#7153 ⟨a, b | bb=aa, abab=ab⟩

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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

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.

1ababbab2abaab2babb3ab3(ba)2bab2b4b5b6b7
11ababbab2abaab2babb3ab3(ba)2bab2b4b5b6b7
aab2abb3abaab2bab2b4abab3b5abaab2b7b4b5b6
bbbab2babab2b3(ba)2bab2ab3b4b3ab2b7b5b6b7b4
abababaab2abb4ab3abaab2b5b7ab3b4b6b4b5b6b7
babab3babb4(ba)2bab2b7b5babb3b6(ba)2bab2b4b5b6b7
b2b2ab2b3ab3bab2b4ab2b7b3b5b4bab2b4b6b7b4b5
abaabaab3abb7abaab2b6b4abab3b5abaab2b7b4b5b6
ab2ab2b4ab3b5ab2b7b4b6ab3b4b7ab2b7b5b6b7b4
babbab(ba)2bab2babb5b3(ba)2bab2b6b4b3b5b7b5b6b7b4
b3b3bab2b4b3b7b5bab2b4b4b6b5b7b5b7b4b5b6
ab3ab3ab2b7ab3b6b4ab2b7b7b5b4b6b4b6b7b4b5
(ba)2(ba)2b3babb4(ba)2bab2b7b5babb3b6(ba)2bab2b4b5b6b7
bab2bab2b5b3b6bab2b4b5b7b3b5b4bab2b4b6b7b4b5
b4b4b7b5b4b4b6b7b5b5b7b6b4b6b4b5b6b7
b5b5b4b6b5b5b7b4b6b6b4b7b5b7b5b6b7b4
b6b6b5b7b6b6b4b5b7b7b5b4b6b4b6b7b4b5
b7b7b6b4b7b7b5b6b4b4b6b5b7b5b7b4b5b6

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. b8 ⇒ b4 [7]
2. ab4 ⇒ b7 [6]
3. bab3 ⇒ b3 [4]
4. b2a ⇒ ab2 [3]
5. a2 ⇒ b2 [1]
6. (ab)2 ⇒ ab [2]
# ab:bb=aa,abab=ab b/a
bbbbbbbb=bbbb
abbbb=bbbbbbb
babbb=bbb
bba=abb
aa=bb
abab=ab

Same cardinality

16 unique, 71 total

Σ#PresentationDescriptionRelated
104375⟨a, b | aaaa=b, abbbb=1⟩Isomorphic to ℤ1739 iso
106428⟨a, b | aab=b, babbb=a⟩Finite non-commutative monoid with 17 elements2 iso
106514⟨a, b | aba=b, baaab=a⟩Finite non-commutative monoid with 17 elements
107152⟨a, b | bb=aa, abab=aa⟩Finite non-commutative monoid with 17 elements
107154⟨a, b | bb=aa, abab=ba⟩Finite non-commutative monoid with 17 elements
1113141⟨a, b | aab=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113142⟨a, b | aab=aaa, bbb=ba⟩Finite non-commutative monoid with 17 elements
1113151⟨a, b | aba=aaa, baa=bb⟩Finite non-commutative monoid with 17 elements
1113156⟨a, b | aba=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113218⟨a, b | abb=aba, bbb=aa⟩Finite non-commutative monoid with 17 elements
1114395⟨a, b | aaaa=b, abbbb=a⟩Isomorphic to ℕ(17 = 1)2 iso
1114396⟨a, b | aaaa=b, abbbb=b⟩Isomorphic to ℕ(17 = 4)2 iso
1114698⟨a, b | aabb=a, baaaa=b⟩Finite non-commutative monoid with 17 elements1 iso, 6 anti-iso
1115488⟨a, b | aaa=ab, babbb=b⟩Finite non-commutative monoid with 17 elements3 iso
1116143⟨a, b | aab=aa, bbbb=ba⟩Finite non-commutative monoid with 17 elements
1116521⟨a, b | aab=bb, bab=aaa⟩Finite non-commutative monoid with 17 elements