#258 ⟨a, b | aa=b, bbb=b⟩

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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.

1aa2a3a4a5
11aa2a3a4a5
aaa2a3a4a5a2
a2a2a3a4a5a2a3
a3a3a4a5a2a3a4
a4a4a5a2a3a4a5
a5a5a2a3a4a5a2

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 ⇒ a2 [2]
2. b ⇒ a2 [1]
# ab:aa=b,bbb=b a/b
aaaaaa=aa
b=aa

Same cardinality

19 unique, 1870 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
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

Isomorphic instances

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

55 total

Σ#PresentationMapping
8653⟨a, b | bb=aa, aaa=b⟩φ(a) = a, φ(b) = aaa
8654⟨a, b | bb=aa, aab=a⟩φ(a) = aaa, φ(b) = a
8907⟨a, b | aa=b, aabb=b⟩φ(a) = a, φ(b) = aa
8909⟨a, b | aa=b, abab=b⟩φ(a) = a, φ(b) = aa
8911⟨a, b | aa=b, abba=b⟩φ(a) = a, φ(b) = aa
8915⟨a, b | aa=b, baab=b⟩φ(a) = a, φ(b) = aa
8989⟨a, b | aa=b, bbb=aa⟩φ(a) = a, φ(b) = aa
92899⟨a, b | aa=b, aaaab=b⟩φ(a) = a, φ(b) = aa
92901⟨a, b | aa=b, aaaba=b⟩φ(a) = a, φ(b) = aa
92905⟨a, b | aa=b, aabaa=b⟩φ(a) = a, φ(b) = aa
93047⟨a, b | aa=b, aabb=aa⟩φ(a) = a, φ(b) = aa
93051⟨a, b | aa=b, abab=aa⟩φ(a) = a, φ(b) = aa
93055⟨a, b | aa=b, abba=aa⟩φ(a) = a, φ(b) = aa
93062⟨a, b | aa=b, baab=aa⟩φ(a) = a, φ(b) = aa
105095⟨a, b | aab=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
105223⟨a, b | aba=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
106565⟨a, b | aaa=b, aaab=aa⟩φ(a) = a, φ(b) = aaa
106569⟨a, b | aaa=b, aaba=aa⟩φ(a) = a, φ(b) = aaa
106605⟨a, b | aab=a, aaab=bb⟩φ(a) = aaa, φ(b) = a
106609⟨a, b | aab=a, aaba=bb⟩φ(a) = aaa, φ(b) = a
106617⟨a, b | aab=a, abaa=bb⟩φ(a) = aaa, φ(b) = a
106951⟨a, b | ab=aa, aaaaa=b⟩φ(a) = a, φ(b) = aaaaa
106953⟨a, b | ab=aa, aaaab=b⟩φ(a) = a, φ(b) = aaaaa
106955⟨a, b | ab=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaaa
106957⟨a, b | ab=aa, aaabb=b⟩φ(a) = a, φ(b) = aaaaa
106959⟨a, b | ab=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaaa
106961⟨a, b | ab=aa, aabab=b⟩φ(a) = a, φ(b) = aaaaa
106963⟨a, b | ab=aa, aabba=b⟩φ(a) = a, φ(b) = aaaaa
106965⟨a, b | ab=aa, aabbb=b⟩φ(a) = a, φ(b) = aaaaa
106967⟨a, b | ab=aa, abaaa=b⟩φ(a) = a, φ(b) = aaaaa
106969⟨a, b | ab=aa, abaab=b⟩φ(a) = a, φ(b) = aaaaa
106971⟨a, b | ab=aa, ababa=b⟩φ(a) = a, φ(b) = aaaaa
106973⟨a, b | ab=aa, ababb=b⟩φ(a) = a, φ(b) = aaaaa
106975⟨a, b | ab=aa, abbaa=b⟩φ(a) = a, φ(b) = aaaaa
106977⟨a, b | ab=aa, abbab=b⟩φ(a) = a, φ(b) = aaaaa
106979⟨a, b | ab=aa, abbba=b⟩φ(a) = a, φ(b) = aaaaa
106981⟨a, b | ab=aa, abbbb=b⟩φ(a) = a, φ(b) = aaaaa
108641⟨a, b | aa=b, aaaaaa=b⟩φ(a) = a, φ(b) = aa
108915⟨a, b | aa=b, aaaab=aa⟩φ(a) = a, φ(b) = aa
108919⟨a, b | aa=b, aaaba=aa⟩φ(a) = a, φ(b) = aa
108927⟨a, b | aa=b, aabaa=aa⟩φ(a) = a, φ(b) = aa
1112433⟨a, b | aabb=aa, abab=b⟩φ(a) = a, φ(b) = aa
1112435⟨a, b | aabb=aa, abba=b⟩φ(a) = a, φ(b) = aa
1112596⟨a, b | abba=bb, baab=a⟩φ(a) = aa, φ(b) = a
1112598⟨a, b | abba=bb, baba=a⟩φ(a) = aa, φ(b) = a
1112602⟨a, b | abba=bb, bbaa=a⟩φ(a) = aa, φ(b) = a
1112629⟨a, b | baba=aa, bbaa=b⟩φ(a) = a, φ(b) = aa
1115548⟨a, b | aab=aa, aaaab=b⟩φ(a) = a, φ(b) = aaaa
1115550⟨a, b | aab=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaa
1115554⟨a, b | aab=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaa
1115562⟨a, b | aab=aa, abaaa=b⟩φ(a) = a, φ(b) = aaaa
1115806⟨a, b | aba=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaa
1115810⟨a, b | aba=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaa
1119234⟨a, b | aaa=a, aaaaa=bb⟩φ(a) = aa, φ(b) = a
1124724⟨a, b | aa=b, aaaaaa=aa⟩φ(a) = a, φ(b) = aa