#5009 ⟨a, b | aba=bb, aabbb=1⟩

Up:Monoid enumeration
Prev:#4969 ⟨a, b | aba=aa, bbbbb=1⟩
Next:#5032 ⟨a, b | aaa=aa, abba=a⟩

Contents

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

Properties

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Cayley table

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10ab10b11ab11
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10ab10b11ab11
aab9abb10ab2b11ab31ab4bab5b2ab6b3ab7b4ab8b5ab9b6ab10b7ab11b8
bbab5b2ab6b3ab7b4ab8b5ab9b6ab10b7ab11b8ab9abb10ab2b11ab31ab4
ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10ab10b11ab111ab
b2b2ab10b3ab11b4ab5abb6ab2b7ab3b8ab4b9ab5b10ab6b11ab71ab8bab9
ab2ab2b7ab3b8ab4b9ab5b10ab6b11ab71ab8bab9b2ab10b3ab11b4ab5abb6
b3b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10ab10b11ab111ababb2ab2
ab3ab31ab4bab5b2ab6b3ab7b4ab8b5ab9b6ab10b7ab11b8ab9abb10ab2b11
b4b4ab8b5ab9b6ab10b7ab11b8ab9abb10ab2b11ab31ab4bab5b2ab6b3ab7
ab4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10ab10b11ab111ababb2ab2b3ab3b4
b5b5abb6ab2b7ab3b8ab4b9ab5b10ab6b11ab71ab8bab9b2ab10b3ab11b4a
ab5ab5b10ab6b11ab71ab8bab9b2ab10b3ab11b4ab5abb6ab2b7ab3b8ab4b9
b6b6ab6b7ab7b8ab8b9ab9b10ab10b11ab111ababb2ab2b3ab3b4ab4b5ab5
ab6ab6b3ab7b4ab8b5ab9b6ab10b7ab11b8ab9abb10ab2b11ab31ab4bab5b2
b7b7ab11b8ab9abb10ab2b11ab31ab4bab5b2ab6b3ab7b4ab8b5ab9b6ab10
ab7ab7b8ab8b9ab9b10ab10b11ab111ababb2ab2b3ab3b4ab4b5ab5b6ab6b7
b8b8ab4b9ab5b10ab6b11ab71ab8bab9b2ab10b3ab11b4ab5abb6ab2b7ab3
ab8ab8bab9b2ab10b3ab11b4ab5abb6ab2b7ab3b8ab4b9ab5b10ab6b11ab71
b9b9ab9b10ab10b11ab111ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab9ab9b6ab10b7ab11b8ab9abb10ab2b11ab31ab4bab5b2ab6b3ab7b4ab8b5
b10b10ab2b11ab31ab4bab5b2ab6b3ab7b4ab8b5ab9b6ab10b7ab11b8ab9ab
ab10ab10b11ab111ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8b9ab9b10
b11b11ab71ab8bab9b2ab10b3ab11b4ab5abb6ab2b7ab3b8ab4b9ab5b10ab6
ab11ab11b4ab5abb6ab2b7ab3b8ab4b9ab5b10ab6b11ab71ab8bab9b2ab10b3

Right Cayley graph

Left 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. b12 ⇒ 1 [8]
2. ba ⇒ ab5 [6]
3. a2 ⇒ b9 [9]
# ab:aba=bb,aabbb=1 b/a - -
bbbbbbbbbbbb=1
ba=abbbbb
aa=bbbbbbbbb

Same cardinality

6 unique, 47 total

Σ#PresentationPropertiesDescriptionφ
8425⟨a, b | aab=ba, bbb=1⟩Noc 2421
106935⟨a, b | bb=aa, aaabab=1⟩Grp 242
1111515⟨a, b | ababa=b, abbaa=1⟩Grp 2410
1113372⟨a, b | aaaab=1, bbbbbb=1⟩Grp Com 24ℤ248
1119638⟨a, b | aba=a, aaaab=bb⟩Noc 24
1120252⟨a, b | aba=b, aaaa=bab⟩Noc 24

Isomorphic instances

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

30 total

Σ#PresentationMapping
105014⟨a, b | aba=bb, abbba=1⟩φ(a) = a, φ(b) = b
105017⟨a, b | aba=bb, baabb=1⟩φ(a) = a, φ(b) = b
1111327⟨a, b | aabaa=b, aabba=1⟩φ(a) = b, φ(b) = a
1111337⟨a, b | aabaa=b, baaab=1⟩φ(a) = b, φ(b) = a
1111341⟨a, b | aabaa=b, bbaaa=1⟩φ(a) = b, φ(b) = a
1113466⟨a, b | aaabb=1, abbbab=1⟩φ(a) = b, φ(b) = a
1113480⟨a, b | aaabb=1, bababb=1⟩φ(a) = b, φ(b) = a
1113483⟨a, b | aaabb=1, babbba=1⟩φ(a) = b, φ(b) = a
1113490⟨a, b | aaabb=1, bbabab=1⟩φ(a) = b, φ(b) = a
1113495⟨a, b | aaabb=1, bbbaba=1⟩φ(a) = b, φ(b) = a
1113624⟨a, b | aabba=1, ababbb=1⟩φ(a) = b, φ(b) = a
1113630⟨a, b | aabba=1, abbbab=1⟩φ(a) = b, φ(b) = a
1113644⟨a, b | aabba=1, bababb=1⟩φ(a) = b, φ(b) = a
1113647⟨a, b | aabba=1, babbba=1⟩φ(a) = b, φ(b) = a
1113654⟨a, b | aabba=1, bbabab=1⟩φ(a) = b, φ(b) = a
1113659⟨a, b | aabba=1, bbbaba=1⟩φ(a) = b, φ(b) = a
1113770⟨a, b | abbba=1, aaabab=1⟩φ(a) = a, φ(b) = b
1113774⟨a, b | abbba=1, aababa=1⟩φ(a) = a, φ(b) = b
1113780⟨a, b | abbba=1, abaaab=1⟩φ(a) = a, φ(b) = b
1115370⟨a, b | aba=bb, aaabab=1⟩φ(a) = a, φ(b) = b
1115374⟨a, b | aba=bb, aababa=1⟩φ(a) = a, φ(b) = b
1115380⟨a, b | aba=bb, abaaab=1⟩φ(a) = a, φ(b) = b
1118631⟨a, b | aba=b, aaabbbb=1⟩φ(a) = bb, φ(b) = a
1118641⟨a, b | aba=b, aabbabb=1⟩φ(a) = bb, φ(b) = a
1118644⟨a, b | aba=b, aabbbba=1⟩φ(a) = bb, φ(b) = a
1118659⟨a, b | aba=b, abbaabb=1⟩φ(a) = bb, φ(b) = a
1118661⟨a, b | aba=b, abbabba=1⟩φ(a) = bb, φ(b) = a
1118671⟨a, b | aba=b, baaabbb=1⟩φ(a) = bb, φ(b) = a
1118674⟨a, b | aba=b, baabbab=1⟩φ(a) = bb, φ(b) = a
1118682⟨a, b | aba=b, bbaaabb=1⟩φ(a) = bb, φ(b) = a