#898 ⟨a, b | aa=a, bbbb=a⟩

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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.

1bb2b3b4b5b6b7
11bb2b3b4b5b6b7
bbb2b3b4b5b6b7b4
b2b2b3b4b5b6b7b4b5
b3b3b4b5b6b7b4b5b6
b4b4b5b6b7b4b5b6b7
b5b5b6b7b4b5b6b7b4
b6b6b7b4b5b6b7b4b5
b7b7b4b5b6b7b4b5b6

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. b8 ⇒ b4 [4]
2. a ⇒ b4 [2]
# ab:aa=a,bbbb=a b/a - -
bbbbbbbb=bbbb
a=bbbb

Same cardinality

33 unique, 1163 total

Σ#PresentationPropertiesDescriptionφ
7188⟨a, b | aab=1, bbbb=1⟩Grp Com 8ℤ8727
7330⟨a, b | aa=1, abba=b⟩Noc 860
8556⟨a, b | aba=b, aabb=1⟩Grp 828
8918⟨a, b | aa=b, bbbb=a⟩Noc Com 8ℕ(8 = 1)34
8919⟨a, b | aa=b, bbbb=b⟩Noc Com 8ℕ(8 = 2)46
8961⟨a, b | aa=a, aba=bb⟩Noc 85
91605⟨a, b | aaa=bb, abb=b⟩Noc Com 8ℕ(8 = 3)55
91606⟨a, b | aaa=bb, bab=a⟩Noc Com 814
91615⟨a, b | aab=aa, baa=b⟩Noc 814
91650⟨a, b | aab=bb, baa=a⟩Noc 815
92206⟨a, b | ab=aa, bbbb=a⟩Noc Com 8ℕ(8 = 5)32
92220⟨a, b | bb=aa, aaab=a⟩Noc Com 822
92247⟨a, b | ab=aa, baa=bb⟩Noc 81
92256⟨a, b | ab=aa, bbb=aa⟩Noc 81
92258⟨a, b | ab=aa, bbb=ba⟩Noc 8
92883⟨a, b | aa=a, abbbb=b⟩Noc 814
93107⟨a, b | ab=a, baaa=bb⟩Noc 89
93123⟨a, b | ab=a, bbaa=bb⟩Noc 810
105191⟨a, b | aab=bb, aaaa=b⟩Noc Com 8ℕ(8 = 6)18
106664⟨a, b | aab=b, aaaa=ba⟩Noc 81
107057⟨a, b | ab=aa, aaaa=bb⟩Noc 87
109380⟨a, b | ab=a, bbbb=baa⟩Noc 82
1112606⟨a, b | abbb=aa, bbbb=a⟩Noc Com 8ℕ(8 = 7)14
1120047⟨a, b | aab=a, bbbb=bbb⟩Noc 8
1120051⟨a, b | aab=b, aaaa=abb⟩Noc Com 81
1124863⟨a, b | ab=a, aaaaaa=bb⟩Noc Com 8
1125112⟨a, b | ab=a, bbbbbb=aa⟩Noc Com 8
1125114⟨a, b | ab=a, bbbbbb=ba⟩Noc 8
1125411⟨a, b | ab=a, aaaaa=bbb⟩Noc Com 8
1125652⟨a, b | ab=a, bbbbb=aaa⟩Noc Com 8
1125658⟨a, b | ab=a, bbbbb=bba⟩Noc 8
1125897⟨a, b | ab=a, bbbb=aaaa⟩Noc Com 8
1125911⟨a, b | ab=a, bbbb=bbba⟩Noc 8

