#950 ⟨a, b | ab=a, bbbb=a⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

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 to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 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#PresentationDescriptionRelated
644⟨a, b | aa=b, abb=1⟩Isomorphic to ℤ51132 iso
7253⟨a, b | aa=b, abb=a⟩Isomorphic to ℕ(5 = 1)71 iso
7254⟨a, b | aa=b, abb=b⟩Isomorphic to ℕ(5 = 2)43 iso
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-iso
8574⟨a, b | aaa=b, aab=b⟩Isomorphic to ℕ(5 = 3)27 iso
8995⟨a, b | ab=a, aaa=bb⟩Finite commutative monoid with 5 elements19 iso
81019⟨a, b | ab=a, bba=bb⟩Finite non-commutative monoid with 5 elements25 iso
81020⟨a, b | ab=a, bbb=aa⟩Finite commutative monoid with 5 elements9 iso
81022⟨a, b | ab=a, bbb=ba⟩Finite non-commutative monoid with 5 elements4 iso
108617⟨a, b | aa=a, abbbba=b⟩Finite commutative monoid with 5 elements3 iso
1115426⟨a, b | aaa=aa, abbba=b⟩Finite commutative monoid with 5 elements

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