#12187 ⟨a, b | aaaa=ab, babb=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. b8 ⇒ b
  2. ba ⇒ b6
  3. a2b ⇒ ab6
  4. a4 ⇒ ab 
# ab:aaaa=ab,babb=b b/a
bbbbbbbb=b
ba=bbbbbb
aab=abbbbbb
aaaa=ab

Cayley table

Idempotents are shown in bold.

1aba2abb2a3ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2a3ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2aba3ab6ab2abab7ab3abab4ab2ab5ab3ab6ab4ab7ab5
bbb6b2b4b7b3b2bb4b2b5b3b6b4b7b5bb6
a2a2a3ab6abab4ab7ab6ab5abab6ab2ab7ab3abab4ab2ab5ab3
ababab6ab2ab4ab7ab3ab2abab4ab2ab5ab3ab6ab4ab7ab5abab6
b2b2b7b3b5bb4b3b2b5b3b6b4b7b5bb6b2b7
a3a3abab4ab6ab2ab5ab4ab3ab6ab4ab7ab5abab6ab2ab7ab3ab
ab2ab2ab7ab3ab5abab4ab3ab2ab5ab3ab6ab4ab7ab5abab6ab2ab7
b3b3bb4b6b2b5b4b3b6b4b7b5bb6b2b7b3b
ab3ab3abab4ab6ab2ab5ab4ab3ab6ab4ab7ab5abab6ab2ab7ab3ab
b4b4b2b5b7b3b6b5b4b7b5bb6b2b7b3bb4b2
ab4ab4ab2ab5ab7ab3ab6ab5ab4ab7ab5abab6ab2ab7ab3abab4ab2
b5b5b3b6bb4b7b6b5bb6b2b7b3bb4b2b5b3
ab5ab5ab3ab6abab4ab7ab6ab5abab6ab2ab7ab3abab4ab2ab5ab3
b6b6b4b7b2b5bb7b6b2b7b3bb4b2b5b3b6b4
ab6ab6ab4ab7ab2ab5abab7ab6ab2ab7ab3abab4ab2ab5ab3ab6ab4
b7b7b5bb3b6b2bb7b3bb4b2b5b3b6b4b7b5
ab7ab7ab5abab3ab6ab2abab7ab3abab4ab2ab5ab3ab6ab4ab7ab5

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
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
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
1112191⟨a, b | aaaa=ab, bbab=b⟩φ(a) = a, φ(b) = b
1112193⟨a, b | aaaa=ab, bbba=b⟩φ(a) = a, φ(b) = b