#1121 ⟨a, b | aa=1, abbba=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 ⇒ ab3
  3. a2 ⇒ 1 
# ab:aa=1,abbba=b b/a
bbbbbbbbb=b
ba=abbb
aa=1

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
aa1abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8
bbab3b2ab4b3ab5b4ab6b5ab7b6ab8b7abb8ab2bab3
ababb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2abb3
b2b2ab6b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5b2ab6
ab2ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab8b4abb5ab2b6
b3b3abb4ab2b5ab3b6ab4b7ab5b8ab6bab7b2ab8b3ab
ab3ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3b
b4b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4
ab4ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4
b5b5ab7b6ab8b7abb8ab2bab3b2ab4b3ab5b4ab6b5ab7
ab5ab5b7ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6ab5b7
b6b6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5abb6ab2
ab6ab6b2ab7b3ab8b4abb5ab2b6ab3b7ab4b8ab5bab6b2
b7b7ab5b8ab6bab7b2ab8b3abb4ab2b5ab3b6ab4b7ab5
ab7ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5
b8b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab8ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 87 total

Σ#PresentationDescriptionRelated
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
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.

17 total

Σ#PresentationMapping
81192⟨a, b | aa=1, abbb=ba⟩φ(a) = a, φ(b) = b
81255⟨a, b | aa=1, bbb=aba⟩φ(a) = a, φ(b) = b
109685⟨a, b | aa=1, aaabbba=b⟩φ(a) = a, φ(b) = b
109725⟨a, b | aa=1, abaabba=b⟩φ(a) = a, φ(b) = b
109731⟨a, b | aa=1, abababa=b⟩φ(a) = a, φ(b) = ab
109957⟨a, b | aa=1, aaabbb=ba⟩φ(a) = a, φ(b) = b
109979⟨a, b | aa=1, aabbba=ab⟩φ(a) = a, φ(b) = b
109995⟨a, b | aa=1, abaabb=ba⟩φ(a) = a, φ(b) = b
109999⟨a, b | aa=1, ababab=ba⟩φ(a) = a, φ(b) = ab
1010011⟨a, b | aa=1, abbaab=ba⟩φ(a) = a, φ(b) = b
1010221⟨a, b | aa=1, aaaba=bbb⟩φ(a) = a, φ(b) = b
1010254⟨a, b | aa=1, aabbb=aba⟩φ(a) = a, φ(b) = b
1010291⟨a, b | aa=1, abbba=aab⟩φ(a) = a, φ(b) = b
1010312⟨a, b | aa=1, baabb=aba⟩φ(a) = a, φ(b) = b
1010320⟨a, b | aa=1, babab=aba⟩φ(a) = a, φ(b) = ab
1010493⟨a, b | aa=1, abbb=aaba⟩φ(a) = a, φ(b) = b
1010501⟨a, b | aa=1, baaa=abbb⟩φ(a) = a, φ(b) = b