#13340 ⟨a, b | aaaab=1, abbbbb=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

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a18a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17

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. a19 ⇒ 1 [8]
2. b ⇒ a15 [7]
# ab:aaaab=1,abbbbb=1 a/b
aaaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Same cardinality

5 unique, 9 total

Σ#PresentationDescriptionRelated
106263⟨a, b | aaa=a, abbba=b⟩Finite non-commutative monoid with 19 elements
106838⟨a, b | aba=b, bab=aaa⟩Finite non-commutative monoid with 19 elements
1114501⟨a, b | aaab=b, abbba=a⟩Finite non-commutative monoid with 19 elements3 iso
1116036⟨a, b | aaa=ab, bbab=bb⟩Finite non-commutative monoid with 19 elements1 iso
1116053⟨a, b | aaa=bb, aaba=ab⟩Finite non-commutative monoid with 19 elements

Isomorphic instances

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

20 total

Σ#PresentationMapping
1113356⟨a, b | aaaab=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113364⟨a, b | aaaab=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113368⟨a, b | aaaab=1, bbbabb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113370⟨a, b | aaaab=1, bbbbab=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113371⟨a, b | aaaab=1, bbbbba=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113404⟨a, b | aaaba=1, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113420⟨a, b | aaaba=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113428⟨a, b | aaaba=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113432⟨a, b | aaaba=1, bbbabb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113434⟨a, b | aaaba=1, bbbbab=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113435⟨a, b | aaaba=1, bbbbba=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113526⟨a, b | aabaa=1, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113533⟨a, b | aabaa=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113535⟨a, b | aabaa=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1117286⟨a, b | aaab=1, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
1117414⟨a, b | aaba=1, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
1118267⟨a, b | aaa=b, abbbbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118281⟨a, b | aaa=b, babbbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118285⟨a, b | aaa=b, bbabbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118286⟨a, b | aaa=b, bbbabbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b