#4637 ⟨a, b | aaaa=b, aabb=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.

1aa2a3a4a5a6a7a8a9
11aa2a3a4a5a6a7a8a9
aaa2a3a4a5a6a7a8a9a4
a2a2a3a4a5a6a7a8a9a4a5
a3a3a4a5a6a7a8a9a4a5a6
a4a4a5a6a7a8a9a4a5a6a7
a5a5a6a7a8a9a4a5a6a7a8
a6a6a7a8a9a4a5a6a7a8a9
a7a7a8a9a4a5a6a7a8a9a4
a8a8a9a4a5a6a7a8a9a4a5
a9a9a4a5a6a7a8a9a4a5a6

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. a10 ⇒ a4 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,aabb=b a/b - -
aaaaaaaaaa=aaaa
b=aaaa

Same cardinality

31 unique, 899 total

Σ#PresentationPropertiesDescriptionφ
7332⟨a, b | aa=1, abbb=b⟩Noc 1085
8507⟨a, b | aaa=b, abbb=1⟩Grp Com 10ℤ10536
8656⟨a, b | bb=aa, aba=a⟩Noc 101
8657⟨a, b | bb=aa, aba=b⟩Noc 101
8970⟨a, b | aa=a, bbb=ab⟩Noc 103
91654⟨a, b | aab=bb, bba=a⟩Noc 106
92012⟨a, b | aaa=b, abbb=a⟩Noc Com 10ℕ(10 = 1)59
92013⟨a, b | aaa=b, abbb=b⟩Noc Com 10ℕ(10 = 3)35
92277⟨a, b | bb=aa, aba=aa⟩Noc 101
92894⟨a, b | aa=a, bbbbb=a⟩Noc Com 10ℕ(10 = 5)17
92935⟨a, b | aa=b, bbbbb=b⟩Noc Com 10ℕ(10 = 2)57
105346⟨a, b | aaa=bb, abb=ab⟩Noc 103
105356⟨a, b | aab=aa, aba=bb⟩Noc 10
106587⟨a, b | aaa=b, abbb=bb⟩Noc Com 10ℕ(10 = 6)3
106633⟨a, b | aab=a, baaa=bb⟩Noc 1021
107089⟨a, b | ab=aa, baaa=bb⟩Noc 103
108619⟨a, b | aa=a, abbbbb=b⟩Noc 105
109051⟨a, b | ab=a, baaaa=bb⟩Noc 104
1110729⟨a, b | aaab=baa, abab=1⟩Grp 103
1112157⟨a, b | aaaa=aa, abba=b⟩Noc 10
1112181⟨a, b | aaaa=ab, baaa=b⟩Noc 101
1112196⟨a, b | aaaa=bb, aaab=a⟩Noc Com 107
1112197⟨a, b | aaaa=bb, aaab=b⟩Noc Com 101
1112452⟨a, b | aabb=aa, bbbb=a⟩Noc Com 10ℕ(10 = 8)3
1115738⟨a, b | aab=bb, aaaaa=b⟩Noc Com 10ℕ(10 = 7)7
1119564⟨a, b | aab=b, abbbb=aa⟩Noc Com 105
1119899⟨a, b | aaa=b, abbb=bbb⟩Noc Com 10ℕ(10 = 9)1
1120052⟨a, b | aab=b, aaaa=baa⟩Noc 10
1120253⟨a, b | aba=b, aaaa=bbb⟩Noc Com 10
1125905⟨a, b | ab=a, bbbb=baaa⟩Noc 10
1125909⟨a, b | ab=a, bbbb=bbaa⟩Noc 10

Isomorphic instances

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

20 total

Σ#PresentationMapping
104639⟨a, b | aaaa=b, abab=b⟩φ(a) = a, φ(b) = aaaa
104641⟨a, b | aaaa=b, abba=b⟩φ(a) = a, φ(b) = aaaa
104645⟨a, b | aaaa=b, baab=b⟩φ(a) = a, φ(b) = aaaa
106585⟨a, b | aaa=b, abbb=ab⟩φ(a) = a, φ(b) = aaa
106586⟨a, b | aaa=b, abbb=ba⟩φ(a) = a, φ(b) = aaa
106592⟨a, b | aaa=b, babb=ab⟩φ(a) = a, φ(b) = aaa
106593⟨a, b | aaa=b, babb=ba⟩φ(a) = a, φ(b) = aaa
106658⟨a, b | aab=a, bbbb=aa⟩φ(a) = aaaaa, φ(b) = a
108983⟨a, b | aa=b, bbbbb=bb⟩φ(a) = a, φ(b) = aa
1124781⟨a, b | aa=b, aabbbb=bb⟩φ(a) = a, φ(b) = aa
1124804⟨a, b | aa=b, ababbb=bb⟩φ(a) = a, φ(b) = aa
1124812⟨a, b | aa=b, abbabb=bb⟩φ(a) = a, φ(b) = aa
1124816⟨a, b | aa=b, abbbab=bb⟩φ(a) = a, φ(b) = aa
1124819⟨a, b | aa=b, abbbba=bb⟩φ(a) = a, φ(b) = aa
1124838⟨a, b | aa=b, baabbb=bb⟩φ(a) = a, φ(b) = aa
1124842⟨a, b | aa=b, bababb=bb⟩φ(a) = a, φ(b) = aa
1124845⟨a, b | aa=b, babbab=bb⟩φ(a) = a, φ(b) = aa
1124852⟨a, b | aa=b, bbaabb=bb⟩φ(a) = a, φ(b) = aa
1125399⟨a, b | aa=b, bbbbb=aab⟩φ(a) = a, φ(b) = aa
1125400⟨a, b | aa=b, bbbbb=aba⟩φ(a) = a, φ(b) = aa