#2074 ⟨a, b | aab=b, babb=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.

1aba2abb2ab2
11aba2abb2ab2
aaa2ababab2b2
bbabb2bab2a2a
a2a2aba2abb2ab2
ababbab2abb2aa2
b2b2ab2a2b2abab
ab2ab2b2aab2a2abb

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 [5]
2. a3 ⇒ a [3]
3. a2b ⇒ b [1]
4. b3 ⇒ a2 [6]
# ab:aab=b,babb=a ab
ba=ab
aaa=a
aab=b
bbb=aa

Same cardinality

22 unique, 1203 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8577⟨a, b | aaa=b, abb=a⟩Isomorphic to ℕ(7 = 1)59 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
104162⟨a, b | abb=aab, bbb=a⟩Isomorphic to ℕ(7 = 5)42 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
108987⟨a, b | ab=a, aaaaa=bb⟩Finite commutative monoid with 7 elements5 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Isomorphic instances

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

20 total

Σ#PresentationMapping
92078⟨a, b | aab=b, bbab=a⟩φ(a) = a, φ(b) = b
92116⟨a, b | aba=b, abbb=a⟩φ(a) = a, φ(b) = b
92120⟨a, b | aba=b, babb=a⟩φ(a) = a, φ(b) = b
1118982⟨a, b | aab=b, aababb=a⟩φ(a) = a, φ(b) = b
1118986⟨a, b | aab=b, aabbab=a⟩φ(a) = a, φ(b) = b
1119002⟨a, b | aab=b, ababab=a⟩φ(a) = a, φ(b) = b
1119030⟨a, b | aab=b, baaabb=a⟩φ(a) = a, φ(b) = b
1119034⟨a, b | aab=b, baabab=a⟩φ(a) = a, φ(b) = b
1119042⟨a, b | aab=b, babaab=a⟩φ(a) = a, φ(b) = b
1119058⟨a, b | aab=b, bbaaab=a⟩φ(a) = a, φ(b) = b
1119174⟨a, b | aba=b, aaabbb=a⟩φ(a) = a, φ(b) = b
1119180⟨a, b | aba=b, aababb=a⟩φ(a) = a, φ(b) = b
1119184⟨a, b | aba=b, aabbab=a⟩φ(a) = a, φ(b) = b
1119186⟨a, b | aba=b, aabbba=a⟩φ(a) = a, φ(b) = b
1119194⟨a, b | aba=b, abaabb=a⟩φ(a) = a, φ(b) = b
1119196⟨a, b | aba=b, ababab=a⟩φ(a) = a, φ(b) = b
1119198⟨a, b | aba=b, ababba=a⟩φ(a) = a, φ(b) = b
1119202⟨a, b | aba=b, abbaab=a⟩φ(a) = a, φ(b) = b
1119214⟨a, b | aba=b, baaabb=a⟩φ(a) = a, φ(b) = b
1119216⟨a, b | aba=b, baabab=a⟩φ(a) = a, φ(b) = b