#5521 ⟨a, b | aaab=1, bbbbbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

The order of x is the least n such that xn = 1.

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a171aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a171aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a171aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a171aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a171aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a171aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a171aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a171aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a171aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a171aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a171aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a171aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a171aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a171aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16

Right Cayley graph

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
#RuleProof
1. a18 ⇒ 1 [9]
2. b ⇒ a15 [8]
# ab:aaab=1,bbbbbb=1 a/b
aaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Same cardinality

17 unique, 71 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
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
1120914⟨a, b | bb=aa, ababa=aa⟩Finite non-commutative monoid with 18 elements1 iso

Isomorphic instances

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

33 total

Σ#PresentationMapping
105585⟨a, b | aaba=1, bbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaaa
106037⟨a, b | aaa=b, bbbbbb=1⟩φ(a) = a, φ(b) = aaa
107259⟨a, b | aaa=1, abbbbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107273⟨a, b | aaa=1, babbbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107277⟨a, b | aaa=1, bbabbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107278⟨a, b | aaa=1, bbbabbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107550⟨a, b | aaa=1, bbbbbb=a⟩φ(a) = aaaaaa, φ(b) = a
1111685⟨a, b | aaaa=bb, abbbb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1111689⟨a, b | aaaa=bb, babbb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1111690⟨a, b | aaaa=bb, bbabb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1113600⟨a, b | aabab=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113728⟨a, b | abaab=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113764⟨a, b | ababa=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113851⟨a, b | aaaa=b, aabbbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113857⟨a, b | aaaa=b, ababbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113859⟨a, b | aaaa=b, abbabb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113860⟨a, b | aaaa=b, abbbab=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113861⟨a, b | aaaa=b, abbbba=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113866⟨a, b | aaaa=b, baabbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113867⟨a, b | aaaa=b, bababb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113868⟨a, b | aaaa=b, babbab=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113870⟨a, b | aaaa=b, bbaabb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1121177⟨a, b | aaa=1, aabbbbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121210⟨a, b | aaa=1, abbabbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121216⟨a, b | aaa=1, abbbbabb=1⟩φ(a) = aaaaaa, φ(b) = a
1121218⟨a, b | aaa=1, abbbbbba=1⟩φ(a) = aaaaaa, φ(b) = a
1121234⟨a, b | aaa=1, baabbbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121242⟨a, b | aaa=1, babbabbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121244⟨a, b | aaa=1, babbbbab=1⟩φ(a) = aaaaaa, φ(b) = a
1121249⟨a, b | aaa=1, bbaabbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121251⟨a, b | aaa=1, bbabbabb=1⟩φ(a) = aaaaaa, φ(b) = a
1121253⟨a, b | aaa=1, bbbaabbb=1⟩φ(a) = aaaaaa, φ(b) = a
1122325⟨a, b | aaa=1, bbbbbb=aa⟩φ(a) = aaaaaaaaaaaa, φ(b) = a