#5344 ⟨a, b | aaa=bb, aba=bb⟩

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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. a7 ⇒ a6
  2. ba4 ⇒ a5
  3. aba ⇒ a3
  4. a3b ⇒ ba3
  5. b2 ⇒ a3 
# ab:aaa=bb,aba=bb a/b
aaaaaaa=aaaaaa
baaaa=aaaaa
aba=aaa
aaab=baaa
bb=aaa

Cayley table

Idempotents are shown in bold.

1aba2abbaa3a2bba2baba4ba3ba2ba5a6
11aba2abbaa3a2bba2baba4ba3ba2ba5a6
aaa2aba3a2ba3a4ba3a4ba3a5a5a5a6a6
bbbaa3ba2baba4ba3ba2ba5a5a5a6a6a6a6
a2a2a3a2ba4ba3a4a5a5a5a5a6a6a6a6a6
ababa3a4a4ba3a5a5a5a6a6a6a6a6a6a6
bababa2babba3ba2bba3a5a6a5a6a6a6a6a6a6
a3a3a4ba3a5a5a5a6a6a6a6a6a6a6a6a6
a2ba2ba4a5a5a5a6a6a6a6a6a6a6a6a6a6
ba2ba2ba3ba2ba5a6a5a6a6a6a6a6a6a6a6a6
babbabba3a5a5a6a6a6a6a6a6a6a6a6a6a6
a4a4a5a5a6a6a6a6a6a6a6a6a6a6a6a6
ba3ba3a5a6a6a6a6a6a6a6a6a6a6a6a6a6
ba2bba2ba5a6a6a6a6a6a6a6a6a6a6a6a6a6
a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6
a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 188 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
91993⟨a, b | aaa=a, abbb=b⟩Finite non-commutative monoid with 15 elements4 iso
105342⟨a, b | aaa=bb, aba=aa⟩Finite non-commutative monoid with 15 elements1 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.

2 total

Σ#PresentationMapping
1116486⟨a, b | aaa=bb, aba=aaa⟩φ(a) = a, φ(b) = b
1116557⟨a, b | bab=aa, bbb=bab⟩φ(a) = b, φ(b) = a