#16042 ⟨a, b | aaa=ab, bbbb=ab⟩

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Properties

Element profile

Complete 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
  1. a9 ⇒ a4
  2. ba3 ⇒ a5
  3. ab ⇒ a3
  4. b4 ⇒ a3 
# ab:aaa=ab,bbbb=ab a/b
aaaaaaaaa=aaaa
baaa=aaaaa
ab=aaa
bbbb=aaa

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3a4b2a2b3aa5b3a2a6a7a8
11aba2bab2a3ba2b2ab3a4b2a2b3aa5b3a2a6a7a8
aaa2a3a3a4a5a4a5a6a7a5a7a8a6a4a7a8a4
bbbab2ba2b2ab3a5b2a2b3aa3a6b3a2a4a7a5a8a4a5
a2a2a3a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
bababa2a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
b2b2b2ab3b2a2b3aa3a7b3a2a4a5a8a5a6a4a7a5a6a7
a3a3a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
ba2ba2a5a6a6a7a8a7a8a4a5a8a5a6a4a7a5a6a7
b2ab2ab2a2a7a7a8a4a8a4a5a6a4a6a7a5a8a6a7a8
b3b3b3aa3b3a2a4a5a4a5a6a7a5a7a8a6a4a7a8a4
a4a4a5a6a6a7a8a7a8a4a5a8a5a6a4a7a5a6a7
b2a2b2a2a7a8a8a4a5a4a5a6a7a5a7a8a6a4a7a8a4
b3ab3ab3a2a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
a5a5a6a7a7a8a4a8a4a5a6a4a6a7a5a8a6a7a8
b3a2b3a2a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
a6a6a7a8a8a4a5a4a5a6a7a5a7a8a6a4a7a8a4
a7a7a8a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
a8a8a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 104 total

Σ#PresentationDescriptionRelated
81121⟨a, b | aa=1, abbba=b⟩Finite non-commutative monoid with 18 elements17 iso
93387⟨a, b | aa=1, abbbbb=b⟩Finite non-commutative monoid with 18 elements22 iso, 9 anti-iso
105521⟨a, b | aaab=1, bbbbbb=1⟩Isomorphic to ℤ1833 iso
106732⟨a, b | aba=a, aaab=bb⟩Finite non-commutative monoid with 18 elements
106788⟨a, b | aba=b, baab=aa⟩Finite non-commutative monoid with 18 elements
106795⟨a, b | aba=b, bbbb=aa⟩Finite non-commutative monoid with 18 elements
107039⟨a, b | bb=aa, aaaba=b⟩Finite non-commutative monoid with 18 elements2 iso
108910⟨a, b | aa=a, bbbbb=ab⟩Finite non-commutative monoid with 18 elements1 iso
1112187⟨a, b | aaaa=ab, babb=b⟩Finite non-commutative monoid with 18 elements2 iso
1115797⟨a, b | aab=bb, bbbba=a⟩Finite non-commutative monoid with 18 elements
1116043⟨a, b | aaa=ab, bbbb=ba⟩Finite non-commutative monoid with 18 elements
1116313⟨a, b | aab=bb, baba=aa⟩Finite non-commutative monoid with 18 elements
1118758⟨a, b | aaa=a, bbbbbb=a⟩Isomorphic to ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Isomorphic to ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Isomorphic to ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Finite non-commutative monoid with 18 elements
1120914⟨a, b | bb=aa, ababa=aa⟩Finite non-commutative monoid with 18 elements1 iso