#15488 ⟨a, b | aaa=ab, babbb=b⟩

Up:Monoid enumeration
Prev:#15487 ⟨a, b | aaa=ab, babbb=a⟩
Next:#15491 ⟨a, b | aaa=ab, bbaab=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.
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.

1aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2ababab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4
bbb5b2b2b6b3b7b4bb5b2b6b3b7b4bb5
a2a2abab5ab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4ab
ababab5ab2ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5
b2b2b6b3b3b7b4bb5b2b6b3b7b4bb5b2b6
ab2ab2ab6ab3ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6
b3b3b7b4b4bb5b2b6b3b7b4bb5b2b6b3b7
ab3ab3ab7ab4ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7
b4b4bb5b5b2b6b3b7b4bb5b2b6b3b7b4b
ab4ab4abab5ab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4ab
b5b5b2b6b6b3b7b4bb5b2b6b3b7b4bb5b2
ab5ab5ab2ab6ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2
b6b6b3b7b7b4bb5b2b6b3b7b4bb5b2b6b3
ab6ab6ab3ab7ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6ab3
b7b7b4bbb5b2b6b3b7b4bb5b2b6b3b7b4
ab7ab7ab4ababab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4

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. b8 ⇒ b [9]
2. ba ⇒ b5 [8]
3. a2b ⇒ ab5 [5]
4. a3 ⇒ ab [1]
# ab:aaa=ab,babbb=b b/a - -
bbbbbbbb=b
ba=bbbbb
aab=abbbbb
aaa=ab

Same cardinality

16 unique, 68 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
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
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.

3 total

Σ#PresentationMapping
1115496⟨a, b | aaa=ab, bbabb=b⟩φ(a) = a, φ(b) = b
1115500⟨a, b | aaa=ab, bbbab=b⟩φ(a) = a, φ(b) = b
1115502⟨a, b | aaa=ab, bbbba=b⟩φ(a) = a, φ(b) = b