#4375 ⟨a, b | aaaa=b, abbbb=1⟩

Up:Monoid enumeration
Prev:#4361 ⟨a, b | aaaa=a, bbbbb=1⟩
Next:#4413 ⟨a, b | aaab=a, bbbbb=1⟩

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

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

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a161
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a161a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a161aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a161aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a161aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a161aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a161aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a161aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a161aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a161aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a161aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a161aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a161aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a161aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a161aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a161aa2a3a4a5a6a7a8a9a10a11a12a13a14a15

Right Cayley graph

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 ⇒ 1 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,abbbb=1 a/b - -
aaaaaaaaaaaaaaaaa=1
b=aaaa

Same cardinality

16 unique, 32 total

Σ#PresentationPropertiesDescriptionφ
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
1114396⟨a, b | aaaa=b, abbbb=b⟩Noc Com 17ℕ(17 = 4)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.

39 total

Σ#PresentationMapping
104379⟨a, b | aaaa=b, babbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
104380⟨a, b | aaaa=b, bbabb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1111121⟨a, b | aaaaa=b, aabbb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111124⟨a, b | aaaaa=b, ababb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111125⟨a, b | aaaaa=b, abbab=1⟩φ(a) = bbbbbbb, φ(b) = b
1111126⟨a, b | aaaaa=b, abbba=1⟩φ(a) = bbbbbbb, φ(b) = b
1111129⟨a, b | aaaaa=b, baabb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111130⟨a, b | aaaaa=b, babab=1⟩φ(a) = bbbbbbb, φ(b) = b
1114998⟨a, b | aaa=bb, abbbbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1115005⟨a, b | aaa=bb, babbbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1115007⟨a, b | aaa=bb, bbabbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1116695⟨a, b | aaab=1, abbbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116727⟨a, b | aaab=1, babbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116743⟨a, b | aaab=1, bbabbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116751⟨a, b | aaab=1, bbbabbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116755⟨a, b | aaab=1, bbbbabb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116757⟨a, b | aaab=1, bbbbbab=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116758⟨a, b | aaab=1, bbbbbba=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116823⟨a, b | aaba=1, abbbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116855⟨a, b | aaba=1, babbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116871⟨a, b | aaba=1, bbabbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116879⟨a, b | aaba=1, bbbabbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116883⟨a, b | aaba=1, bbbbabb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116885⟨a, b | aaba=1, bbbbbab=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116886⟨a, b | aaba=1, bbbbbba=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1117812⟨a, b | aaab=1, bbbbb=aa⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1117940⟨a, b | aaba=1, bbbbb=aa⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1118245⟨a, b | aaa=b, aabbbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118257⟨a, b | aaa=b, ababbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118262⟨a, b | aaa=b, abbabbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118264⟨a, b | aaa=b, abbbabb=1⟩φ(a) = bbbbbb, φ(b) = b
1118265⟨a, b | aaa=b, abbbbab=1⟩φ(a) = bbbbbb, φ(b) = b
1118266⟨a, b | aaa=b, abbbbba=1⟩φ(a) = bbbbbb, φ(b) = b
1118275⟨a, b | aaa=b, baabbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118278⟨a, b | aaa=b, bababbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118279⟨a, b | aaa=b, babbabb=1⟩φ(a) = bbbbbb, φ(b) = b
1118280⟨a, b | aaa=b, babbbab=1⟩φ(a) = bbbbbb, φ(b) = b
1118283⟨a, b | aaa=b, bbaabbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118284⟨a, b | aaa=b, bbababb=1⟩φ(a) = bbbbbb, φ(b) = b