#12179 ⟨a, b | aaaa=ab, abbb=b⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8a9a10
11aa2a3a4a5a6a7a8a9a10
aaa2a3a4a5a6a7a8a9a10a4
a2a2a3a4a5a6a7a8a9a10a4a5
a3a3a4a5a6a7a8a9a10a4a5a6
a4a4a5a6a7a8a9a10a4a5a6a7
a5a5a6a7a8a9a10a4a5a6a7a8
a6a6a7a8a9a10a4a5a6a7a8a9
a7a7a8a9a10a4a5a6a7a8a9a10
a8a8a9a10a4a5a6a7a8a9a10a4
a9a9a10a4a5a6a7a8a9a10a4a5
a10a10a4a5a6a7a8a9a10a4a5a6

Right 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. a11 ⇒ a4 [3]
2. b ⇒ a10 [2]
# ab:aaaa=ab,abbb=b a/b - -
aaaaaaaaaaa=aaaa
b=aaaaaaaaaa

Same cardinality

29 unique, 460 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105111⟨a, b | aab=aa, baaa=b⟩Noc 117
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105335⟨a, b | aaa=ab, bbb=aa⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
108777⟨a, b | ab=a, baaaaa=b⟩Noc 1125
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

20 total

Σ#PresentationMapping
1114376⟨a, b | aaaa=b, aaabb=b⟩φ(a) = a, φ(b) = aaaa
1114380⟨a, b | aaaa=b, aabab=b⟩φ(a) = a, φ(b) = aaaa
1114382⟨a, b | aaaa=b, aabba=b⟩φ(a) = a, φ(b) = aaaa
1114386⟨a, b | aaaa=b, abaab=b⟩φ(a) = a, φ(b) = aaaa
1114388⟨a, b | aaaa=b, ababa=b⟩φ(a) = a, φ(b) = aaaa
1114398⟨a, b | aaaa=b, baaab=b⟩φ(a) = a, φ(b) = aaaa
1119331⟨a, b | aaa=b, aabbb=ab⟩φ(a) = a, φ(b) = aaa
1119332⟨a, b | aaa=b, aabbb=ba⟩φ(a) = a, φ(b) = aaa
1119342⟨a, b | aaa=b, ababb=ab⟩φ(a) = a, φ(b) = aaa
1119343⟨a, b | aaa=b, ababb=ba⟩φ(a) = a, φ(b) = aaa
1119346⟨a, b | aaa=b, abbab=ab⟩φ(a) = a, φ(b) = aaa
1119347⟨a, b | aaa=b, abbab=ba⟩φ(a) = a, φ(b) = aaa
1119350⟨a, b | aaa=b, abbba=ab⟩φ(a) = a, φ(b) = aaa
1119360⟨a, b | aaa=b, baabb=ab⟩φ(a) = a, φ(b) = aaa
1119361⟨a, b | aaa=b, baabb=ba⟩φ(a) = a, φ(b) = aaa
1119364⟨a, b | aaa=b, babab=ab⟩φ(a) = a, φ(b) = aaa
1120040⟨a, b | aab=a, bbbb=aaa⟩φ(a) = aaaaaa, φ(b) = a
1124823⟨a, b | aa=b, abbbbb=bb⟩φ(a) = a, φ(b) = aa
1124849⟨a, b | aa=b, babbbb=bb⟩φ(a) = a, φ(b) = aa
1124856⟨a, b | aa=b, bbabbb=bb⟩φ(a) = a, φ(b) = aa