#20927 ⟨a, b | ab=aa, aaaa=bbb⟩

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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

Properties

Finite structure

Elements in the center commute with all other elements.
The zero element z satisfies zy = yz = z for all y.
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.

1aba2bab2a3ba2b2ab3ba3b2a2b2a3
11aba2bab2a3ba2b2ab3ba3b2a2b2a3
aaa2a2a3a3a3b3b3b3b3b3b3b3
bbbab2ba2b2ab3ba3b2a2b3b3b2a3b3b3
a2a2a3a3b3b3b3b3b3b3b3b3b3b3
bababa2ba2ba3ba3ba3b3b3b3b3b3b3b3
b2b2b2ab3b2a2b3b3b2a3b3b3b3b3b3b3
a3a3b3b3b3b3b3b3b3b3b3b3b3b3
ba2ba2ba3ba3b3b3b3b3b3b3b3b3b3b3
b2ab2ab2a2b2a2b2a3b2a3b2a3b3b3b3b3b3b3b3
b3b3b3b3b3b3b3b3b3b3b3b3b3b3
ba3ba3b3b3b3b3b3b3b3b3b3b3b3b3
b2a2b2a2b2a3b2a3b3b3b3b3b3b3b3b3b3b3
b2a3b2a3b3b3b3b3b3b3b3b3b3b3b3b3

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. ab ⇒ a2 [1]
2. a4 ⇒ b3 [2]
3. b3a ⇒ b3 [4]
4. b4 ⇒ b3 [5]
# ab:ab=aa,aaaa=bbb ab - -
ab=aa
aaaa=bbb
bbba=bbb
bbbb=bbb

Same cardinality

26 unique, 296 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
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
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.

7 total

Σ#PresentationMapping
1120935⟨a, b | ab=aa, aaab=bbb⟩φ(a) = a, φ(b) = b
1120943⟨a, b | ab=aa, aaba=bbb⟩φ(a) = a, φ(b) = b
1120951⟨a, b | ab=aa, aabb=bbb⟩φ(a) = a, φ(b) = b
1120959⟨a, b | ab=aa, abaa=bbb⟩φ(a) = a, φ(b) = b
1120967⟨a, b | ab=aa, abab=bbb⟩φ(a) = a, φ(b) = b
1120975⟨a, b | ab=aa, abba=bbb⟩φ(a) = a, φ(b) = b
1120983⟨a, b | ab=aa, abbb=bbb⟩φ(a) = a, φ(b) = b