#5300 ⟨a, b | aaa=aa, aba=bb⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality

Properties

Elements

Elements in the center commute with all other 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.

Right Cayley graph

Left Cayley graph

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. a3 ⇒ a2 [1]
2. b2 ⇒ aba [2]
3. (ab)2 ⇒ (ba)2 [3]
4. baba2b ⇒ (a2b)2a [4]
5. ab(a2b)2 ⇒ b(a2b)2a [5]
# ab:aaa=aa,aba=bb a/b
aaa=aa
bb=aba
abab=baba
babaab=aabaaba
abaabaab=baabaaba

Same cardinality

6 unique, 93 total

Σ#PresentationDescriptionRelated
8450⟨a, b | bab=aa, bbb=1⟩Finite non-commutative monoid with 27 elements27 iso
8752⟨a, b | aaa=1, babb=a⟩Finite non-Abelian group with 27 elements59 iso
91695⟨a, b | bab=aa, bbb=b⟩Finite non-commutative monoid with 27 elements1 iso
1115514⟨a, b | aaa=bb, aabaa=b⟩Finite non-commutative monoid with 27 elements
1118739⟨a, b | aaa=a, abbbbb=b⟩Finite non-commutative monoid with 27 elements
1120314⟨a, b | aba=b, bbbb=aaa⟩Finite non-commutative monoid with 27 elements