#16143 ⟨a, b | aab=aa, bbbb=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.

1aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2aba2a2ab2a2ab3a2ab4a2ab5a2ab6a2ab7a2
bbb4b2b7b5b3b6b4b7b5b7b6b7b7b7b7b7
a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2
ababab4ab2ab7ab5ab3ab6ab4ab7ab5ab7ab6ab7ab7ab7ab7ab7
b2b2b5b3b7b6b4b7b5b7b6b7b7b7b7b7b7b7
ab2ab2ab5ab3ab7ab6ab4ab7ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7
b3b3b6b4b7b7b5b7b6b7b7b7b7b7b7b7b7b7
ab3ab3ab6ab4ab7ab7ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7
b4b4b7b5b7b7b6b7b7b7b7b7b7b7b7b7b7b7
ab4ab4ab7ab5ab7ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b5b5b7b6b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab5ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b6b6b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab6ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7

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 ⇒ b7 [4]
2. ba ⇒ b4 [2]
3. a2b ⇒ a2 [1]
4. a3 ⇒ a2 [3]
# ab:aab=aa,bbbb=ba b/a
bbbbbbbb=bbbbbbb
ba=bbbb
aab=aa
aaa=aa

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
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
1116521⟨a, b | aab=bb, bab=aaa⟩Finite non-commutative monoid with 17 elements