#918 ⟨a, b | aa=b, bbbb=a⟩

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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
aaa2a3a4a5a6a7a
a2a2a3a4a5a6a7aa2
a3a3a4a5a6a7aa2a3
a4a4a5a6a7aa2a3a4
a5a5a6a7aa2a3a4a5
a6a6a7aa2a3a4a5a6
a7a7aa2a3a4a5a6a7

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

Same cardinality

33 unique, 1177 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
8919⟨a, b | aa=b, bbbb=b⟩Isomorphic to ℕ(8 = 2)46 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.

34 total

Σ#PresentationMapping
92006⟨a, b | aaa=b, aabb=a⟩φ(a) = bbbbb, φ(b) = b
92008⟨a, b | aaa=b, abab=a⟩φ(a) = bbbbb, φ(b) = b
92010⟨a, b | aaa=b, abba=a⟩φ(a) = bbbbb, φ(b) = b
92014⟨a, b | aaa=b, baab=a⟩φ(a) = bbbbb, φ(b) = b
92910⟨a, b | aa=b, aabbb=a⟩φ(a) = bbbb, φ(b) = b
92916⟨a, b | aa=b, ababb=a⟩φ(a) = bbbb, φ(b) = b
92918⟨a, b | aa=b, abbab=a⟩φ(a) = bbbb, φ(b) = b
92920⟨a, b | aa=b, abbba=a⟩φ(a) = bbbb, φ(b) = b
92926⟨a, b | aa=b, baabb=a⟩φ(a) = bbbb, φ(b) = b
92928⟨a, b | aa=b, babab=a⟩φ(a) = bbbb, φ(b) = b
108646⟨a, b | aa=b, aaaabb=a⟩φ(a) = bbbb, φ(b) = b
108650⟨a, b | aa=b, aaabab=a⟩φ(a) = bbbb, φ(b) = b
108652⟨a, b | aa=b, aaabba=a⟩φ(a) = bbbb, φ(b) = b
108656⟨a, b | aa=b, aabaab=a⟩φ(a) = bbbb, φ(b) = b
108658⟨a, b | aa=b, aababa=a⟩φ(a) = bbbb, φ(b) = b
108662⟨a, b | aa=b, aabbaa=a⟩φ(a) = bbbb, φ(b) = b
108670⟨a, b | aa=b, abaaab=a⟩φ(a) = bbbb, φ(b) = b
108672⟨a, b | aa=b, abaaba=a⟩φ(a) = bbbb, φ(b) = b
108692⟨a, b | aa=b, baaaab=a⟩φ(a) = bbbb, φ(b) = b
1114371⟨a, b | aaaa=b, aaaab=a⟩φ(a) = bb, φ(b) = b
1114373⟨a, b | aaaa=b, aaaba=a⟩φ(a) = bb, φ(b) = b
1114377⟨a, b | aaaa=b, aabaa=a⟩φ(a) = bb, φ(b) = b
1114410⟨a, b | aaab=a, aaaaa=b⟩φ(a) = bbb, φ(b) = b
1114538⟨a, b | aaba=a, aaaaa=b⟩φ(a) = bbb, φ(b) = b
1118762⟨a, b | aaa=b, aaaaab=a⟩φ(a) = bbbbb, φ(b) = b
1118764⟨a, b | aaa=b, aaaaba=a⟩φ(a) = bbbbb, φ(b) = b
1118768⟨a, b | aaa=b, aaabaa=a⟩φ(a) = bbbbb, φ(b) = b
1118833⟨a, b | aab=a, aaaaaa=b⟩φ(a) = bbbbbb, φ(b) = b
1119089⟨a, b | aba=a, aaaaaa=b⟩φ(a) = bbbbbb, φ(b) = b
1124190⟨a, b | aa=b, aaaaaab=a⟩φ(a) = bbbb, φ(b) = b
1124192⟨a, b | aa=b, aaaaaba=a⟩φ(a) = bbbb, φ(b) = b
1124196⟨a, b | aa=b, aaaabaa=a⟩φ(a) = bbbb, φ(b) = b
1124204⟨a, b | aa=b, aaabaaa=a⟩φ(a) = bbbb, φ(b) = b
1124333⟨a, b | ab=a, aaaaaaa=b⟩φ(a) = b, φ(b) = bbbbbbb