#2300 ⟨a, b | aaa=1, ababbb=1⟩

Up:Monoids with two generators and two relations
Prev:#2298 ⟨a, b | aaa=1, ababab=1⟩
Next:#2302 ⟨a, b | aaa=1, abbabb=1⟩

Properties

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

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. b12 ⇒ 1
  2. b2a ⇒ ab2
  3. a2 ⇒ bab3
  4. aba ⇒ b9 
# ab:aaa=1,ababbb=1 b/a
bbbbbbbbbbbb=1
bba=abb
aa=babbb
aba=bbbbbbbbb

Right Cayley graph

Left Cayley graph

Others with same cardinality

4 unique, 18 total

Σ#PresentationDescriptionRelated
91345⟨a, b | aaab=a, bbbb=1⟩Finite non-commutative monoid with 36 elements5 iso, 8 anti-iso
105073⟨a, b | aaa=ab, bbbb=b⟩Finite non-commutative monoid with 36 elements1 iso
1116044⟨a, b | aaa=ab, bbbb=bb⟩Finite non-commutative monoid with 36 elements
1118959⟨a, b | aab=a, bbbbbb=b⟩Finite non-commutative monoid with 36 elements

Other isomorphic instances

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

23 total

Σ#PresentationMapping
92303⟨a, b | aaa=1, abbbab=1⟩φ(a) = a, φ(b) = b
92310⟨a, b | aaa=1, bababb=1⟩φ(a) = a, φ(b) = b
106946⟨a, b | bb=aa, ababab=1⟩φ(a) = abbb, φ(b) = b
107538⟨a, b | aaa=1, baabbb=a⟩φ(a) = abbbb, φ(b) = b
107814⟨a, b | aaa=1, babbb=aa⟩φ(a) = a, φ(b) = b
1113821⟨a, b | aaaa=a, ababbb=1⟩φ(a) = a, φ(b) = b
1113824⟨a, b | aaaa=a, abbbab=1⟩φ(a) = a, φ(b) = b
1113831⟨a, b | aaaa=a, bababb=1⟩φ(a) = a, φ(b) = b
1113960⟨a, b | aaab=b, ababbb=1⟩φ(a) = a, φ(b) = b
1113966⟨a, b | aaab=b, abbbab=1⟩φ(a) = a, φ(b) = b
1113980⟨a, b | aaab=b, bababb=1⟩φ(a) = a, φ(b) = b
1113983⟨a, b | aaab=b, babbba=1⟩φ(a) = a, φ(b) = b
1113990⟨a, b | aaab=b, bbabab=1⟩φ(a) = a, φ(b) = b
1113995⟨a, b | aaab=b, bbbaba=1⟩φ(a) = a, φ(b) = b
1121157⟨a, b | aaa=1, aabaabbb=1⟩φ(a) = abbbb, φ(b) = b
1121171⟨a, b | aaa=1, aabbbaab=1⟩φ(a) = abbbb, φ(b) = b
1121188⟨a, b | aaa=1, abaabbba=1⟩φ(a) = abbbb, φ(b) = b
1121228⟨a, b | aaa=1, baabaabb=1⟩φ(a) = abbbb, φ(b) = b
1121692⟨a, b | aaa=1, aababbb=a⟩φ(a) = a, φ(b) = b
1121702⟨a, b | aaa=1, aabbbab=a⟩φ(a) = a, φ(b) = b
1121724⟨a, b | aaa=1, abababb=a⟩φ(a) = a, φ(b) = b
1121728⟨a, b | aaa=1, ababbba=a⟩φ(a) = a, φ(b) = b
1121736⟨a, b | aaa=1, abbabab=a⟩φ(a) = a, φ(b) = b