#57 ⟨a, b | aa=a, bb=a⟩

Up:Monoids with two generators and two relations
Prev:#56 ⟨a, b | aa=a, ab=b⟩
Next:#58 ⟨a, b | aa=a, bb=b⟩

Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3
11bb2b3
bbb2b3b2
b2b2b3b2b3
b3b3b2b3b2

Right Cayley graph

Idempotents are shown in bold.

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
#RuleProof
1. b4 ⇒ b2 [4]
2. a ⇒ b2 [2]
# ab:aa=a,bb=a b/a
bbbb=bb
a=bb

Same cardinality

8 unique, 1570 total

Σ#PresentationDescriptionRelated
57⟨a, b | aa=b, bb=1⟩Isomorphic to ℤ41419 iso
661⟨a, b | aa=b, bb=a⟩Isomorphic to ℕ(4 = 1)72 iso
7158⟨a, b | ab=aa, ba=b⟩Finite non-commutative monoid with 4 elements8 iso, 6 anti-iso
7159⟨a, b | ab=aa, bb=a⟩Isomorphic to ℕ(4 = 3)16 iso
7242⟨a, b | aa=a, abb=b⟩Finite non-commutative monoid with 4 elements14 iso
7280⟨a, b | ab=a, bb=aa⟩Finite commutative monoid with 4 elements17 iso
92881⟨a, b | aa=a, abbba=b⟩Finite commutative monoid with 4 elements8 iso
105033⟨a, b | aaa=aa, abba=b⟩Finite commutative monoid with 4 elements2 iso

Isomorphic instances

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

37 total

Σ#PresentationMapping
7250⟨a, b | aa=b, aab=b⟩φ(a) = b, φ(b) = bb
7252⟨a, b | aa=b, aba=b⟩φ(a) = b, φ(b) = bb
7276⟨a, b | aa=a, bb=aa⟩φ(a) = bb, φ(b) = b
7278⟨a, b | aa=b, bb=aa⟩φ(a) = b, φ(b) = bb
8631⟨a, b | ab=aa, aaa=b⟩φ(a) = b, φ(b) = bbb
8633⟨a, b | ab=aa, aab=b⟩φ(a) = b, φ(b) = bbb
8635⟨a, b | ab=aa, aba=b⟩φ(a) = b, φ(b) = bbb
8637⟨a, b | ab=aa, abb=b⟩φ(a) = b, φ(b) = bbb
8901⟨a, b | aa=b, aaaa=b⟩φ(a) = b, φ(b) = bb
8954⟨a, b | aa=a, aaa=bb⟩φ(a) = bb, φ(b) = b
8975⟨a, b | aa=b, aab=aa⟩φ(a) = b, φ(b) = bb
8979⟨a, b | aa=b, aba=aa⟩φ(a) = b, φ(b) = bb
91611⟨a, b | aab=aa, aba=b⟩φ(a) = b, φ(b) = bb
93002⟨a, b | aa=a, aaaa=bb⟩φ(a) = bb, φ(b) = b
93036⟨a, b | aa=b, aaaa=aa⟩φ(a) = b, φ(b) = bb
105101⟨a, b | aab=aa, aabb=b⟩φ(a) = b, φ(b) = bb
105105⟨a, b | aab=aa, abab=b⟩φ(a) = b, φ(b) = bb
105107⟨a, b | aab=aa, abba=b⟩φ(a) = b, φ(b) = bb
105233⟨a, b | aba=aa, abba=b⟩φ(a) = b, φ(b) = bb
106562⟨a, b | aaa=b, aaaa=aa⟩φ(a) = b, φ(b) = bbb
106613⟨a, b | aab=a, aabb=bb⟩φ(a) = bbb, φ(b) = b
106621⟨a, b | aab=a, abab=bb⟩φ(a) = bbb, φ(b) = b
106625⟨a, b | aab=a, abba=bb⟩φ(a) = bbb, φ(b) = b
106666⟨a, b | aab=b, aaab=aa⟩φ(a) = b, φ(b) = bbb
106765⟨a, b | aba=b, aaab=aa⟩φ(a) = b, φ(b) = bbb
106769⟨a, b | aba=b, aaba=aa⟩φ(a) = b, φ(b) = bbb
108842⟨a, b | aa=a, aaaaa=bb⟩φ(a) = bb, φ(b) = b
1115560⟨a, b | aab=aa, aabbb=b⟩φ(a) = b, φ(b) = bb
1115568⟨a, b | aab=aa, ababb=b⟩φ(a) = b, φ(b) = bb
1115572⟨a, b | aab=aa, abbab=b⟩φ(a) = b, φ(b) = bb
1115574⟨a, b | aab=aa, abbba=b⟩φ(a) = b, φ(b) = bb
1115826⟨a, b | aba=aa, abbba=b⟩φ(a) = b, φ(b) = bb
1119508⟨a, b | aab=b, aaaab=aa⟩φ(a) = b, φ(b) = bb
1119707⟨a, b | aba=b, aaaab=aa⟩φ(a) = b, φ(b) = bb
1119711⟨a, b | aba=b, aaaba=aa⟩φ(a) = b, φ(b) = bb
1119719⟨a, b | aba=b, aabaa=aa⟩φ(a) = b, φ(b) = bb
1124590⟨a, b | aa=a, aaaaaa=bb⟩φ(a) = bb, φ(b) = b