#13142 ⟨a, b | aab=aaa, bbb=ba⟩

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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.
A left zero element x satisfies xy = x for all y.
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.

1aba2abb2a2bab2b3a2b2ab3b4ab4b5ab5b6ab6
11aba2abb2a2bab2b3a2b2ab3b4ab4b5ab5b6ab6
aaa2aba2ba2bab2a2b2a2b2ab3a2b2a2b2ab4a2b2ab5a2b2ab6a2b2
bbb3b2b5b4b3b6b5b4b6b6b5b6b6b6b6b6
a2a2a2ba2ba2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2
ababab3ab2ab5ab4ab3ab6ab5ab4ab6ab6ab5ab6ab6ab6ab6ab6
b2b2b4b3b6b5b4b6b6b5b6b6b6b6b6b6b6b6
a2ba2ba2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2
ab2ab2ab4ab3ab6ab5ab4ab6ab6ab5ab6ab6ab6ab6ab6ab6ab6ab6
b3b3b5b4b6b6b5b6b6b6b6b6b6b6b6b6b6b6
a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2
ab3ab3ab5ab4ab6ab6ab5ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6
b4b4b6b5b6b6b6b6b6b6b6b6b6b6b6b6b6b6
ab4ab4ab6ab5ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6
b5b5b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
ab5ab5ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6
b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6ab6

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. b7 ⇒ b6 [4]
2. ba ⇒ b3 [2]
3. a2b3 ⇒ a2b2 [3]
4. a3 ⇒ a2b [1]
# ab:aab=aaa,bbb=ba b/a
bbbbbbb=bbbbbb
ba=bbb
aabbb=aabb
aaa=aab

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
107153⟨a, b | bb=aa, abab=ab⟩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
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