#4630 ⟨a, b | aaaa=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.

1bb2b3b4b5b6b7b8b9b10b11b12b13b14b15
11bb2b3b4b5b6b7b8b9b10b11b12b13b14b15
bbb2b3b4b5b6b7b8b9b10b11b12b13b14b15b4
b2b2b3b4b5b6b7b8b9b10b11b12b13b14b15b4b5
b3b3b4b5b6b7b8b9b10b11b12b13b14b15b4b5b6
b4b4b5b6b7b8b9b10b11b12b13b14b15b4b5b6b7
b5b5b6b7b8b9b10b11b12b13b14b15b4b5b6b7b8
b6b6b7b8b9b10b11b12b13b14b15b4b5b6b7b8b9
b7b7b8b9b10b11b12b13b14b15b4b5b6b7b8b9b10
b8b8b9b10b11b12b13b14b15b4b5b6b7b8b9b10b11
b9b9b10b11b12b13b14b15b4b5b6b7b8b9b10b11b12
b10b10b11b12b13b14b15b4b5b6b7b8b9b10b11b12b13
b11b11b12b13b14b15b4b5b6b7b8b9b10b11b12b13b14
b12b12b13b14b15b4b5b6b7b8b9b10b11b12b13b14b15
b13b13b14b15b4b5b6b7b8b9b10b11b12b13b14b15b4
b14b14b15b4b5b6b7b8b9b10b11b12b13b14b15b4b5
b15b15b4b5b6b7b8b9b10b11b12b13b14b15b4b5b6

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. b16 ⇒ b4 [4]
2. a ⇒ b4 [2]
# ab:aaaa=a,bbbb=a b/a - -
bbbbbbbbbbbbbbbb=bbbb
a=bbbb

Same cardinality

20 unique, 174 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
103808⟨a, b | aaab=ba, abab=1⟩Grp 167
104648⟨a, b | aaaa=b, bbbb=a⟩Noc Com 16ℕ(16 = 1)5
106205⟨a, b | aba=b, aaaabb=1⟩Grp 163
1112164⟨a, b | aaaa=aa, bbbb=a⟩Noc Com 16ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Noc Com 16ℕ(16 = 5)2
1112212⟨a, b | aaaa=bb, bbbb=a⟩Noc Com 16ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Noc 166
1113251⟨a, b | bab=aab, bbb=aa⟩Noc 16
1113259⟨a, b | bab=aba, bbb=aa⟩Noc 16
1116028⟨a, b | aaa=ab, babb=bb⟩Noc 16
1116032⟨a, b | aaa=ab, bbaa=bb⟩Noc 16
1116060⟨a, b | aaa=bb, abab=aa⟩Noc 16
1116371⟨a, b | aba=aa, bbbb=ab⟩Noc 16
1118811⟨a, b | aaa=b, abbbbb=b⟩Noc Com 16ℕ(16 = 3)2
1120251⟨a, b | aba=b, aaaa=abb⟩Noc 16
1121039⟨a, b | ab=aa, bbba=bbb⟩Noc 16
1124660⟨a, b | aa=a, ababab=bb⟩Noc 16

Isomorphic instances

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

1 total

Σ#PresentationMapping
1112213⟨a, b | aaaa=bb, bbbb=b⟩φ(a) = b, φ(b) = bbbbbbbb