#5079 ⟨a, b | aaa=bb, aaba=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.

1aba2abb2a2bab2a2b2
11aba2abb2a2bab2a2b2
aaa2abb2a2bab2ba2b2b2
bbabb2a2bab2ba2b2aba2b
a2a2b2a2bab2ba2b2abb2ab2
ababa2bab2ba2b2abb2a2bb
b2b2ab2ba2b2abb2a2bab2a2b2
a2ba2bba2b2abb2a2bab2bab
ab2ab2a2b2abb2a2bab2ba2b2b2
a2b2a2b2b2a2bab2ba2b2abb2ab2

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. ba ⇒ ab [12]
2. a3 ⇒ b2 [1]
3. b3 ⇒ b [13]
# ab:aaa=bb,aaba=b ab
ba=ab
aaa=bb
bbb=b

Same cardinality

26 unique, 934 total

Σ#PresentationDescriptionRelated
7118⟨a, b | aaa=b, bbb=1⟩Isomorphic to ℤ9562 iso
8568⟨a, b | aaa=a, abb=b⟩Finite non-commutative monoid with 9 elements5 iso
8571⟨a, b | aaa=a, bbb=a⟩Isomorphic to ℕ(9 = 3)48 iso
8581⟨a, b | aaa=b, bbb=a⟩Isomorphic to ℕ(9 = 1)62 iso
8601⟨a, b | aba=b, bab=a⟩Finite non-commutative monoid with 9 elements3 iso
91584⟨a, b | aaa=aa, bbb=a⟩Isomorphic to ℕ(9 = 6)16 iso
91595⟨a, b | aaa=ab, bab=b⟩Finite non-commutative monoid with 9 elements4 iso, 5 anti-iso
91598⟨a, b | aaa=ab, bbb=a⟩Isomorphic to ℕ(9 = 4)40 iso
91608⟨a, b | aaa=bb, bbb=a⟩Isomorphic to ℕ(9 = 2)70 iso
91991⟨a, b | aaa=a, abba=b⟩Finite non-commutative monoid with 9 elements3 iso
92064⟨a, b | aab=b, abba=a⟩Finite non-commutative monoid with 9 elements14 iso, 24 anti-iso
92255⟨a, b | ab=aa, bba=bb⟩Finite non-commutative monoid with 9 elements
104125⟨a, b | aab=aaa, baa=b⟩Finite non-commutative monoid with 9 elements1 iso, 1 anti-iso
104130⟨a, b | aab=aaa, bbb=a⟩Isomorphic to ℕ(9 = 7)1 iso
104156⟨a, b | abb=aaa, bbb=a⟩Isomorphic to ℕ(9 = 5)12 iso
105035⟨a, b | aaa=aa, abbb=b⟩Finite non-commutative monoid with 9 elements3 iso
105124⟨a, b | aab=aa, bbbb=a⟩Isomorphic to ℕ(9 = 8)5 iso
105339⟨a, b | aaa=bb, aab=ab⟩Finite non-commutative monoid with 9 elements1 iso
106824⟨a, b | aab=b, bbb=aaa⟩Finite commutative monoid with 9 elements5 iso
107139⟨a, b | bb=aa, aaaa=ab⟩Finite non-commutative monoid with 9 elements3 iso
109327⟨a, b | ab=a, baaa=bbb⟩Finite non-commutative monoid with 9 elements3 iso
109359⟨a, b | ab=a, bbaa=bbb⟩Finite non-commutative monoid with 9 elements2 iso
1119506⟨a, b | aab=b, aaaaa=ba⟩Finite non-commutative monoid with 9 elements
1119507⟨a, b | aab=b, aaaaa=bb⟩Finite commutative monoid with 9 elements
1120723⟨a, b | ab=aa, aaaaa=bb⟩Finite non-commutative monoid with 9 elements15 iso
1125656⟨a, b | ab=a, bbbbb=baa⟩Finite non-commutative monoid with 9 elements

Isomorphic instances

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

8 total

Σ#PresentationMapping
1112967⟨a, b | abb=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1112969⟨a, b | abb=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1112973⟨a, b | abb=aaa, abaa=b⟩φ(a) = a, φ(b) = ab
1112979⟨a, b | abb=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113083⟨a, b | bab=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1113085⟨a, b | bab=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1113093⟨a, b | bab=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113097⟨a, b | bab=aaa, babb=b⟩φ(a) = a, φ(b) = ab