〈a, b | aa=a, abba=b〉

Monoid presentation of length 8

Properties

Finite commutative monoid with 3 elements
Isomorphic to commutative presentation: 〈a, b | 2b=b, a+b=b, 2a=a〉
Grothendieck group is the trivial group

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a² ⇒ a
ab ⇒ b
ba ⇒ b
b² ⇒ b

Idempotents

3 elements

1
a
b

Cayley table

Idempotents are shown in bold.

	1	a	b
1	1	a	b
a	a	a	b
b	b	b	b

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 2195 total

Length:	Presentation:	Description:	Related:
5	〈a, b \| aa=b, ab=1〉	Isomorphic to ℤ₃	2029 isomorphic
6	〈a, b \| aa=b, ab=a〉	Finite commutative monoid with 3 elements	61 isomorphic
6	〈a, b \| aa=b, ab=b〉	Finite commutative monoid with 3 elements	23 isomorphic
6	〈a, b \| ab=a, ba=b〉	Finite non-commutative monoid with 3 elements	61 isomorphic
9	〈a, b \| aba=a, aabb=b〉	Finite non-commutative monoid with 3 elements	16 isomorphic

Other isomorphic instances

12 total

Length:	Presentation:
9	〈a, b \| aa=a, aabba=b〉
9	〈a, b \| aa=a, ababa=b〉
10	〈a, b \| aa=a, aaabba=b〉
10	〈a, b \| aa=a, aababa=b〉
10	〈a, b \| aa=a, aabbaa=b〉
10	〈a, b \| aa=a, abaaba=b〉
11	〈a, b \| aa=a, aaaabba=b〉
11	〈a, b \| aa=a, aaababa=b〉
11	〈a, b \| aa=a, aaabbaa=b〉
11	〈a, b \| aa=a, aabaaba=b〉
11	〈a, b \| aa=a, aababaa=b〉
11	〈a, b \| aa=a, abaaaba=b〉