#9747 ⟨a, b | aa=1, abbabba=b⟩

Up:Monoids with two generators and two relations
Prev:#9739 ⟨a, b | aa=1, ababbbb=b⟩
Next:#9749 ⟨a, b | aa=1, abbabbb=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. b13 ⇒ b
  2. bab12 ⇒ ba
  3. b3a ⇒ ab3
  4. a2 ⇒ 1
  5. aba ⇒ b2ab2
  6. ab2a ⇒ bab10 
# ab:aa=1,abbabba=b b/a
bbbbbbbbbbbbb=b
babbbbbbbbbbbb=ba
bbba=abbb
aa=1
aba=bbabb
abba=babbbbbbbbbb

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 5 total

Σ#PresentationDescriptionRelated
109757⟨a, b | aa=1, abbbbba=b⟩Finite non-commutative monoid with 50 elements2 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.

5 total

Σ#PresentationMapping
1010015⟨a, b | aa=1, abbabb=ba⟩φ(a) = a, φ(b) = b
1010334⟨a, b | aa=1, bbabb=aba⟩φ(a) = a, φ(b) = b
1126593⟨a, b | aa=1, ababbaba=b⟩φ(a) = a, φ(b) = ab
1127120⟨a, b | aa=1, ababbab=ba⟩φ(a) = a, φ(b) = ab
1127736⟨a, b | aa=1, babbab=aba⟩φ(a) = a, φ(b) = ab