21ab
b2b2ab1ab2ba
ab2ab2bab2ab1

Right Cayley graph

Left Cayley graph

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. b3 ⇒ 1 [10]
2. ba ⇒ ab2 [9]
3. a2 ⇒ 1 [12]
# ab:abba=b,baba=1 b/a
bbb=1
ba=abb
aa=1

Same cardinality

19 unique, 1859 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-iso
7245⟨a, b | aa=a, bbb=a⟩Isomorphic to ℕ(6 = 3)49 iso
7257⟨a, b | aa=b, bbb=a⟩Isomorphic to ℕ(6 = 1)61 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Isomorphic instances

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

66 total

Σ#PresentationMapping
91513⟨a, b | aab=ba, abab=1⟩φ(a) = b, φ(b) = a
91518⟨a, b | aab=ba, baba=1⟩φ(a) = b, φ(b) = a
91806⟨a, b | abab=1, aabba=1⟩φ(a) = b, φ(b) = a
91810⟨a, b | abab=1, abbaa=1⟩φ(a) = b, φ(b) = a
91811⟨a, b | abab=1, abbba=1⟩φ(a) = a, φ(b) = b
91815⟨a, b | abab=1, bbaaa=1⟩φ(a) = b, φ(b) = a
104889⟨a, b | aab=ba, aaabb=1⟩φ(a) = b, φ(b) = a
104892⟨a, b | aab=ba, aabba=1⟩φ(a) = b, φ(b) = a
104898⟨a, b | aab=ba, abbaa=1⟩φ(a) = b, φ(b) = a
104903⟨a, b | aab=ba, baaab=1⟩φ(a) = b, φ(b) = a
104910⟨a, b | aab=ba, bbaaa=1⟩φ(a) = b, φ(b) = a
1110701⟨a, b | aaab=aba, abab=1⟩φ(a) = b, φ(b) = a
1110706⟨a, b | aaab=aba, baba=1⟩φ(a) = b, φ(b) = a
1110836⟨a, b | aaba=baa, abab=1⟩φ(a) = b, φ(b) = a
1110841⟨a, b | aaba=baa, baba=1⟩φ(a) = b, φ(b) = a
1110916⟨a, b | aabb=aba, baba=1⟩φ(a) = a, φ(b) = b
1111020⟨a, b | abba=aab, baba=1⟩φ(a) = a, φ(b) = b
1111043⟨a, b | abba=baa, baba=1⟩φ(a) = a, φ(b) = b
1111048⟨a, b | abba=bab, baba=1⟩φ(a) = b, φ(b) = a
1111344⟨a, b | aabab=a, aabba=1⟩φ(a) = b, φ(b) = a
1111350⟨a, b | aabab=a, abbaa=1⟩φ(a) = b, φ(b) = a
1111352⟨a, b | aabab=a, abbba=1⟩φ(a) = a, φ(b) = b
1111355⟨a, b | aabab=a, baaab=1⟩φ(a) = b, φ(b) = a
1111357⟨a, b | aabab=a, baabb=1⟩φ(a) = a, φ(b) = b
1111362⟨a, b | aabab=a, bbaaa=1⟩φ(a) = b, φ(b) = a
1111363⟨a, b | aabab=a, bbaab=1⟩φ(a) = a, φ(b) = b
1111366⟨a, b | aabab=a, bbbaa=1⟩φ(a) = a, φ(b) = b
1111499⟨a, b | ababa=a, abbaa=1⟩φ(a) = b, φ(b) = a
1111501⟨a, b | ababa=a, abbba=1⟩φ(a) = a, φ(b) = b
1111504⟨a, b | ababa=a, baaab=1⟩φ(a) = b, φ(b) = a
1111506⟨a, b | ababa=a, baabb=1⟩φ(a) = a, φ(b) = b
1111572⟨a, b | abbba=b, baaab=1⟩φ(a) = b, φ(b) = a
1111579⟨a, b | abbba=b, bbaaa=1⟩φ(a) = b, φ(b) = a
1111586⟨a, b | baaab=a, baabb=1⟩φ(a) = a, φ(b) = b
1113455⟨a, b | aaabb=1, abaaba=1⟩φ(a) = b, φ(b) = a
1113473⟨a, b | aaabb=1, baabaa=1⟩φ(a) = b, φ(b) = a
1113610⟨a, b | aabba=1, aabaab=1⟩φ(a) = b, φ(b) = a
1113619⟨a, b | aabba=1, abaaba=1⟩φ(a) = b, φ(b) = a
1113637⟨a, b | aabba=1, baabaa=1⟩φ(a) = b, φ(b) = a
1113787⟨a, b | abbba=1, abbabb=1⟩φ(a) = a, φ(b) = b
1113796⟨a, b | abbba=1, babbab=1⟩φ(a) = a, φ(b) = b
1115146⟨a, b | aab=ba, aabaab=1⟩φ(a) = b, φ(b) = a
1115155⟨a, b | aab=ba, abaaba=1⟩φ(a) = b, φ(b) = a
1115173⟨a, b | aab=ba, baabaa=1⟩φ(a) = b, φ(b) = a
1116975⟨a, b | abab=1, aabbaab=1⟩φ(a) = a, φ(b) = b
1116985⟨a, b | abab=1, abaabba=1⟩φ(a) = a, φ(b) = b
1116993⟨a, b | abab=1, abbaaba=1⟩φ(a) = a, φ(b) = b
1116995⟨a, b | abab=1, abbabba=1⟩φ(a) = b, φ(b) = a
1117006⟨a, b | abab=1, baabbaa=1⟩φ(a) = a, φ(b) = b
1117013⟨a, b | abab=1, bbaabaa=1⟩φ(a) = a, φ(b) = b
1117502⟨a, b | abab=1, aabbaa=a⟩φ(a) = b, φ(b) = a
1117510⟨a, b | abab=1, abaaab=a⟩φ(a) = b, φ(b) = a
1117519⟨a, b | abab=1, abbaaa=a⟩φ(a) = b, φ(b) = a
1117521⟨a, b | abab=1, abbaab=a⟩φ(a) = a, φ(b) = b
1117524⟨a, b | abab=1, abbbaa=a⟩φ(a) = a, φ(b) = b
1117530⟨a, b | abab=1, baaaba=a⟩φ(a) = b, φ(b) = a
1117534⟨a, b | abab=1, baabba=a⟩φ(a) = a, φ(b) = b
1117539⟨a, b | abab=1, babbaa=b⟩φ(a) = b, φ(b) = a
1117540⟨a, b | abab=1, bbaaaa=a⟩φ(a) = b, φ(b) = a
1117543⟨a, b | abab=1, bbbaaa=a⟩φ(a) = a, φ(b) = b
1118033⟨a, b | abab=1, aabba=ab⟩φ(a) = a, φ(b) = b
1118040⟨a, b | abab=1, abaab=aa⟩φ(a) = b, φ(b) = a
1118042⟨a, b | abab=1, abaab=ba⟩φ(a) = a, φ(b) = b
1118050⟨a, b | abab=1, abbaa=ba⟩φ(a) = a, φ(b) = b
1118060⟨a, b | abab=1, baaba=aa⟩φ(a) = b, φ(b) = a
1118061⟨a, b | abab=1, baaba=ab⟩φ(a) = a, φ(b) = b