#12441 ⟨a, b | aabb=aa, baab=b⟩

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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.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1aba2abbaa3ababa2a4aba2ba3a5aba3
11aba2abbaa3ababa2a4aba2ba3a5aba3
aaa2aba3a4abaa4a5aba2a5a2aba3a2a3
bbbaba2ba2ba3ba3ba3bbbbabababa2
a2a2a3a4a4a5a5a5a2a2a2a3a3a3a4
abababaaba2aba2aba3aba3aba3ababababaabaabaaba2
bababa2ba3ba3bbbbabababa2ba2ba2ba3
a3a3a4a5a5a2a2a2a3a3a3a4a4a4a5
abaabaaba2aba3aba3ababababaabaabaaba2aba2aba2aba3
ba2ba2ba3bbbabababa2ba2ba2ba3ba3ba3b
a4a4a5a2a2a3a3a3a4a4a4a5a5a5a2
aba2aba2aba3abababaabaabaaba2aba2aba2aba3aba3aba3ab
ba3ba3bbababa2ba2ba2ba3ba3ba3bbbba
a5a5a2a3a3a4a4a4a5a5a5a2a2a2a3
aba3aba3ababaabaaba2aba2aba2aba3aba3aba3ababababa

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. a6 ⇒ a2 [7]
2. ba4 ⇒ b [5]
3. a2b ⇒ a4 [6]
4. b2 ⇒ ba2 [4]
5. bab ⇒ ba3 [8]
# ab:aabb=aa,baab=b a/b
aaaaaa=aa
baaaa=b
aab=aaaa
bb=baa
bab=baaa

Same cardinality

30 unique, 313 total

Σ#PresentationDescriptionRelated
81123⟨a, b | aa=1, abbbb=b⟩Finite non-commutative monoid with 14 elements42 iso, 17 anti-iso
91682⟨a, b | aba=bb, bab=a⟩Finite non-commutative monoid with 14 elements1 iso
93034⟨a, b | aa=a, bbbb=ab⟩Finite non-commutative monoid with 14 elements2 iso
103773⟨a, b | aaaa=bb, abbb=1⟩Isomorphic to ℤ14165 iso
105218⟨a, b | aab=bb, bbba=a⟩Finite non-commutative monoid with 14 elements5 iso, 1 anti-iso
105404⟨a, b | aab=bb, aba=aa⟩Finite non-commutative monoid with 14 elements1 iso
106718⟨a, b | aab=b, bbba=aa⟩Finite non-commutative monoid with 14 elements2 iso
1112183⟨a, b | aaaa=ab, baab=b⟩Finite non-commutative monoid with 14 elements3 iso
1112206⟨a, b | aaaa=bb, abbb=a⟩Finite commutative monoid with 14 elements1 iso
1112207⟨a, b | aaaa=bb, abbb=b⟩Finite commutative monoid with 14 elements1 iso
1112499⟨a, b | abab=aa, abba=b⟩Finite non-commutative monoid with 14 elements
1114383⟨a, b | aaaa=b, aabbb=a⟩Isomorphic to ℕ(14 = 1)15 iso
1114384⟨a, b | aaaa=b, aabbb=b⟩Isomorphic to ℕ(14 = 4)5 iso
1115532⟨a, b | aaa=bb, abbbb=b⟩Isomorphic to ℕ(14 = 3)11 iso
1115539⟨a, b | aaa=bb, babbb=a⟩Finite commutative monoid with 14 elements
1116020⟨a, b | aaa=ab, baab=bb⟩Finite non-commutative monoid with 14 elements1 iso
1116079⟨a, b | aaa=bb, bbbb=ab⟩Finite non-commutative monoid with 14 elements
1116293⟨a, b | aab=bb, abab=aa⟩Finite non-commutative monoid with 14 elements
1116470⟨a, b | aba=bb, bbbb=aa⟩Finite non-commutative monoid with 14 elements
1119552⟨a, b | aab=b, abbaa=aa⟩Finite non-commutative monoid with 14 elements2 iso
1120844⟨a, b | ab=aa, bbbbb=aa⟩Finite non-commutative monoid with 14 elements1 iso
1120846⟨a, b | ab=aa, bbbbb=ba⟩Finite non-commutative monoid with 14 elements
1121023⟨a, b | ab=aa, bbaa=bbb⟩Finite non-commutative monoid with 14 elements1 iso
1121040⟨a, b | ab=aa, bbbb=aaa⟩Finite non-commutative monoid with 14 elements3 iso
1121044⟨a, b | ab=aa, bbbb=baa⟩Finite non-commutative monoid with 14 elements1 iso
1121046⟨a, b | ab=aa, bbbb=bba⟩Finite non-commutative monoid with 14 elements
1121110⟨a, b | bb=aa, abab=aaa⟩Finite non-commutative monoid with 14 elements2 anti-iso
1124186⟨a, b | aa=a, bbbbbbb=a⟩Isomorphic to ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Isomorphic to ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Finite non-commutative monoid with 14 elements

Isomorphic instances

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

3 total

Σ#PresentationMapping
1112443⟨a, b | aabb=aa, baba=b⟩φ(a) = a, φ(b) = b
1112447⟨a, b | aabb=aa, bbaa=b⟩φ(a) = a, φ(b) = b
1115578⟨a, b | aab=aa, baaaa=b⟩φ(a) = a, φ(b) = baa