#950 ⟨a, b | ab=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

The zero element z satisfies zy = yz = z for all y.
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.

1bb2b3b4
11bb2b3b4
bbb2b3b4b4
b2b2b3b4b4b4
b3b3b4b4b4b4
b4b4b4b4b4b4

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. b5 ⇒ b4 [4]
2. a ⇒ b4 [2]
# ab:ab=a,bbbb=a b/a - -
bbbbb=bbbb
a=bbbb

Same cardinality

11 unique, 1430 total

Σ#PresentationPropertiesDescriptionφ
644⟨a, b | aa=b, abb=1⟩Grp Com 5ℤ51132
7253⟨a, b | aa=b, abb=a⟩Noc Com 5ℕ(5 = 1)71
7254⟨a, b | aa=b, abb=b⟩Noc Com 5ℕ(5 = 2)43
7268⟨a, b | ab=a, baa=b⟩Noc 586
8574⟨a, b | aaa=b, aab=b⟩Noc Com 5ℕ(5 = 3)27
8995⟨a, b | ab=a, aaa=bb⟩Noc Com 519
81019⟨a, b | ab=a, bba=bb⟩Noc 525
81020⟨a, b | ab=a, bbb=aa⟩Noc Com 59
81022⟨a, b | ab=a, bbb=ba⟩Noc 54
108617⟨a, b | aa=a, abbbba=b⟩Noc Com 53
1115426⟨a, b | aaa=aa, abbba=b⟩Noc Com 5

Isomorphic instances

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

32 total

Σ#PresentationMapping
8985⟨a, b | aa=b, abb=bb⟩φ(a) = b, φ(b) = bb
8988⟨a, b | aa=b, bab=bb⟩φ(a) = b, φ(b) = bb
91688⟨a, b | abb=ab, bbb=a⟩φ(a) = bbb, φ(b) = b
91690⟨a, b | abb=ba, bbb=a⟩φ(a) = bbb, φ(b) = b
91696⟨a, b | bab=ab, bbb=a⟩φ(a) = bbb, φ(b) = b
93042⟨a, b | aa=b, aaab=bb⟩φ(a) = b, φ(b) = bb
93046⟨a, b | aa=b, aaba=bb⟩φ(a) = b, φ(b) = bb
93133⟨a, b | ab=a, bbbb=ab⟩φ(a) = bbbb, φ(b) = b
93158⟨a, b | aa=b, abb=aab⟩φ(a) = b, φ(b) = bb
93159⟨a, b | aa=b, abb=aba⟩φ(a) = b, φ(b) = bb
93161⟨a, b | aa=b, baa=abb⟩φ(a) = b, φ(b) = bb
93163⟨a, b | aa=b, bab=aab⟩φ(a) = b, φ(b) = bb
93164⟨a, b | aa=b, bab=aba⟩φ(a) = b, φ(b) = bb
108914⟨a, b | aa=b, aaaaa=bb⟩φ(a) = b, φ(b) = bb
109187⟨a, b | aa=b, aaaa=abb⟩φ(a) = b, φ(b) = bb
109188⟨a, b | aa=b, aaaa=bab⟩φ(a) = b, φ(b) = bb
109191⟨a, b | aa=b, aaab=aab⟩φ(a) = b, φ(b) = bb
109192⟨a, b | aa=b, aaab=aba⟩φ(a) = b, φ(b) = bb
109194⟨a, b | aa=b, aaab=baa⟩φ(a) = b, φ(b) = bb
109199⟨a, b | aa=b, aaba=aab⟩φ(a) = b, φ(b) = bb
109200⟨a, b | aa=b, aaba=aba⟩φ(a) = b, φ(b) = bb
109202⟨a, b | aa=b, aaba=baa⟩φ(a) = b, φ(b) = bb
109379⟨a, b | ab=a, bbbb=abb⟩φ(a) = bbbb, φ(b) = b
1114370⟨a, b | aaaa=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbb
1119305⟨a, b | aaa=b, aaaaa=ab⟩φ(a) = b, φ(b) = bbb
1119849⟨a, b | aaa=b, aaaa=aab⟩φ(a) = b, φ(b) = bbb
1119850⟨a, b | aaa=b, aaaa=aba⟩φ(a) = b, φ(b) = bbb
1125261⟨a, b | aa=b, aaaaa=aab⟩φ(a) = b, φ(b) = bb
1125262⟨a, b | aa=b, aaaaa=aba⟩φ(a) = b, φ(b) = bb
1125726⟨a, b | aa=b, aaab=aaaa⟩φ(a) = b, φ(b) = bb
1125727⟨a, b | aa=b, aaba=aaaa⟩φ(a) = b, φ(b) = bb
1125904⟨a, b | ab=a, bbbb=abbb⟩φ(a) = bbbb, φ(b) = b