#1584 ⟨a, b | aaa=aa, bbb=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.

1bb2b3b4b5b6b7b8
11bb2b3b4b5b6b7b8
bbb2b3b4b5b6b7b8b6
b2b2b3b4b5b6b7b8b6b7
b3b3b4b5b6b7b8b6b7b8
b4b4b5b6b7b8b6b7b8b6
b5b5b6b7b8b6b7b8b6b7
b6b6b7b8b6b7b8b6b7b8
b7b7b8b6b7b8b6b7b8b6
b8b8b6b7b8b6b7b8b6b7

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. b9 ⇒ b6 [4]
2. a ⇒ b3 [2]
# ab:aaa=aa,bbb=a b/a - -
bbbbbbbbb=bbbbbb
a=bbb

Same cardinality

26 unique, 926 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

16 total

Σ#PresentationMapping
1112997⟨a, b | abb=aab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1113013⟨a, b | abb=aba, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1113042⟨a, b | abb=aba, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1113065⟨a, b | baa=abb, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1113101⟨a, b | bab=aba, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1119318⟨a, b | aaa=b, aaabb=bb⟩φ(a) = b, φ(b) = bbb
1119325⟨a, b | aaa=b, aabab=bb⟩φ(a) = b, φ(b) = bbb
1119329⟨a, b | aaa=b, aabba=bb⟩φ(a) = b, φ(b) = bbb
1119337⟨a, b | aaa=b, abaab=bb⟩φ(a) = b, φ(b) = bbb
1119340⟨a, b | aaa=b, ababa=bb⟩φ(a) = b, φ(b) = bbb
1119358⟨a, b | aaa=b, baaab=bb⟩φ(a) = b, φ(b) = bbb
1119861⟨a, b | aaa=b, aaab=bbb⟩φ(a) = b, φ(b) = bbb
1119869⟨a, b | aaa=b, aaba=bbb⟩φ(a) = b, φ(b) = bbb
1125363⟨a, b | aa=b, abbbb=bbb⟩φ(a) = b, φ(b) = bb
1125391⟨a, b | aa=b, babbb=bbb⟩φ(a) = b, φ(b) = bb
1125397⟨a, b | aa=b, bbabb=bbb⟩φ(a) = b, φ(b) = bb