Isomorphic instances

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

48 total

Σ#PresentationMapping
93033⟨a, b | aa=a, bbbb=aa⟩φ(a) = bbbb, φ(b) = b
93071⟨a, b | aa=b, bbbb=bb⟩φ(a) = b, φ(b) = bb
106574⟨a, b | aaa=b, aabb=ab⟩φ(a) = b, φ(b) = bbb
106575⟨a, b | aaa=b, aabb=ba⟩φ(a) = b, φ(b) = bbb
106578⟨a, b | aaa=b, abab=ab⟩φ(a) = b, φ(b) = bbb
106579⟨a, b | aaa=b, abab=ba⟩φ(a) = b, φ(b) = bbb
106582⟨a, b | aaa=b, abba=ab⟩φ(a) = b, φ(b) = bbb
106589⟨a, b | aaa=b, baab=ab⟩φ(a) = b, φ(b) = bbb
106659⟨a, b | aab=a, bbbb=ab⟩φ(a) = bbbbbbb, φ(b) = b
108941⟨a, b | aa=b, aabbb=bb⟩φ(a) = b, φ(b) = bb
108952⟨a, b | aa=b, ababb=bb⟩φ(a) = b, φ(b) = bb
108956⟨a, b | aa=b, abbab=bb⟩φ(a) = b, φ(b) = bb
108959⟨a, b | aa=b, abbba=bb⟩φ(a) = b, φ(b) = bb
108970⟨a, b | aa=b, baabb=bb⟩φ(a) = b, φ(b) = bb
108973⟨a, b | aa=b, babab=bb⟩φ(a) = b, φ(b) = bb
109178⟨a, b | aa=a, bbbb=aaa⟩φ(a) = bbbb, φ(b) = b
109251⟨a, b | aa=b, bbbb=aab⟩φ(a) = b, φ(b) = bb
109252⟨a, b | aa=b, bbbb=aba⟩φ(a) = b, φ(b) = bb
1114372⟨a, b | aaaa=b, aaaab=b⟩φ(a) = b, φ(b) = bbbb
1114374⟨a, b | aaaa=b, aaaba=b⟩φ(a) = b, φ(b) = bbbb
1114378⟨a, b | aaaa=b, aabaa=b⟩φ(a) = b, φ(b) = bbbb
1124738⟨a, b | aa=b, aaaabb=bb⟩φ(a) = b, φ(b) = bb
1124746⟨a, b | aa=b, aaabab=bb⟩φ(a) = b, φ(b) = bb
1124750⟨a, b | aa=b, aaabba=bb⟩φ(a) = b, φ(b) = bb
1124758⟨a, b | aa=b, aabaab=bb⟩φ(a) = b, φ(b) = bb
1124762⟨a, b | aa=b, aababa=bb⟩φ(a) = b, φ(b) = bb
1124769⟨a, b | aa=b, aabbaa=bb⟩φ(a) = b, φ(b) = bb
1124785⟨a, b | aa=b, abaaab=bb⟩φ(a) = b, φ(b) = bb
1124788⟨a, b | aa=b, abaaba=bb⟩φ(a) = b, φ(b) = bb
1124826⟨a, b | aa=b, baaaab=bb⟩φ(a) = b, φ(b) = bb
1125313⟨a, b | aa=b, aabbb=aab⟩φ(a) = b, φ(b) = bb
1125314⟨a, b | aa=b, aabbb=aba⟩φ(a) = b, φ(b) = bb
1125316⟨a, b | aa=b, aabbb=baa⟩φ(a) = b, φ(b) = bb
1125335⟨a, b | aa=b, ababb=aab⟩φ(a) = b, φ(b) = bb
1125336⟨a, b | aa=b, ababb=aba⟩φ(a) = b, φ(b) = bb
1125338⟨a, b | aa=b, ababb=baa⟩φ(a) = b, φ(b) = bb
1125343⟨a, b | aa=b, abbab=aab⟩φ(a) = b, φ(b) = bb
1125344⟨a, b | aa=b, abbab=aba⟩φ(a) = b, φ(b) = bb
1125346⟨a, b | aa=b, abbab=baa⟩φ(a) = b, φ(b) = bb
1125351⟨a, b | aa=b, abbba=aab⟩φ(a) = b, φ(b) = bb
1125352⟨a, b | aa=b, abbba=aba⟩φ(a) = b, φ(b) = bb
1125371⟨a, b | aa=b, baabb=aab⟩φ(a) = b, φ(b) = bb
1125372⟨a, b | aa=b, baabb=aba⟩φ(a) = b, φ(b) = bb
1125374⟨a, b | aa=b, baabb=baa⟩φ(a) = b, φ(b) = bb
1125379⟨a, b | aa=b, babab=aab⟩φ(a) = b, φ(b) = bb
1125380⟨a, b | aa=b, babab=aba⟩φ(a) = b, φ(b) = bb
1125717⟨a, b | aa=a, bbbb=aaaa⟩φ(a) = bbbb, φ(b) = b
1125783⟨a, b | aa=b, bbbb=aaaa⟩φ(a) = b, φ(b) = bb