#160 ⟨a, b | ab=aa, bb=b⟩

Up:Monoids with two generators and two relations
Prev:#159 ⟨a, b | ab=aa, bb=a⟩
Next:#162 ⟨a, b | ba=ab, bb=b⟩

Properties

Element profile

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. ab ⇒ a2
  2. b2 ⇒ b
  3. a3 ⇒ a2 
# ab:ab=aa,bb=b ab
ab=aa
bb=b
aaa=aa

Cayley table

Idempotents are shown in bold.

1aba2baba2
11aba2baba2
aaa2a2a2a2a2
bbbabba2baba2
a2a2a2a2a2a2a2
bababa2ba2ba2ba2ba2
ba2ba2ba2ba2ba2ba2ba2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 1921 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7245⟨a, b | aa=a, bbb=a⟩Isomorphic to ℕ(6 = 3)49 iso
7257⟨a, b | aa=b, bbb=a⟩Isomorphic to ℕ(6 = 1)61 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91427⟨a, b | abba=b, baba=1⟩Finite non-Abelian group with 6 elements66 iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Other anti-isomorphic instances

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

4 total

Σ#PresentationMapping
8958⟨a, b | aa=a, aab=bb⟩φ(a) = b, φ(b) = a
93006⟨a, b | aa=a, aaab=bb⟩φ(a) = b, φ(b) = a
108846⟨a, b | aa=a, aaaab=bb⟩φ(a) = b, φ(b) = a
1124594⟨a, b | aa=a, aaaaab=bb⟩φ(a) = b, φ(b) = a