#16032 ⟨a, b | aaa=ab, bbaa=bb⟩

Up:Monoids with two generators and two relations
Prev:#16031 ⟨a, b | aaa=ab, bbaa=ba⟩
Next:#16033 ⟨a, b | aaa=ab, bbab=aa⟩

Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Elements

Elements in the center commute with all other 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.

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aa4ba3a5ba4a6ba5ba6
11aba2bab2a3ba2b2aa4ba3a5ba4a6ba5ba6
aaa2a3a3a4a5a4a5a6a5a6a6a5a5a6a5
bbbab2ba2b2ab2ba3b2b2aba4b2aba5b2ba6b2ab2
a2a2a3a4a4a5a6a5a6a5a6a5a5a6a6a5a6
bababa2ba3ba3ba4ba5ba4ba5ba6ba5ba6ba6ba5ba5ba6ba5
b2b2b2ab2b2b2ab2b2ab2b2ab2b2ab2ab2b2b2ab2
a3a3a4a5a5a6a5a6a5a6a5a6a6a5a5a6a5
ba2ba2ba3ba4ba4ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6
b2ab2ab2b2ab2ab2b2ab2b2ab2b2ab2b2b2ab2ab2b2a
a4a4a5a6a6a5a6a5a6a5a6a5a5a6a6a5a6
ba3ba3ba4ba5ba5ba6ba5ba6ba5ba6ba5ba6ba6ba5ba5ba6ba5
a5a5a6a5a5a6a5a6a5a6a5a6a6a5a5a6a5
ba4ba4ba5ba6ba6ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6
a6a6a5a6a6a5a6a5a6a5a6a5a5a6a6a5a6
ba5ba5ba6ba5ba5ba6ba5ba6ba5ba6ba5ba6ba6ba5ba5ba6ba5
ba6ba6ba5ba6ba6ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6

Right Cayley graph

Idempotents are shown in bold.

Left 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. a7 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2a2 ⇒ b2 [2]
4. b3 ⇒ b2 [4]
# ab:aaa=ab,bbaa=bb a/b
aaaaaaa=aaaaa
ab=aaa
bbaa=bb
bbb=bb

Same cardinality

20 unique, 175 total

Σ#PresentationDescriptionRelated
8628⟨a, b | bb=aa, abab=1⟩Finite non-Abelian group with 16 elements58 iso
91331⟨a, b | aaaa=b, bbbb=1⟩Isomorphic to ℤ1667 iso
92051⟨a, b | aab=a, bbbb=b⟩Finite non-commutative monoid with 16 elements4 anti-iso
103808⟨a, b | aaab=ba, abab=1⟩Finite non-Abelian group with 16 elements7 iso
104630⟨a, b | aaaa=a, bbbb=a⟩Isomorphic to ℕ(16 = 4)1 iso
104648⟨a, b | aaaa=b, bbbb=a⟩Isomorphic to ℕ(16 = 1)5 iso
106205⟨a, b | aba=b, aaaabb=1⟩Finite non-Abelian group with 16 elements3 iso
1112164⟨a, b | aaaa=aa, bbbb=a⟩Isomorphic to ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Isomorphic to ℕ(16 = 5)2 iso
1112212⟨a, b | aaaa=bb, bbbb=a⟩Isomorphic to ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Finite non-commutative monoid with 16 elements6 iso
1113251⟨a, b | bab=aab, bbb=aa⟩Finite non-commutative monoid with 16 elements
1113259⟨a, b | bab=aba, bbb=aa⟩Finite non-commutative monoid with 16 elements
1116028⟨a, b | aaa=ab, babb=bb⟩Finite non-commutative monoid with 16 elements
1116060⟨a, b | aaa=bb, abab=aa⟩Finite non-commutative monoid with 16 elements
1116371⟨a, b | aba=aa, bbbb=ab⟩Finite non-commutative monoid with 16 elements
1118811⟨a, b | aaa=b, abbbbb=b⟩Isomorphic to ℕ(16 = 3)2 iso
1120251⟨a, b | aba=b, aaaa=abb⟩Finite non-commutative monoid with 16 elements
1121039⟨a, b | ab=aa, bbba=bbb⟩Finite non-commutative monoid with 16 elements
1124660⟨a, b | aa=a, ababab=bb⟩Finite non-commutative monoid with 16 elements