#995 ⟨a, b | ab=a, aaa=bb⟩

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Properties

Element profile

Complete 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
  1. a4 ⇒ a
  2. ba ⇒ a
  3. ab ⇒ a
  4. b2 ⇒ a3 
# ab:ab=a,aaa=bb a/b
aaaa=a
ba=a
ab=a
bb=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2a3
11aba2a3
aaa2aa3a
bbaa3a2a3
a2a2a3a2aa2
a3a3aa3a2a3

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

11 unique, 1443 total

Σ#PresentationDescriptionRelated
644⟨a, b | aa=b, abb=1⟩Isomorphic to ℤ51132 iso
7253⟨a, b | aa=b, abb=a⟩Isomorphic to ℕ(5 = 1)71 iso
7254⟨a, b | aa=b, abb=b⟩Isomorphic to ℕ(5 = 2)43 iso
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-iso
8574⟨a, b | aaa=b, aab=b⟩Isomorphic to ℕ(5 = 3)27 iso
8950⟨a, b | ab=a, bbbb=a⟩Isomorphic to ℕ(5 = 4)32 iso
81019⟨a, b | ab=a, bba=bb⟩Finite non-commutative monoid with 5 elements25 iso
81020⟨a, b | ab=a, bbb=aa⟩Finite commutative monoid with 5 elements9 iso
81022⟨a, b | ab=a, bbb=ba⟩Finite non-commutative monoid with 5 elements4 iso
108617⟨a, b | aa=a, abbbba=b⟩Finite commutative monoid with 5 elements3 iso
1115426⟨a, b | aaa=aa, abbba=b⟩Finite commutative monoid with 5 elements

Other isomorphic instances

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

19 total

Σ#PresentationMapping
93079⟨a, b | ab=a, aaab=bb⟩φ(a) = a, φ(b) = b
93083⟨a, b | ab=a, aaba=bb⟩φ(a) = a, φ(b) = b
93091⟨a, b | ab=a, abaa=bb⟩φ(a) = a, φ(b) = b
108999⟨a, b | ab=a, aaabb=bb⟩φ(a) = a, φ(b) = b
109007⟨a, b | ab=a, aabab=bb⟩φ(a) = a, φ(b) = b
109011⟨a, b | ab=a, aabba=bb⟩φ(a) = a, φ(b) = b
109023⟨a, b | ab=a, abaab=bb⟩φ(a) = a, φ(b) = b
109027⟨a, b | ab=a, ababa=bb⟩φ(a) = a, φ(b) = b
109035⟨a, b | ab=a, abbaa=bb⟩φ(a) = a, φ(b) = b
1124891⟨a, b | ab=a, aaabbb=bb⟩φ(a) = a, φ(b) = b
1124907⟨a, b | ab=a, aababb=bb⟩φ(a) = a, φ(b) = b
1124915⟨a, b | ab=a, aabbab=bb⟩φ(a) = a, φ(b) = b
1124919⟨a, b | ab=a, aabbba=bb⟩φ(a) = a, φ(b) = b
1124939⟨a, b | ab=a, abaabb=bb⟩φ(a) = a, φ(b) = b
1124947⟨a, b | ab=a, ababab=bb⟩φ(a) = a, φ(b) = b
1124951⟨a, b | ab=a, ababba=bb⟩φ(a) = a, φ(b) = b
1124963⟨a, b | ab=a, abbaab=bb⟩φ(a) = a, φ(b) = b
1124967⟨a, b | ab=a, abbaba=bb⟩φ(a) = a, φ(b) = b
1124975⟨a, b | ab=a, abbbaa=bb⟩φ(a) = a, φ(b) = b