#16012 ⟨a, b | aaa=ab, abbb=bb⟩

Up:Monoid enumeration
Prev:#16011 ⟨a, b | aaa=ab, abbb=ba⟩
Next:#16017 ⟨a, b | aaa=ab, baab=aa⟩

Contents

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

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.

1aba2baa3ba2a4ba3a5ba4a6a7
11aba2baa3ba2a4ba3a5ba4a6a7
aaa2a3a3a4a4a5a5a6a6a7a7a5
bbbaa7ba2a5ba3a6ba4a7a7a5a5a6
a2a2a3a4a4a5a5a6a6a7a7a5a5a6
bababa2ba3ba3ba4ba4a7a7a5a5a6a6a7
a3a3a4a5a5a6a6a7a7a5a5a6a6a7
ba2ba2ba3ba4ba4a7a7a5a5a6a6a7a7a5
a4a4a5a6a6a7a7a5a5a6a6a7a7a5
ba3ba3ba4a7a7a5a5a6a6a7a7a5a5a6
a5a5a6a7a7a5a5a6a6a7a7a5a5a6
ba4ba4a7a5a5a6a6a7a7a5a5a6a6a7
a6a6a7a5a5a6a6a7a7a5a5a6a6a7
a7a7a5a6a6a7a7a5a5a6a6a7a7a5

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. a8 ⇒ a5 [3]
2. ba5 ⇒ a7 [5]
3. ab ⇒ a3 [1]
4. b2 ⇒ a7 [2]
# ab:aaa=ab,abbb=bb a/b - -
aaaaaaaa=aaaaa
baaaaa=aaaaaaa
ab=aaa
bb=aaaaaaa

Same cardinality

26 unique, 303 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
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
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