#24660 ⟨a, b | aa=a, ababab=bb⟩

Up:Monoids with two generators and two relations
Prev:#24658 ⟨a, b | aa=a, ababab=ab⟩
Next:#24666 ⟨a, b | aa=a, ababbb=ab⟩

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.
A right zero element x satisfies yx = x for all y.
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.

1ababbab2abababb2ab3(ab)2(ba)2b3aa(ba)2b(ab)2(ba)3
11ababbab2abababb2ab3(ab)2(ba)2b3aa(ba)2b(ab)2(ba)3
aaaabababab2aba(ab)2b2ab3(ab)2a(ba)2b3aa(ba)2b2b2a
bbbab2babb2ab3(ba)2b3b3ab3b(ab)2b3ab3a(ba)3b3b3a
abababab2(ab)2b2ab3a(ba)2b3b3ab3b2b3ab3ab2ab3b3a
bababababbab(ba)2b3(ba)2b(ab)2b3ab3b(ab)2(ba)3b3a(ba)3b3b3a
b2b2b2ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
abaabaaba(ab)2(ab)2a(ba)2b3a(ba)2b2b3ab3b2b2ab3ab2ab3b3a
babbab(ba)2b3b(ab)2b3ab3(ba)3b3b3ab3b3b3ab3ab3ab3b3a
b2ab2ab2ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
b3b3b3ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
(ab)2(ab)2a(ba)2b3b2b3ab3b2ab3b3ab3b3b3ab3ab3ab3b3a
(ba)2(ba)2(ba)2b(ab)2b(ab)2(ba)3b3(ba)3b3b3ab3b3b3ab3ab3ab3b3a
b3ab3ab3ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
a(ba)2a(ba)2a(ba)2b2b2b2ab3b2ab3b3ab3b3b3ab3ab3ab3b3a
b(ab)2b(ab)2(ba)3b3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
(ba)3(ba)3(ba)3b3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a

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. a2 ⇒ a [1]
2. ab2 ⇒ b2 [3]
3. b2ab ⇒ b3 [4]
4. b4 ⇒ b3 [5]
5. (ab)3 ⇒ b2 [2]
# ab:aa=a,ababab=bb ab
aa=a
abb=bb
bbab=bbb
bbbb=bbb
ababab=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
1116032⟨a, b | aaa=ab, bbaa=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