#898 ⟨a, b | aa=a, 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.

1bb2b3b4b5b6b7
11bb2b3b4b5b6b7
bbb2b3b4b5b6b7b4
b2b2b3b4b5b6b7b4b5
b3b3b4b5b6b7b4b5b6
b4b4b5b6b7b4b5b6b7
b5b5b6b7b4b5b6b7b4
b6b6b7b4b5b6b7b4b5
b7b7b4b5b6b7b4b5b6

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. b8 ⇒ b4 [4]
2. a ⇒ b4 [2]
# ab:aa=a,bbbb=a b/a
bbbbbbbb=bbbb
a=bbbb

Same cardinality

33 unique, 1163 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
8918⟨a, b | aa=b, bbbb=a⟩Isomorphic to ℕ(8 = 1)34 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.

48 total

Σ#PresentationMapping
93033⟨a, b | aa=a, bbbb=aa⟩φ(a) = bbbb, φ(b) = b
93071⟨a, b | aa=b, bbbb=bb⟩φ(a) = b, φ(b) = bb
106574⟨a, b | aaa=b, aabb=ab⟩φ(a) = b, φ(b) = bbb
106575⟨a, b | aaa=b, aabb=ba⟩φ(a) = b, φ(b) = bbb
106578⟨a, b | aaa=b, abab=ab⟩φ(a) = b, φ(b) = bbb
106579⟨a, b | aaa=b, abab=ba⟩φ(a) = b, φ(b) = bbb
106582⟨a, b | aaa=b, abba=ab⟩φ(a) = b, φ(b) = bbb
106589⟨a, b | aaa=b, baab=ab⟩φ(a) = b, φ(b) = bbb
106659⟨a, b | aab=a, bbbb=ab⟩φ(a) = bbbbbbb, φ(b) = b
108941⟨a, b | aa=b, aabbb=bb⟩φ(a) = b, φ(b) = bb
108952⟨a, b | aa=b, ababb=bb⟩φ(a) = b, φ(b) = bb
108956⟨a, b | aa=b, abbab=bb⟩φ(a) = b, φ(b) = bb
108959⟨a, b | aa=b, abbba=bb⟩φ(a) = b, φ(b) = bb
108970⟨a, b | aa=b, baabb=bb⟩φ(a) = b, φ(b) = bb
108973⟨a, b | aa=b, babab=bb⟩φ(a) = b, φ(b) = bb
109178⟨a, b | aa=a, bbbb=aaa⟩φ(a) = bbbb, φ(b) = b
109251⟨a, b | aa=b, bbbb=aab⟩φ(a) = b, φ(b) = bb
109252⟨a, b | aa=b, bbbb=aba⟩φ(a) = b, φ(b) = bb
1114372⟨a, b | aaaa=b, aaaab=b⟩φ(a) = b, φ(b) = bbbb
1114374⟨a, b | aaaa=b, aaaba=b⟩φ(a) = b, φ(b) = bbbb
1114378⟨a, b | aaaa=b, aabaa=b⟩φ(a) = b, φ(b) = bbbb
1124738⟨a, b | aa=b, aaaabb=bb⟩φ(a) = b, φ(b) = bb
1124746⟨a, b | aa=b, aaabab=bb⟩φ(a) = b, φ(b) = bb
1124750⟨a, b | aa=b, aaabba=bb⟩φ(a) = b, φ(b) = bb
1124758⟨a, b | aa=b, aabaab=bb⟩φ(a) = b, φ(b) = bb
1124762⟨a, b | aa=b, aababa=bb⟩φ(a) = b, φ(b) = bb
1124769⟨a, b | aa=b, aabbaa=bb⟩φ(a) = b, φ(b) = bb
1124785⟨a, b | aa=b, abaaab=bb⟩φ(a) = b, φ(b) = bb
1124788⟨a, b | aa=b, abaaba=bb⟩φ(a) = b, φ(b) = bb
1124826⟨a, b | aa=b, baaaab=bb⟩φ(a) = b, φ(b) = bb
1125313⟨a, b | aa=b, aabbb=aab⟩φ(a) = b, φ(b) = bb
1125314⟨a, b | aa=b, aabbb=aba⟩φ(a) = b, φ(b) = bb
1125316⟨a, b | aa=b, aabbb=baa⟩φ(a) = b, φ(b) = bb
1125335⟨a, b | aa=b, ababb=aab⟩φ(a) = b, φ(b) = bb
1125336⟨a, b | aa=b, ababb=aba⟩φ(a) = b, φ(b) = bb
1125338⟨a, b | aa=b, ababb=baa⟩φ(a) = b, φ(b) = bb
1125343⟨a, b | aa=b, abbab=aab⟩φ(a) = b, φ(b) = bb
1125344⟨a, b | aa=b, abbab=aba⟩φ(a) = b, φ(b) = bb
1125346⟨a, b | aa=b, abbab=baa⟩φ(a) = b, φ(b) = bb
1125351⟨a, b | aa=b, abbba=aab⟩φ(a) = b, φ(b) = bb
1125352⟨a, b | aa=b, abbba=aba⟩φ(a) = b, φ(b) = bb
1125371⟨a, b | aa=b, baabb=aab⟩φ(a) = b, φ(b) = bb
1125372⟨a, b | aa=b, baabb=aba⟩φ(a) = b, φ(b) = bb
1125374⟨a, b | aa=b, baabb=baa⟩φ(a) = b, φ(b) = bb
1125379⟨a, b | aa=b, babab=aab⟩φ(a) = b, φ(b) = bb
1125380⟨a, b | aa=b, babab=aba⟩φ(a) = b, φ(b) = bb
1125717⟨a, b | aa=a, bbbb=aaaa⟩φ(a) = bbbb, φ(b) = b
1125783⟨a, b | aa=b, bbbb=aaaa⟩φ(a) = b, φ(b) = bb