#7039 ⟨a, b | bb=aa, aaaba=b⟩

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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. b9 ⇒ b
  2. ba ⇒ ab5
  3. a2 ⇒ b2 
# ab:bb=aa,aaaba=b b/a
bbbbbbbbb=b
ba=abbbbb
aa=bb

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
aab2abb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2
bbab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5
ababb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7
b2b2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2
ab2ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2abb3ab2b4
b3b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7
ab3ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3b
b4b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4
ab4ab4b6ab5b7ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6
b5b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5ab
ab5ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3
b6b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6
ab6ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6ab5b7ab6b8
b7b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3
ab7ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5
b8b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab8ab8b2abb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 102 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
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
1116042⟨a, b | aaa=ab, bbbb=ab⟩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

Other isomorphic instances

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

2 total

Σ#PresentationMapping
107044⟨a, b | bb=aa, aabab=a⟩φ(a) = b, φ(b) = a
107053⟨a, b | bb=aa, abbba=b⟩φ(a) = a, φ(b) = b