#20914 ⟨a, b | bb=aa, ababa=aa⟩

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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. b10 ⇒ b4
  2. ab2 ⇒ b6
  3. b2a ⇒ b6
  4. a2 ⇒ b2
  5. a(ba)2 ⇒ b2 
# ab:bb=aa,ababa=aa b/a
bbbbbbbbbb=bbbb
abb=bbbbbb
bba=bbbbbb
aa=bb
ababa=bb

Cayley table

Idempotents are shown in bold.

1ababbab2abababb3(ab)2(ba)2b4b(ab)2b5b6b7b8b9
11ababbab2abababb3(ab)2(ba)2b4b(ab)2b5b6b7b8b9
aab2abb3abab6b7(ab)2b7b8b2b8b3b9b4b5b6b7
bbbab2babb6b3(ba)2b7b4b(ab)2b5b5b6b6b7b8b9b4
abababab6(ab)2b4b7b2b5b8b3b9b9b4b4b5b6b7b8
babab3babb4(ba)2b7b8b(ab)2b8b9b3b9b4b4b5b6b7b8
b2b2b6b3b7b7b4b5b8b5b6b6b6b7b7b8b9b4b5
abaabab7(ab)2b8b2b5b6b3b6b7b7b7b8b8b9b4b5b6
babbab(ba)2b7b(ab)2b5b8b3b6b9b4b4b4b5b5b6b7b8b9
b3b3b7b4b8b8b5b6b9b6b7b7b7b8b8b9b4b5b6
(ab)2(ab)2b2b5b3b9b6b7b4b7b8b8b8b9b9b4b5b6b7
(ba)2(ba)2b8b(ab)2b9b3b6b7b4b7b8b8b8b9b9b4b5b6b7
b4b4b8b5b9b9b6b7b4b7b8b8b8b9b9b4b5b6b7
b(ab)2b(ab)2b3b6b4b4b7b8b5b8b9b9b9b4b4b5b6b7b8
b5b5b9b6b4b4b7b8b5b8b9b9b9b4b4b5b6b7b8
b6b6b4b7b5b5b8b9b6b9b4b4b4b5b5b6b7b8b9
b7b7b5b8b6b6b9b4b7b4b5b5b5b6b6b7b8b9b4
b8b8b6b9b7b7b4b5b8b5b6b6b6b7b7b8b9b4b5
b9b9b7b4b8b8b5b6b9b6b7b7b7b8b8b9b4b5b6

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 103 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
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

Other isomorphic instances

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

1 total

Σ#PresentationMapping
1120916⟨a, b | bb=aa, ababa=bb⟩φ(a) = a, φ(b) = b