#245 ⟨a, b | aa=a, bbb=a⟩

Up:Monoids with two generators and two relations
Prev:#244 ⟨a, b | aa=a, bab=b⟩
Next:#246 ⟨a, b | aa=a, bbb=b⟩

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. b6 ⇒ b3
  2. a ⇒ b3 
# ab:aa=a,bbb=a b/a
bbbbbb=bbb
a=bbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5
11bb2b3b4b5
bbb2b3b4b5b3
b2b2b3b4b5b3b4
b3b3b4b5b3b4b5
b4b4b5b3b4b5b3
b5b5b3b4b5b3b4

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 1876 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-iso
7257⟨a, b | aa=b, bbb=a⟩Isomorphic to ℕ(6 = 1)61 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91427⟨a, b | abba=b, baba=1⟩Finite non-Abelian group with 6 elements66 iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Other isomorphic instances

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

49 total

Σ#PresentationMapping
8969⟨a, b | aa=a, bbb=aa⟩φ(a) = bbb, φ(b) = b
8990⟨a, b | aa=b, bbb=ab⟩φ(a) = b, φ(b) = bb
91591⟨a, b | aaa=ab, abb=b⟩φ(a) = b, φ(b) = bbbbb
92003⟨a, b | aaa=b, aaab=b⟩φ(a) = b, φ(b) = bbb
92005⟨a, b | aaa=b, aaba=b⟩φ(a) = b, φ(b) = bbb
93048⟨a, b | aa=b, aabb=ab⟩φ(a) = b, φ(b) = bb
93049⟨a, b | aa=b, aabb=ba⟩φ(a) = b, φ(b) = bb
93052⟨a, b | aa=b, abab=ab⟩φ(a) = b, φ(b) = bb
93053⟨a, b | aa=b, abab=ba⟩φ(a) = b, φ(b) = bb
93056⟨a, b | aa=b, abba=ab⟩φ(a) = b, φ(b) = bb
93063⟨a, b | aa=b, baab=ab⟩φ(a) = b, φ(b) = bb
93149⟨a, b | aa=a, bbb=aaa⟩φ(a) = bbb, φ(b) = b
93167⟨a, b | aa=b, bbb=aaa⟩φ(a) = b, φ(b) = bb
105045⟨a, b | aaa=ab, aaab=b⟩φ(a) = b, φ(b) = bbbbb
105047⟨a, b | aaa=ab, aaba=b⟩φ(a) = b, φ(b) = bbbbb
105051⟨a, b | aaa=ab, abaa=b⟩φ(a) = b, φ(b) = bbbbb
108916⟨a, b | aa=b, aaaab=ab⟩φ(a) = b, φ(b) = bb
108917⟨a, b | aa=b, aaaab=ba⟩φ(a) = b, φ(b) = bb
108920⟨a, b | aa=b, aaaba=ab⟩φ(a) = b, φ(b) = bb
108921⟨a, b | aa=b, aaaba=ba⟩φ(a) = b, φ(b) = bb
108928⟨a, b | aa=b, aabaa=ab⟩φ(a) = b, φ(b) = bb
109117⟨a, b | aa=a, aaaa=bbb⟩φ(a) = bbb, φ(b) = b
109206⟨a, b | aa=b, aabb=aaa⟩φ(a) = b, φ(b) = bb
109214⟨a, b | aa=b, abab=aaa⟩φ(a) = b, φ(b) = bb
109222⟨a, b | aa=b, abba=aaa⟩φ(a) = b, φ(b) = bb
109236⟨a, b | aa=b, baab=aaa⟩φ(a) = b, φ(b) = bb
1112881⟨a, b | aab=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112883⟨a, b | aab=aaa, aaab=b⟩φ(a) = b, φ(b) = bbbb
1112885⟨a, b | aab=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112887⟨a, b | aab=aaa, aabb=b⟩φ(a) = b, φ(b) = bbbb
1112889⟨a, b | aab=aaa, abaa=b⟩φ(a) = b, φ(b) = bbbb
1112891⟨a, b | aab=aaa, abab=b⟩φ(a) = b, φ(b) = bbbb
1112893⟨a, b | aab=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1112895⟨a, b | aab=aaa, abbb=b⟩φ(a) = b, φ(b) = bbbb
1112913⟨a, b | aba=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112917⟨a, b | aba=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112923⟨a, b | aba=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1115442⟨a, b | aaa=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1118761⟨a, b | aaa=b, aaaaaa=b⟩φ(a) = b, φ(b) = bbb
1119854⟨a, b | aaa=b, aaab=aaa⟩φ(a) = b, φ(b) = bbb
1119862⟨a, b | aaa=b, aaba=aaa⟩φ(a) = b, φ(b) = bbb
1119951⟨a, b | aab=a, aabb=bbb⟩φ(a) = bbbbb, φ(b) = b
1119967⟨a, b | aab=a, abab=bbb⟩φ(a) = bbbbb, φ(b) = b
1119975⟨a, b | aab=a, abba=bbb⟩φ(a) = bbbbb, φ(b) = b
1124725⟨a, b | aa=b, aaaaaa=ab⟩φ(a) = b, φ(b) = bb
1125121⟨a, b | aa=a, aaaaa=bbb⟩φ(a) = bbb, φ(b) = b
1125266⟨a, b | aa=b, aaaab=aaa⟩φ(a) = b, φ(b) = bb
1125274⟨a, b | aa=b, aaaba=aaa⟩φ(a) = b, φ(b) = bb
1125290⟨a, b | aa=b, aabaa=aaa⟩φ(a) = b, φ(b) = bb