#2271 ⟨a, b | bb=aa, aaa=ab⟩

Up:Monoid enumeration
Prev:#2270 ⟨a, b | bb=aa, aaa=aa⟩
Next:#2277 ⟨a, b | bb=aa, aba=aa⟩

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
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
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.

Cayley table

Idempotents are shown in bold.

1abbab2b2ab3
11abbab2b2ab3
aab2b2ab3b2ab3b2a
bbbab2b2ab3b2ab3
babab3b2ab3b2ab3b2a
b2b2b2ab3b2ab3b2ab3
b2ab2ab3b2ab3b2ab3b2a
b3b3b2ab3b2ab3b2ab3

Right Cayley graph

Idempotents are shown in bold.

Left 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. b4 ⇒ b3 [4]
2. b3a ⇒ b2a [5]
3. ab ⇒ b2a [2]
4. a2 ⇒ b2 [1]
# ab:bb=aa,aaa=ab reversed:b/a - -
bbbb=bbb
bbba=bba
ab=bba
aa=bb

Same cardinality

22 unique, 1220 total

Σ#PresentationPropertiesDescriptionφ
7116⟨a, b | aaa=b, abb=1⟩Grp Com 7ℤ7763
8577⟨a, b | aaa=b, abb=a⟩Noc Com 7ℕ(7 = 1)59
8578⟨a, b | aaa=b, abb=b⟩Noc Com 7ℕ(7 = 3)44
8913⟨a, b | aa=b, abbb=b⟩Noc Com 7ℕ(7 = 2)82
8937⟨a, b | ab=a, baaa=b⟩Noc 766
91593⟨a, b | aaa=ab, baa=b⟩Noc 73
91601⟨a, b | aaa=bb, aab=b⟩Noc Com 75
91620⟨a, b | aab=aa, bbb=a⟩Noc Com 7ℕ(7 = 6)35
91632⟨a, b | aab=ab, bbb=a⟩Noc Com 7ℕ(7 = 4)50
92074⟨a, b | aab=b, babb=a⟩Noc Com 720
92231⟨a, b | ab=aa, aaa=bb⟩Noc 73
93197⟨a, b | ab=a, bbb=baa⟩Noc 75
104162⟨a, b | abb=aab, bbb=a⟩Noc Com 7ℕ(7 = 5)42
106251⟨a, b | aaa=a, aabba=b⟩Noc 71
108987⟨a, b | ab=a, aaaaa=bb⟩Noc Com 75
109108⟨a, b | ab=a, bbbbb=aa⟩Noc Com 72
109110⟨a, b | ab=a, bbbbb=ba⟩Noc 71
109263⟨a, b | ab=a, aaaa=bbb⟩Noc Com 74
109375⟨a, b | ab=a, bbba=bbb⟩Noc 73
109376⟨a, b | ab=a, bbbb=aaa⟩Noc Com 73
109382⟨a, b | ab=a, bbbb=bba⟩Noc 71
1114843⟨a, b | abba=b, aabab=a⟩Noc 71

Isomorphic instances

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

1 total

Σ#PresentationMapping
92275⟨a, b | bb=aa, aab=ba⟩φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

2 total

Σ#PresentationMapping
92274⟨a, b | bb=aa, aab=ab⟩φ(a) = b, φ(b) = a
92278⟨a, b | bb=aa, aba=ab⟩φ(a) = b, φ(b) = a