#9757 ⟨a, b | aa=1, abbbbba=b⟩

Up:Monoids with two generators and two relations
Prev:#9755 ⟨a, b | aa=1, abbbbab=b⟩
Next:#9759 ⟨a, b | aa=1, abbbbbb=b⟩

Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Complete 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
  1. b25 ⇒ b
  2. ba ⇒ ab5
  3. a2 ⇒ 1 
# ab:aa=1,abbbbba=b b/a
bbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbbbb
aa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 8 total

Σ#PresentationDescriptionRelated
109747⟨a, b | aa=1, abbabba=b⟩Finite non-commutative monoid with 50 elements5 iso
1010301⟨a, b | aa=1, abbbb=bab⟩Finite non-commutative monoid with 50 elements1 iso

Other isomorphic instances

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

2 total

Σ#PresentationMapping
1010026⟨a, b | aa=1, abbbbb=ba⟩φ(a) = a, φ(b) = b
1010340⟨a, b | aa=1, bbbbb=aba⟩φ(a) = a, φ(b) = b