#644 ⟨a, b | ab=aa, bbb=a⟩

Up:Monoids with two generators and two relations
Prev:#642 ⟨a, b | ab=aa, bba=a⟩
Next:#645 ⟨a, b | ab=aa, bbb=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.

1bb2b3b4b5
11bb2b3b4b5
bbb2b3b4b5b4
b2b2b3b4b5b4b5
b3b3b4b5b4b5b4
b4b4b5b4b5b4b5
b5b5b4b5b4b5b4

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. b6 ⇒ b4 [4]
2. a ⇒ b3 [2]
# ab:ab=aa,bbb=a b/a
bbbbbb=bbbb
a=bbb

Same cardinality

19 unique, 1879 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
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-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
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

Isomorphic instances

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

46 total

Σ#PresentationMapping
8991⟨a, b | aa=b, bbb=bb⟩φ(a) = b, φ(b) = bb
92053⟨a, b | aab=b, aaaa=b⟩φ(a) = b, φ(b) = bbbb
92105⟨a, b | aba=b, aaaa=b⟩φ(a) = b, φ(b) = bbbb
93050⟨a, b | aa=b, aabb=bb⟩φ(a) = b, φ(b) = bb
93054⟨a, b | aa=b, abab=bb⟩φ(a) = b, φ(b) = bb
93057⟨a, b | aa=b, abba=bb⟩φ(a) = b, φ(b) = bb
93064⟨a, b | aa=b, baab=bb⟩φ(a) = b, φ(b) = bb
93168⟨a, b | aa=b, bbb=aab⟩φ(a) = b, φ(b) = bb
93169⟨a, b | aa=b, bbb=aba⟩φ(a) = b, φ(b) = bb
106564⟨a, b | aaa=b, aaaa=bb⟩φ(a) = b, φ(b) = bbb
106566⟨a, b | aaa=b, aaab=ab⟩φ(a) = b, φ(b) = bbb
106567⟨a, b | aaa=b, aaab=ba⟩φ(a) = b, φ(b) = bbb
106570⟨a, b | aaa=b, aaba=ab⟩φ(a) = b, φ(b) = bbb
106571⟨a, b | aaa=b, aaba=ba⟩φ(a) = b, φ(b) = bbb
106663⟨a, b | aab=b, aaaa=ab⟩φ(a) = b, φ(b) = bbbbb
106763⟨a, b | aba=b, aaaa=ab⟩φ(a) = b, φ(b) = bbbbb
108918⟨a, b | aa=b, aaaab=bb⟩φ(a) = b, φ(b) = bb
108922⟨a, b | aa=b, aaaba=bb⟩φ(a) = b, φ(b) = bb
108929⟨a, b | aa=b, aabaa=bb⟩φ(a) = b, φ(b) = bb
109189⟨a, b | aa=b, aaaa=bbb⟩φ(a) = b, φ(b) = bb
109207⟨a, b | aa=b, aabb=aab⟩φ(a) = b, φ(b) = bb
109208⟨a, b | aa=b, aabb=aba⟩φ(a) = b, φ(b) = bb
109210⟨a, b | aa=b, aabb=baa⟩φ(a) = b, φ(b) = bb
109215⟨a, b | aa=b, abab=aab⟩φ(a) = b, φ(b) = bb
109216⟨a, b | aa=b, abab=aba⟩φ(a) = b, φ(b) = bb
109218⟨a, b | aa=b, abab=baa⟩φ(a) = b, φ(b) = bb
109223⟨a, b | aa=b, abba=aab⟩φ(a) = b, φ(b) = bb
109224⟨a, b | aa=b, abba=aba⟩φ(a) = b, φ(b) = bb
109237⟨a, b | aa=b, baab=aab⟩φ(a) = b, φ(b) = bb
109238⟨a, b | aa=b, baab=aba⟩φ(a) = b, φ(b) = bb
1120049⟨a, b | aab=b, aaaa=aab⟩φ(a) = b, φ(b) = bbbb
1120249⟨a, b | aba=b, aaaa=aab⟩φ(a) = b, φ(b) = bbbb
1120250⟨a, b | aba=b, aaaa=aba⟩φ(a) = b, φ(b) = bbbb
1124726⟨a, b | aa=b, aaaaaa=bb⟩φ(a) = b, φ(b) = bb
1125267⟨a, b | aa=b, aaaab=aab⟩φ(a) = b, φ(b) = bb
1125268⟨a, b | aa=b, aaaab=aba⟩φ(a) = b, φ(b) = bb
1125270⟨a, b | aa=b, aaaab=baa⟩φ(a) = b, φ(b) = bb
1125275⟨a, b | aa=b, aaaba=aab⟩φ(a) = b, φ(b) = bb
1125276⟨a, b | aa=b, aaaba=aba⟩φ(a) = b, φ(b) = bb
1125278⟨a, b | aa=b, aaaba=baa⟩φ(a) = b, φ(b) = bb
1125291⟨a, b | aa=b, aabaa=aab⟩φ(a) = b, φ(b) = bb
1125292⟨a, b | aa=b, aabaa=aba⟩φ(a) = b, φ(b) = bb
1125729⟨a, b | aa=b, aabb=aaaa⟩φ(a) = b, φ(b) = bb
1125735⟨a, b | aa=b, abab=aaaa⟩φ(a) = b, φ(b) = bb
1125740⟨a, b | aa=b, abba=aaaa⟩φ(a) = b, φ(b) = bb
1125756⟨a, b | aa=b, baab=aaaa⟩φ(a) = b, φ(b) = bb