#1993 ⟨a, b | aaa=a, abbb=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. b5 ⇒ b
  2. ab ⇒ b3
  3. a3 ⇒ a 
# ab:aaa=a,abbb=b b/a
bbbbb=b
ab=bbb
aaa=a

Cayley table

Idempotents are shown in bold.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2
aaa2b3ab3ab4b3a2b4abb4a2bab2ba2b2ab2a2
bbbab2ba2b2ab3b2a2b3ab4b3a2b4abb4a2baba2
a2a2aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2
bababa2b4bab4abb4a2bab2ba2b2ab3b2a2b3ab3a2
b2b2b2ab3b2a2b3ab4b3a2b4abb4a2bab2ba2b2ab2a2
ba2ba2bab2ba2b2ab3b2a2b3ab4b3a2b4abb4a2baba2
b2ab2ab2a2bb2abab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2
b3b3b3ab4b3a2b4abb4a2bab2ba2b2ab3b2a2b3ab3a2
b2a2b2a2b2ab3b2a2b3ab4b3a2b4abb4a2bab2ba2b2ab2a2
b3ab3ab3a2b2b3ab2ab3b2a2b3ab4b3a2b4abb4a2baba2
b4b4b4abb4a2bab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2
b3a2b3a2b3ab4b3a2b4abb4a2bab2ba2b2ab3b2a2b3ab3a2
b4ab4ab4a2b3b4ab3ab4b3a2b4abb4a2bab2ba2b2ab2a2
b4a2b4a2b4abb4a2bab2ba2b2ab3b2a2b3ab4b3a2b4ab4a2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 186 total

Σ#PresentationDescriptionRelated
91602⟨a, b | aaa=bb, aba=a⟩Finite non-commutative monoid with 15 elements2 iso
91603⟨a, b | aaa=bb, aba=b⟩Finite non-commutative monoid with 15 elements1 iso
91751⟨a, b | aaab=1, bbbbb=1⟩Isomorphic to ℤ15138 iso
105342⟨a, b | aaa=bb, aba=aa⟩Finite non-commutative monoid with 15 elements1 iso
105344⟨a, b | aaa=bb, aba=bb⟩Finite non-commutative monoid with 15 elements2 iso
106276⟨a, b | aaa=a, bbbbb=a⟩Isomorphic to ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Isomorphic to ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Isomorphic to ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Finite non-commutative monoid with 15 elements3 iso
1112985⟨a, b | abb=aaa, baba=b⟩Finite non-commutative monoid with 15 elements7 iso
1113123⟨a, b | aab=aaa, aba=bb⟩Finite non-commutative monoid with 15 elements
1113191⟨a, b | abb=aaa, bbb=ab⟩Finite non-commutative monoid with 15 elements1 iso
1113192⟨a, b | abb=aaa, bbb=ba⟩Finite non-commutative monoid with 15 elements
1115439⟨a, b | aaa=aa, bbbbb=a⟩Isomorphic to ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Isomorphic to ℕ(15 = 6)1 iso
1115543⟨a, b | aaa=bb, bbbbb=a⟩Isomorphic to ℕ(15 = 2)1 iso
1116041⟨a, b | aaa=ab, bbbb=aa⟩Finite non-commutative monoid with 15 elements
1116142⟨a, b | aab=aa, bbbb=ab⟩Finite non-commutative monoid with 15 elements
1119374⟨a, b | aaa=b, bbbbb=ab⟩Isomorphic to ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Finite non-commutative monoid with 15 elements1 anti-iso
1120144⟨a, b | aab=b, bbaa=aaa⟩Finite non-commutative monoid with 15 elements
1120819⟨a, b | ab=aa, bbaaa=bb⟩Finite non-commutative monoid with 15 elements6 iso

Other isomorphic instances

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

4 total

Σ#PresentationMapping
1118703⟨a, b | aaa=a, aaabbb=b⟩φ(a) = a, φ(b) = b
1118723⟨a, b | aaa=a, abaabb=b⟩φ(a) = a, φ(b) = b
1118725⟨a, b | aaa=a, ababab=b⟩φ(a) = a, φ(b) = b
1118731⟨a, b | aaa=a, abbaab=b⟩φ(a) = a, φ(b) = b