#20885 ⟨a, b | bb=aa, aaaaa=ab⟩

Up:Monoids with two generators and two relations
Prev:#20884 ⟨a, b | bb=aa, aaaaa=aa⟩
Next:#20914 ⟨a, b | bb=aa, ababa=aa⟩

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 ⇒ a3
  2. ba2 ⇒ a6
  3. ab ⇒ a5
  4. b2 ⇒ a2 
# ab:bb=aa,aaaaa=ab a/b
aaaaaaaaa=aaa
baa=aaaaaa
ab=aaaaa
bb=aa

Cayley table

Idempotents are shown in bold.

1aba2baa3a4a5a6a7a8
11aba2baa3a4a5a6a7a8
aaa2a5a3a6a4a5a6a7a8a3
bbbaa2a6a3a7a8a3a4a5a6
a2a2a3a6a4a7a5a6a7a8a3a4
babaa6a3a7a4a8a3a4a5a6a7
a3a3a4a7a5a8a6a7a8a3a4a5
a4a4a5a8a6a3a7a8a3a4a5a6
a5a5a6a3a7a4a8a3a4a5a6a7
a6a6a7a4a8a5a3a4a5a6a7a8
a7a7a8a5a3a6a4a5a6a7a8a3
a8a8a3a6a4a7a5a6a7a8a3a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

29 unique, 465 total

Σ#PresentationDescriptionRelated
91472⟨a, b | aaa=bb, abbb=1⟩Isomorphic to ℤ11259 iso
104155⟨a, b | abb=aaa, bba=b⟩Finite non-commutative monoid with 11 elements4 iso
105086⟨a, b | aaa=bb, abbb=a⟩Finite commutative monoid with 11 elements11 iso
105087⟨a, b | aaa=bb, abbb=b⟩Finite commutative monoid with 11 elements3 iso
105111⟨a, b | aab=aa, baaa=b⟩Finite non-commutative monoid with 11 elements7 iso
105322⟨a, b | aaa=ab, abb=bb⟩Finite non-commutative monoid with 11 elements3 iso
105335⟨a, b | aaa=ab, bbb=aa⟩Finite non-commutative monoid with 11 elements2 iso, 1 anti-iso
105410⟨a, b | aab=bb, bab=aa⟩Finite non-commutative monoid with 11 elements2 iso
106292⟨a, b | aaa=b, aabbb=a⟩Isomorphic to ℕ(11 = 1)35 iso
106293⟨a, b | aaa=b, aabbb=b⟩Isomorphic to ℕ(11 = 3)11 iso
106380⟨a, b | aab=a, bbbbb=a⟩Isomorphic to ℕ(11 = 5)1 iso
106834⟨a, b | aba=b, abb=aaa⟩Finite non-commutative monoid with 11 elements
107114⟨a, b | ab=aa, bbbb=aa⟩Finite non-commutative monoid with 11 elements1 iso
107116⟨a, b | ab=aa, bbbb=ba⟩Finite non-commutative monoid with 11 elements
108691⟨a, b | aa=b, abbbbb=b⟩Isomorphic to ℕ(11 = 2)32 iso
108777⟨a, b | ab=a, baaaaa=b⟩Finite non-commutative monoid with 11 elements20 iso, 5 anti-iso
109083⟨a, b | ab=a, bbaaa=bb⟩Finite non-commutative monoid with 11 elements4 iso
1112179⟨a, b | aaaa=ab, abbb=b⟩Isomorphic to ℕ(11 = 4)20 iso
1115607⟨a, b | aab=aa, bbbbb=a⟩Isomorphic to ℕ(11 = 10)1 iso
1115671⟨a, b | aab=ab, bbbbb=a⟩Isomorphic to ℕ(11 = 6)8 iso
1116148⟨a, b | aab=ab, aaaa=bb⟩Finite non-commutative monoid with 11 elements1 iso
1116212⟨a, b | aab=ba, aaaa=bb⟩Finite non-commutative monoid with 11 elements
1118715⟨a, b | aaa=a, aabbba=b⟩Finite non-commutative monoid with 11 elements
1119070⟨a, b | aab=b, bbabbb=a⟩Finite commutative monoid with 11 elements4 iso
1120096⟨a, b | aab=b, abba=aaa⟩Finite non-commutative monoid with 11 elements1 iso
1120112⟨a, b | aab=b, baaa=aaa⟩Finite non-commutative monoid with 11 elements
1120168⟨a, b | aab=b, bbbb=aaa⟩Finite commutative monoid with 11 elements
1125539⟨a, b | ab=a, baaaa=bbb⟩Finite non-commutative monoid with 11 elements
1125635⟨a, b | ab=a, bbbaa=bbb⟩Finite non-commutative monoid with 11 elements

Other isomorphic instances

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

7 total

Σ#PresentationMapping
1120889⟨a, b | bb=aa, aaaab=ba⟩φ(a) = b, φ(b) = a
1120893⟨a, b | bb=aa, aaaba=ba⟩φ(a) = b, φ(b) = a
1120896⟨a, b | bb=aa, aaabb=ab⟩φ(a) = a, φ(b) = b
1120903⟨a, b | bb=aa, aabab=ab⟩φ(a) = a, φ(b) = b
1120907⟨a, b | bb=aa, aabba=ab⟩φ(a) = a, φ(b) = b
1120911⟨a, b | bb=aa, abaab=ab⟩φ(a) = a, φ(b) = b
1120915⟨a, b | bb=aa, ababa=ab⟩φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

8 total

Σ#PresentationMapping
1120888⟨a, b | bb=aa, aaaab=ab⟩φ(a) = b, φ(b) = a
1120892⟨a, b | bb=aa, aaaba=ab⟩φ(a) = b, φ(b) = a
1120897⟨a, b | bb=aa, aaabb=ba⟩φ(a) = a, φ(b) = b
1120900⟨a, b | bb=aa, aabaa=ab⟩φ(a) = b, φ(b) = a
1120904⟨a, b | bb=aa, aabab=ba⟩φ(a) = a, φ(b) = b
1120908⟨a, b | bb=aa, aabba=ba⟩φ(a) = a, φ(b) = b
1120912⟨a, b | bb=aa, abaab=ba⟩φ(a) = a, φ(b) = b
1120918⟨a, b | bb=aa, abbba=ab⟩φ(a) = b, φ(b) = a