#5346 ⟨a, b | aaa=bb, abb=ab⟩

Up:Monoid enumeration
Prev:#5345 ⟨a, b | aaa=bb, abb=aa⟩
Next:#5348 ⟨a, b | aaa=bb, abb=bb⟩

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.

1aba2bab2ba2b2ab3b2a2
11aba2bab2ba2b2ab3b2a2
aaa2b2ab2b2a2b2ab3b2a2b2ab3
bbbab2ba2b2ab3b2a2b2ab3b2a2
a2a2b2b2a2b2ab3b2a2b2ab3b2a2b2a
bababa2b2ab3b2a2b2ab3b2a2b2ab3
b2b2b2ab3b2a2b2ab3b2a2b2ab3b2a2
ba2ba2b3b2a2b2ab3b2a2b2ab3b2a2b2a
b2ab2ab2a2b2ab3b2a2b2ab3b2a2b2ab3
b3b3b2ab3b2a2b2ab3b2a2b2ab3b2a2
b2a2b2a2b3b2a2b2ab3b2a2b2ab3b2a2b2a

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 [3]
4. a3 ⇒ b2 [1]
# ab:aaa=bb,abb=ab reversed:b/a - -
bbbb=bbb
bbba=bba
ab=bba
aaa=bb

Same cardinality

31 unique, 916 total

Σ#PresentationPropertiesDescriptionφ
7332⟨a, b | aa=1, abbb=b⟩Noc 1085
8507⟨a, b | aaa=b, abbb=1⟩Grp Com 10ℤ10536
8656⟨a, b | bb=aa, aba=a⟩Noc 101
8657⟨a, b | bb=aa, aba=b⟩Noc 101
8970⟨a, b | aa=a, bbb=ab⟩Noc 103
91654⟨a, b | aab=bb, bba=a⟩Noc 106
92012⟨a, b | aaa=b, abbb=a⟩Noc Com 10ℕ(10 = 1)59
92013⟨a, b | aaa=b, abbb=b⟩Noc Com 10ℕ(10 = 3)35
92277⟨a, b | bb=aa, aba=aa⟩Noc 101
92894⟨a, b | aa=a, bbbbb=a⟩Noc Com 10ℕ(10 = 5)17
92935⟨a, b | aa=b, bbbbb=b⟩Noc Com 10ℕ(10 = 2)57
104637⟨a, b | aaaa=b, aabb=b⟩Noc Com 10ℕ(10 = 4)20
105356⟨a, b | aab=aa, aba=bb⟩Noc 10
106587⟨a, b | aaa=b, abbb=bb⟩Noc Com 10ℕ(10 = 6)3
106633⟨a, b | aab=a, baaa=bb⟩Noc 1021
107089⟨a, b | ab=aa, baaa=bb⟩Noc 103
108619⟨a, b | aa=a, abbbbb=b⟩Noc 105
109051⟨a, b | ab=a, baaaa=bb⟩Noc 104
1110729⟨a, b | aaab=baa, abab=1⟩Grp 103
1112157⟨a, b | aaaa=aa, abba=b⟩Noc 10
1112181⟨a, b | aaaa=ab, baaa=b⟩Noc 101
1112196⟨a, b | aaaa=bb, aaab=a⟩Noc Com 107
1112197⟨a, b | aaaa=bb, aaab=b⟩Noc Com 101
1112452⟨a, b | aabb=aa, bbbb=a⟩Noc Com 10ℕ(10 = 8)3
1115738⟨a, b | aab=bb, aaaaa=b⟩Noc Com 10ℕ(10 = 7)7
1119564⟨a, b | aab=b, abbbb=aa⟩Noc Com 105
1119899⟨a, b | aaa=b, abbb=bbb⟩Noc Com 10ℕ(10 = 9)1
1120052⟨a, b | aab=b, aaaa=baa⟩Noc 10
1120253⟨a, b | aba=b, aaaa=bbb⟩Noc Com 10
1125905⟨a, b | ab=a, bbbb=baaa⟩Noc 10
1125909⟨a, b | ab=a, bbbb=bbaa⟩Noc 10

Isomorphic instances

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

2 total

Σ#PresentationMapping
105350⟨a, b | aaa=bb, bab=ab⟩φ(a) = a, φ(b) = b
1116046⟨a, b | aaa=bb, aaaa=ab⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
105347⟨a, b | aaa=bb, abb=ba⟩φ(a) = a, φ(b) = b