#919 ⟨a, b | aa=b, bbbb=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.

1aa2a3a4a5a6a7
11aa2a3a4a5a6a7
aaa2a3a4a5a6a7a2
a2a2a3a4a5a6a7a2a3
a3a3a4a5a6a7a2a3a4
a4a4a5a6a7a2a3a4a5
a5a5a6a7a2a3a4a5a6
a6a6a7a2a3a4a5a6a7
a7a7a2a3a4a5a6a7a2

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. a8 ⇒ a2 [2]
2. b ⇒ a2 [1]
# ab:aa=b,bbbb=b a/b
aaaaaaaa=aa
b=aa

Same cardinality

33 unique, 1165 total

Σ#PresentationDescriptionRelated
7188⟨a, b | aab=1, bbbb=1⟩Isomorphic to ℤ8727 iso
7330⟨a, b | aa=1, abba=b⟩Finite non-commutative monoid with 8 elements60 iso
8556⟨a, b | aba=b, aabb=1⟩Finite non-Abelian group with 8 elements28 iso
8898⟨a, b | aa=a, bbbb=a⟩Isomorphic to ℕ(8 = 4)48 iso
8918⟨a, b | aa=b, bbbb=a⟩Isomorphic to ℕ(8 = 1)34 iso
8961⟨a, b | aa=a, aba=bb⟩Finite non-commutative monoid with 8 elements5 iso
91605⟨a, b | aaa=bb, abb=b⟩Isomorphic to ℕ(8 = 3)55 iso
91606⟨a, b | aaa=bb, bab=a⟩Finite commutative monoid with 8 elements14 iso
91615⟨a, b | aab=aa, baa=b⟩Finite non-commutative monoid with 8 elements9 iso, 5 anti-iso
91650⟨a, b | aab=bb, baa=a⟩Finite non-commutative monoid with 8 elements4 iso, 11 anti-iso
92206⟨a, b | ab=aa, bbbb=a⟩Isomorphic to ℕ(8 = 5)32 iso
92220⟨a, b | bb=aa, aaab=a⟩Finite commutative monoid with 8 elements22 iso
92247⟨a, b | ab=aa, baa=bb⟩Finite non-commutative monoid with 8 elements1 iso
92256⟨a, b | ab=aa, bbb=aa⟩Finite non-commutative monoid with 8 elements1 iso
92258⟨a, b | ab=aa, bbb=ba⟩Finite non-commutative monoid with 8 elements
92883⟨a, b | aa=a, abbbb=b⟩Finite non-commutative monoid with 8 elements14 iso
93107⟨a, b | ab=a, baaa=bb⟩Finite non-commutative monoid with 8 elements9 iso
93123⟨a, b | ab=a, bbaa=bb⟩Finite non-commutative monoid with 8 elements10 iso
105191⟨a, b | aab=bb, aaaa=b⟩Isomorphic to ℕ(8 = 6)18 iso
106664⟨a, b | aab=b, aaaa=ba⟩Finite non-commutative monoid with 8 elements1 iso
107057⟨a, b | ab=aa, aaaa=bb⟩Finite non-commutative monoid with 8 elements7 iso
109380⟨a, b | ab=a, bbbb=baa⟩Finite non-commutative monoid with 8 elements2 iso
1112606⟨a, b | abbb=aa, bbbb=a⟩Isomorphic to ℕ(8 = 7)14 iso
1120047⟨a, b | aab=a, bbbb=bbb⟩Finite non-commutative monoid with 8 elements
1120051⟨a, b | aab=b, aaaa=abb⟩Finite commutative monoid with 8 elements1 iso
1124863⟨a, b | ab=a, aaaaaa=bb⟩Finite commutative monoid with 8 elements
1125112⟨a, b | ab=a, bbbbbb=aa⟩Finite commutative monoid with 8 elements
1125114⟨a, b | ab=a, bbbbbb=ba⟩Finite non-commutative monoid with 8 elements
1125411⟨a, b | ab=a, aaaaa=bbb⟩Finite commutative monoid with 8 elements
1125652⟨a, b | ab=a, bbbbb=aaa⟩Finite commutative monoid with 8 elements
1125658⟨a, b | ab=a, bbbbb=bba⟩Finite non-commutative monoid with 8 elements
1125897⟨a, b | ab=a, bbbb=aaaa⟩Finite commutative monoid with 8 elements
1125911⟨a, b | ab=a, bbbb=bbba⟩Finite non-commutative monoid with 8 elements

