#4642 ⟨a, b | aaaa=b, abbb=a⟩

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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.

1aa2a3a4a5a6a7a8a9a10a11a12
11aa2a3a4a5a6a7a8a9a10a11a12
aaa2a3a4a5a6a7a8a9a10a11a12a
a2a2a3a4a5a6a7a8a9a10a11a12aa2
a3a3a4a5a6a7a8a9a10a11a12aa2a3
a4a4a5a6a7a8a9a10a11a12aa2a3a4
a5a5a6a7a8a9a10a11a12aa2a3a4a5
a6a6a7a8a9a10a11a12aa2a3a4a5a6
a7a7a8a9a10a11a12aa2a3a4a5a6a7
a8a8a9a10a11a12aa2a3a4a5a6a7a8
a9a9a10a11a12aa2a3a4a5a6a7a8a9
a10a10a11a12aa2a3a4a5a6a7a8a9a10
a11a11a12aa2a3a4a5a6a7a8a9a10a11
a12a12aa2a3a4a5a6a7a8a9a10a11a12

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. a13 ⇒ a [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,abbb=a a/b - -
aaaaaaaaaaaaa=a
b=aaaa

Same cardinality

26 unique, 293 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
105340⟨a, b | aaa=bb, aab=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1112324⟨a, b | aaab=bb, bbbb=a⟩Noc Com 13ℕ(13 = 2)14
1114647⟨a, b | aaba=b, babbb=a⟩Noc Com 132
1115520⟨a, b | aaa=bb, aabbb=b⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133

Isomorphic instances

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

10 total

Σ#PresentationMapping
104646⟨a, b | aaaa=b, babb=a⟩φ(a) = a, φ(b) = aaaa
106304⟨a, b | aaa=b, abbbb=a⟩φ(a) = a, φ(b) = aaa
106312⟨a, b | aaa=b, babbb=a⟩φ(a) = a, φ(b) = aaa
106314⟨a, b | aaa=b, bbabb=a⟩φ(a) = a, φ(b) = aaa
1119086⟨a, b | aab=b, bbbbbb=a⟩φ(a) = aaaaaa, φ(b) = a
1119230⟨a, b | aba=b, bbbbbb=a⟩φ(a) = aaaaaa, φ(b) = a
1124290⟨a, b | aa=b, abbbbbb=a⟩φ(a) = a, φ(b) = aa
1124318⟨a, b | aa=b, babbbbb=a⟩φ(a) = a, φ(b) = aa
1124326⟨a, b | aa=b, bbabbbb=a⟩φ(a) = a, φ(b) = aa
1124328⟨a, b | aa=b, bbbabbb=a⟩φ(a) = a, φ(b) = aa