#1998 ⟨a, b | aaa=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.

1bb2b3b4b5b6b7b8b9b10b11
11bb2b3b4b5b6b7b8b9b10b11
bbb2b3b4b5b6b7b8b9b10b11b4
b2b2b3b4b5b6b7b8b9b10b11b4b5
b3b3b4b5b6b7b8b9b10b11b4b5b6
b4b4b5b6b7b8b9b10b11b4b5b6b7
b5b5b6b7b8b9b10b11b4b5b6b7b8
b6b6b7b8b9b10b11b4b5b6b7b8b9
b7b7b8b9b10b11b4b5b6b7b8b9b10
b8b8b9b10b11b4b5b6b7b8b9b10b11
b9b9b10b11b4b5b6b7b8b9b10b11b4
b10b10b11b4b5b6b7b8b9b10b11b4b5
b11b11b4b5b6b7b8b9b10b11b4b5b6

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. b12 ⇒ b4 [4]
2. a ⇒ b4 [2]
# ab:aaa=a,bbbb=a b/a - -
bbbbbbbbbbbb=bbbb
a=bbbb

Same cardinality

37 unique, 653 total

Σ#PresentationPropertiesDescriptionφ
7124⟨a, b | aab=a, bbb=1⟩Noc 1261
8458⟨a, b | aaaa=1, abbb=1⟩Grp Com 12ℤ12325
8645⟨a, b | ab=aa, bbb=b⟩Noc 121
91646⟨a, b | aab=bb, aba=a⟩Noc 122
91963⟨a, b | aba=b, aaabb=1⟩Grp 1230
92018⟨a, b | aaa=b, bbbb=a⟩Noc Com 12ℕ(12 = 1)27
92019⟨a, b | aaa=b, bbbb=b⟩Noc Com 12ℕ(12 = 3)23
92259⟨a, b | ab=aa, bbb=bb⟩Noc 12
105040⟨a, b | aaa=aa, bbbb=a⟩Noc Com 12ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Noc Com 12ℕ(12 = 5)7
105092⟨a, b | aaa=bb, bbbb=a⟩Noc Com 12ℕ(12 = 2)43
105202⟨a, b | aab=bb, abba=a⟩Noc 1241
105330⟨a, b | aaa=ab, bab=bb⟩Noc 121
106597⟨a, b | aaa=b, bbbb=bb⟩Noc Com 12ℕ(12 = 6)4
106660⟨a, b | aab=a, bbbb=ba⟩Noc 121
106661⟨a, b | aab=a, bbbb=bb⟩Noc 121
107105⟨a, b | ab=aa, bbaa=bb⟩Noc 122
1112897⟨a, b | aab=aaa, baaa=b⟩Noc 125
1112910⟨a, b | aab=aaa, bbbb=a⟩Noc Com 12ℕ(12 = 9)2
1115428⟨a, b | aaa=aa, abbbb=b⟩Noc 12
1115996⟨a, b | aaa=ab, aabb=bb⟩Noc 122
1116057⟨a, b | aaa=bb, aabb=ab⟩Noc 124
1116104⟨a, b | aab=aa, abab=bb⟩Noc 12
1116274⟨a, b | aab=bb, aaaa=ab⟩Noc 121
1116275⟨a, b | aab=bb, aaaa=ba⟩Noc 12
1116448⟨a, b | aba=bb, aabb=aa⟩Noc 122
1119773⟨a, b | aba=b, bbbbb=aa⟩Noc Com 12
1119917⟨a, b | aaa=b, bbbb=abb⟩Noc Com 12ℕ(12 = 7)2
1120054⟨a, b | aab=b, aaaa=bba⟩Noc 12
1120686⟨a, b | bb=aa, aaaaab=a⟩Noc Com 1215
1120787⟨a, b | ab=aa, baaaa=bb⟩Noc 127
1121086⟨a, b | bb=aa, aaaa=aba⟩Noc 124
1121092⟨a, b | bb=aa, aaab=aba⟩Noc 123
1121112⟨a, b | bb=aa, abab=aba⟩Noc 12
1124147⟨a, b | aa=a, abbbbbb=b⟩Noc 12
1124991⟨a, b | ab=a, baaaaa=bb⟩Noc 12
1125603⟨a, b | ab=a, bbaaa=bbb⟩Noc 12

Isomorphic instances

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

6 total

Σ#PresentationMapping
106596⟨a, b | aaa=b, bbbb=ab⟩φ(a) = b, φ(b) = bbb
1112201⟨a, b | aaaa=bb, aabb=b⟩φ(a) = b, φ(b) = bbbbbb
1112203⟨a, b | aaaa=bb, abab=b⟩φ(a) = b, φ(b) = bbbbbb
1112205⟨a, b | aaaa=bb, abba=b⟩φ(a) = b, φ(b) = bbbbbb
1119842⟨a, b | aaa=a, bbbb=aaa⟩φ(a) = bbbb, φ(b) = b
1124859⟨a, b | aa=b, bbbbbb=bb⟩φ(a) = b, φ(b) = bb