#4683 ⟨a, b | aaab=b, abba=a⟩

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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.

1abbab2b2ab3b3ab4b4ab5b5ab6
11abbab2b2ab3b3ab4b4ab5b5ab6
aab4ab5b5ab6abbab2b2ab3b3ab4
bbbab2b2ab3b3ab4b4ab5b5ab6ab
babab5ab6abbab2b2ab3b3ab4b4ab5
b2b2b2ab3b3ab4b4ab5b5ab6abbab2
b2ab2aabbab2b2ab3b3ab4b4ab5b5ab6
b3b3b3ab4b4ab5b5ab6abbab2b2ab3
b3ab3abab2b2ab3b3ab4b4ab5b5ab6ab
b4b4b4ab5b5ab6abbab2b2ab3b3ab4
b4ab4ab2ab3b3ab4b4ab5b5ab6abbab2
b5b5b5ab6abbab2b2ab3b3ab4b4ab5
b5ab5ab3ab4b4ab5b5ab6abbab2b2ab3
b6b6abbab2b2ab3b3ab4b4ab5b5ab6

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. b7 ⇒ b [15]
2. ab ⇒ b5 [16]
3. b6a ⇒ a [12]
4. a2 ⇒ b4a [8]
# ab:aaab=b,abba=a b/a - -
bbbbbbb=b
ab=bbbbb
bbbbbba=a
aa=bbbba

Same cardinality

26 unique, 287 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
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
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
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

Isomorphic instances

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

15 total

Σ#PresentationMapping
104689⟨a, b | aaab=b, baba=a⟩φ(a) = a, φ(b) = b
104693⟨a, b | aaab=b, bbaa=a⟩φ(a) = a, φ(b) = b
106410⟨a, b | aab=b, abbba=a⟩φ(a) = bbbbba, φ(b) = b
106426⟨a, b | aab=b, babba=a⟩φ(a) = bbbbba, φ(b) = b
106434⟨a, b | aab=b, bbaba=a⟩φ(a) = bbbbba, φ(b) = b
106438⟨a, b | aab=b, bbbaa=a⟩φ(a) = bbbbba, φ(b) = b
1114485⟨a, b | aaab=b, aabba=a⟩φ(a) = bbbba, φ(b) = b
1114497⟨a, b | aaab=b, abbaa=a⟩φ(a) = bbbba, φ(b) = b
1114509⟨a, b | aaab=b, baaba=a⟩φ(a) = bbbba, φ(b) = b
1114521⟨a, b | aaab=b, bbaaa=a⟩φ(a) = bbbba, φ(b) = b
1119004⟨a, b | aab=b, ababba=a⟩φ(a) = bbb, φ(b) = a
1119012⟨a, b | aab=b, abbaba=a⟩φ(a) = bbb, φ(b) = a
1119044⟨a, b | aab=b, bababa=a⟩φ(a) = bbb, φ(b) = a
1119048⟨a, b | aab=b, babbaa=a⟩φ(a) = bbb, φ(b) = a
1119064⟨a, b | aab=b, bbabaa=a⟩φ(a) = bbb, φ(b) = a

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1124461⟨a, b | ab=a, baaaaaa=b⟩φ(a) = b, φ(b) = bba