#1907 ⟨a, b | aab=a, bbbbb=1⟩

Up:Monoids with two generators and two relations
Prev:#1855 ⟨a, b | aaa=a, bbbbb=1⟩
Next:#1908 ⟨a, b | aab=b, aaaaa=1⟩

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. a6 ⇒ a
  2. ab ⇒ a5
  3. b5 ⇒ 1 
# ab:aab=a,bbbbb=1 a/b
aaaaaa=a
ab=aaaaa
bbbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

8 unique, 108 total

Σ#PresentationDescriptionRelated
91470⟨a, b | aaa=bb, abab=1⟩Finite non-Abelian group with 30 elements73 iso
92443⟨a, b | aaa=1, abbbb=b⟩Finite non-commutative monoid with 30 elements14 iso, 5 anti-iso
106560⟨a, b | aaa=a, bbbb=ab⟩Finite non-commutative monoid with 30 elements
107013⟨a, b | ab=aa, bbbbb=b⟩Finite non-commutative monoid with 30 elements
1111058⟨a, b | abba=bbb, baba=1⟩Finite non-Abelian group with 30 elements8 iso
1113115⟨a, b | bbb=aaa, aaba=b⟩Finite non-commutative monoid with 30 elements
1115979⟨a, b | aaa=aa, bbbb=ab⟩Finite non-commutative monoid with 30 elements
1120847⟨a, b | ab=aa, bbbbb=bb⟩Finite non-commutative monoid with 30 elements

Other isomorphic instances

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

11 total

Σ#PresentationMapping
92155⟨a, b | ab=aa, bbbbb=1⟩φ(a) = bba, φ(b) = b
104541⟨a, b | aabb=a, bbbbb=1⟩φ(a) = aa, φ(b) = b
104573⟨a, b | abab=a, bbbbb=1⟩φ(a) = aa, φ(b) = b
105001⟨a, b | aba=ab, bbbbb=1⟩φ(a) = ba, φ(b) = b
1111448⟨a, b | aabbb=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1111533⟨a, b | ababb=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1111549⟨a, b | abbab=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1112096⟨a, b | abba=ab, aaaaa=1⟩φ(a) = b, φ(b) = a
1112127⟨a, b | abba=ab, bbbbb=1⟩φ(a) = a, φ(b) = b
1112772⟨a, b | abb=aba, aaaaa=1⟩φ(a) = b, φ(b) = bba
1112803⟨a, b | abb=aba, bbbbb=1⟩φ(a) = bba, φ(b) = b

Other anti-isomorphic instances

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

4 total

Σ#PresentationMapping
104918⟨a, b | aab=bb, aaaaa=1⟩φ(a) = bb, φ(b) = a
1111788⟨a, b | aaab=bb, aaaaa=1⟩φ(a) = b, φ(b) = aa
1111980⟨a, b | aabb=ab, aaaaa=1⟩φ(a) = b, φ(b) = a
1112756⟨a, b | abb=aab, aaaaa=1⟩φ(a) = bbbb, φ(b) = a