#8987 ⟨a, b | ab=a, aaaaa=bb⟩

Up:Monoids with two generators and two relations
Prev:#8986 ⟨a, b | ab=a, aaaaa=ba⟩
Next:#9048 ⟨a, b | ab=a, baaaa=aa⟩

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.

1aba2a3a4a5
11aba2a3a4a5
aaa2aa3a4a5a
bbaa5a2a3a4a5
a2a2a3a2a4a5aa2
a3a3a4a3a5aa2a3
a4a4a5a4aa2a3a4
a5a5aa5a2a3a4a5

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. a6 ⇒ a [3]
2. ba ⇒ a [5]
3. ab ⇒ a [1]
4. b2 ⇒ a5 [2]
# ab:ab=a,aaaaa=bb a/b
aaaaaa=a
ba=a
ab=a
bb=aaaaa

Same cardinality

22 unique, 1218 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8577⟨a, b | aaa=b, abb=a⟩Isomorphic to ℕ(7 = 1)59 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92074⟨a, b | aab=b, babb=a⟩Finite commutative monoid with 7 elements20 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
104162⟨a, b | abb=aab, bbb=a⟩Isomorphic to ℕ(7 = 5)42 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Isomorphic instances

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

5 total

Σ#PresentationMapping
1124867⟨a, b | ab=a, aaaaab=bb⟩φ(a) = a, φ(b) = b
1124871⟨a, b | ab=a, aaaaba=bb⟩φ(a) = a, φ(b) = b
1124879⟨a, b | ab=a, aaabaa=bb⟩φ(a) = a, φ(b) = b
1124895⟨a, b | ab=a, aabaaa=bb⟩φ(a) = a, φ(b) = b
1124927⟨a, b | ab=a, abaaaa=bb⟩φ(a) = a, φ(b) = b