#14396 ⟨a, b | aaaa=b, abbbb=b⟩

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Contents

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

Properties

Commutative structure

Finite structure

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a4
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a4a5
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a4a5a6
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a4a5a6a7
a5a5a6a7a8a9a10a11a12a13a14a15a16a4a5a6a7a8
a6a6a7a8a9a10a11a12a13a14a15a16a4a5a6a7a8a9
a7a7a8a9a10a11a12a13a14a15a16a4a5a6a7a8a9a10
a8a8a9a10a11a12a13a14a15a16a4a5a6a7a8a9a10a11
a9a9a10a11a12a13a14a15a16a4a5a6a7a8a9a10a11a12
a10a10a11a12a13a14a15a16a4a5a6a7a8a9a10a11a12a13
a11a11a12a13a14a15a16a4a5a6a7a8a9a10a11a12a13a14
a12a12a13a14a15a16a4a5a6a7a8a9a10a11a12a13a14a15
a13a13a14a15a16a4a5a6a7a8a9a10a11a12a13a14a15a16
a14a14a15a16a4a5a6a7a8a9a10a11a12a13a14a15a16a4
a15a15a16a4a5a6a7a8a9a10a11a12a13a14a15a16a4a5
a16a16a4a5a6a7a8a9a10a11a12a13a14a15a16a4a5a6

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a17 ⇒ a4 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,abbbb=b a/b - -
aaaaaaaaaaaaaaaaa=aaaa
b=aaaa

Same cardinality

16 unique, 69 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106428⟨a, b | aab=b, babbb=a⟩Noc 172
106514⟨a, b | aba=b, baaab=a⟩Noc 17
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩Noc 17
107154⟨a, b | bb=aa, abab=ba⟩Noc 17
1113141⟨a, b | aab=aaa, bbb=ab⟩Noc 17
1113142⟨a, b | aab=aaa, bbb=ba⟩Noc 17
1113151⟨a, b | aba=aaa, baa=bb⟩Noc 17
1113156⟨a, b | aba=aaa, bbb=ab⟩Noc 17
1113218⟨a, b | abb=aba, bbb=aa⟩Noc 17
1114395⟨a, b | aaaa=b, abbbb=a⟩Noc Com 17ℕ(17 = 1)2
1114698⟨a, b | aabb=a, baaaa=b⟩Noc 177
1115488⟨a, b | aaa=ab, babbb=b⟩Noc 173
1116143⟨a, b | aab=aa, bbbb=ba⟩Noc 17
1116521⟨a, b | aab=bb, bab=aaa⟩Noc 17

Isomorphic instances

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

2 total

Σ#PresentationMapping
1114404⟨a, b | aaaa=b, babbb=b⟩φ(a) = a, φ(b) = aaaa
1114406⟨a, b | aaaa=b, bbabb=b⟩φ(a) = a, φ(b) = aaaa