#20723 ⟨a, b | ab=aa, aaaaa=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. a6 ⇒ a3
  2. ba3 ⇒ a4
  3. ab ⇒ a2
  4. b2 ⇒ a5 
# ab:ab=aa,aaaaa=bb a/b
aaaaaa=aaa
baaa=aaaa
ab=aa
bb=aaaaa

Cayley table

Idempotents are shown in bold.

1aba2baa3ba2a4a5
11aba2baa3ba2a4a5
aaa2a2a3a3a4a4a5a3
bbbaa5ba2a3a4a4a5a3
a2a2a3a3a4a4a5a5a3a4
bababa2ba2a4a4a5a5a3a4
a3a3a4a4a5a5a3a3a4a5
ba2ba2a4a4a5a5a3a3a4a5
a4a4a5a5a3a3a4a4a5a3
a5a5a3a3a4a4a5a5a3a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 927 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
105079⟨a, b | aaa=bb, aaba=b⟩Finite commutative monoid with 9 elements8 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
1125656⟨a, b | ab=a, bbbbb=baa⟩Finite non-commutative monoid with 9 elements

Other isomorphic instances

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

15 total

Σ#PresentationMapping
1120727⟨a, b | ab=aa, aaaab=bb⟩φ(a) = a, φ(b) = b
1120731⟨a, b | ab=aa, aaaba=bb⟩φ(a) = a, φ(b) = b
1120735⟨a, b | ab=aa, aaabb=bb⟩φ(a) = a, φ(b) = b
1120739⟨a, b | ab=aa, aabaa=bb⟩φ(a) = a, φ(b) = b
1120743⟨a, b | ab=aa, aabab=bb⟩φ(a) = a, φ(b) = b
1120747⟨a, b | ab=aa, aabba=bb⟩φ(a) = a, φ(b) = b
1120751⟨a, b | ab=aa, aabbb=bb⟩φ(a) = a, φ(b) = b
1120755⟨a, b | ab=aa, abaaa=bb⟩φ(a) = a, φ(b) = b
1120759⟨a, b | ab=aa, abaab=bb⟩φ(a) = a, φ(b) = b
1120763⟨a, b | ab=aa, ababa=bb⟩φ(a) = a, φ(b) = b
1120767⟨a, b | ab=aa, ababb=bb⟩φ(a) = a, φ(b) = b
1120771⟨a, b | ab=aa, abbaa=bb⟩φ(a) = a, φ(b) = b
1120775⟨a, b | ab=aa, abbab=bb⟩φ(a) = a, φ(b) = b
1120779⟨a, b | ab=aa, abbba=bb⟩φ(a) = a, φ(b) = b
1120783⟨a, b | ab=aa, abbbb=bb⟩φ(a) = a, φ(b) = b