Isomorphic instances

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

46 total

Σ#PresentationMapping
92218⟨a, b | bb=aa, aaaa=a⟩φ(a) = aaaa, φ(b) = a
92219⟨a, b | bb=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
92224⟨a, b | bb=aa, aabb=a⟩φ(a) = aaaa, φ(b) = a
92225⟨a, b | bb=aa, abab=a⟩φ(a) = aaaa, φ(b) = a
92226⟨a, b | bb=aa, abba=a⟩φ(a) = aaaa, φ(b) = a
92227⟨a, b | bb=aa, abba=b⟩φ(a) = a, φ(b) = aaaa
92911⟨a, b | aa=b, aabbb=b⟩φ(a) = a, φ(b) = aa
92917⟨a, b | aa=b, ababb=b⟩φ(a) = a, φ(b) = aa
92919⟨a, b | aa=b, abbab=b⟩φ(a) = a, φ(b) = aa
92921⟨a, b | aa=b, abbba=b⟩φ(a) = a, φ(b) = aa
92927⟨a, b | aa=b, baabb=b⟩φ(a) = a, φ(b) = aa
92929⟨a, b | aa=b, babab=b⟩φ(a) = a, φ(b) = aa
93069⟨a, b | aa=b, bbbb=aa⟩φ(a) = a, φ(b) = aa
106573⟨a, b | aaa=b, aabb=aa⟩φ(a) = a, φ(b) = aaa
106577⟨a, b | aaa=b, abab=aa⟩φ(a) = a, φ(b) = aaa
106581⟨a, b | aaa=b, abba=aa⟩φ(a) = a, φ(b) = aaa
106588⟨a, b | aaa=b, baab=aa⟩φ(a) = a, φ(b) = aaa
106601⟨a, b | aab=a, aaaa=bb⟩φ(a) = aaaaa, φ(b) = a
108647⟨a, b | aa=b, aaaabb=b⟩φ(a) = a, φ(b) = aa
108651⟨a, b | aa=b, aaabab=b⟩φ(a) = a, φ(b) = aa
108653⟨a, b | aa=b, aaabba=b⟩φ(a) = a, φ(b) = aa
108657⟨a, b | aa=b, aabaab=b⟩φ(a) = a, φ(b) = aa
108659⟨a, b | aa=b, aababa=b⟩φ(a) = a, φ(b) = aa
108663⟨a, b | aa=b, aabbaa=b⟩φ(a) = a, φ(b) = aa
108671⟨a, b | aa=b, abaaab=b⟩φ(a) = a, φ(b) = aa
108673⟨a, b | aa=b, abaaba=b⟩φ(a) = a, φ(b) = aa
108693⟨a, b | aa=b, baaaab=b⟩φ(a) = a, φ(b) = aa
108938⟨a, b | aa=b, aabbb=aa⟩φ(a) = a, φ(b) = aa
108949⟨a, b | aa=b, ababb=aa⟩φ(a) = a, φ(b) = aa
108953⟨a, b | aa=b, abbab=aa⟩φ(a) = a, φ(b) = aa
108957⟨a, b | aa=b, abbba=aa⟩φ(a) = a, φ(b) = aa
108967⟨a, b | aa=b, baabb=aa⟩φ(a) = a, φ(b) = aa
108971⟨a, b | aa=b, babab=aa⟩φ(a) = a, φ(b) = aa
1124191⟨a, b | aa=b, aaaaaab=b⟩φ(a) = a, φ(b) = aa
1124193⟨a, b | aa=b, aaaaaba=b⟩φ(a) = a, φ(b) = aa
1124197⟨a, b | aa=b, aaaabaa=b⟩φ(a) = a, φ(b) = aa
1124205⟨a, b | aa=b, aaabaaa=b⟩φ(a) = a, φ(b) = aa
1124735⟨a, b | aa=b, aaaabb=aa⟩φ(a) = a, φ(b) = aa
1124743⟨a, b | aa=b, aaabab=aa⟩φ(a) = a, φ(b) = aa
1124747⟨a, b | aa=b, aaabba=aa⟩φ(a) = a, φ(b) = aa
1124755⟨a, b | aa=b, aabaab=aa⟩φ(a) = a, φ(b) = aa
1124759⟨a, b | aa=b, aababa=aa⟩φ(a) = a, φ(b) = aa
1124767⟨a, b | aa=b, aabbaa=aa⟩φ(a) = a, φ(b) = aa
1124782⟨a, b | aa=b, abaaab=aa⟩φ(a) = a, φ(b) = aa
1124786⟨a, b | aa=b, abaaba=aa⟩φ(a) = a, φ(b) = aa
1124824⟨a, b | aa=b, baaaab=aa⟩φ(a) = a, φ(b) = aa