#4623 ⟨a, b | aaaa=a, abba=b⟩

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Quick links

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

Properties

Elements

Elements in the center commute with all other elements.
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. ba2 ⇒ ab2 [6]
2. b2a ⇒ a2b [5]
3. a4 ⇒ a [1]
4. a3b ⇒ b [3]
5. bab2 ⇒ a2ba [7]
6. b5 ⇒ a(ba)2 [9]
7. (ba)3 ⇒ (ab)3 [10]
# ab:aaaa=a,abba=b ab
baa=abb
bba=aab
aaaa=a
aaab=b
babb=aaba
bbbbb=ababa
bababa=ababab

Same cardinality

2 unique, 7 total

Σ#PresentationDescriptionRelated
104625⟨a, b | aaaa=a, abbb=b⟩Finite non-commutative monoid with 28 elements2 iso
104691⟨a, b | aaab=b, babb=a⟩Finite non-commutative monoid with 28 elements3 iso

Isomorphic instances

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

1 total

Σ#PresentationMapping
1114348⟨a, b | aaaa=a, ababa=b⟩φ(a) = a, φ(b) = aab