〈a, b | aaa=aa, abba=b〉

Monoid presentation of length 10

Properties

Finite commutative monoid with 4 elements
Isomorphic to commutative presentation: 〈a, b | 2b=b, a+b=b, 3a=2a〉
Grothendieck group is the trivial group

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

ab ⇒ b
ba ⇒ b
b² ⇒ b
a³ ⇒ a²

Idempotents

3 elements

1
b
a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²
1	1	a	b	a²
a	a	a²	b	a²
b	b	b	b	b
a²	a²	a²	b	a²

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1605 total

Length:	Presentation:	Description:	Related:
5	〈a, b \| aa=b, bb=1〉	Isomorphic to ℤ₄	1419 isomorphic
6	〈a, b \| aa=a, bb=a〉	Finite commutative monoid with 4 elements	37 isomorphic
6	〈a, b \| aa=b, bb=a〉	Finite commutative monoid with 4 elements	72 isomorphic
7	〈a, b \| ab=aa, ba=b〉	Finite non-commutative monoid with 4 elements	8 isomorphic, 6 anti-isomorphic
7	〈a, b \| ab=aa, bb=a〉	Finite commutative monoid with 4 elements	16 isomorphic
7	〈a, b \| aa=a, abb=b〉	Finite non-commutative monoid with 4 elements	14 isomorphic
7	〈a, b \| ab=a, bb=aa〉	Finite commutative monoid with 4 elements	17 isomorphic
9	〈a, b \| aa=a, abbba=b〉	Finite commutative monoid with 4 elements	8 isomorphic

Other isomorphic instances

2 total

Length:	Presentation:
11	〈a, b \| aaa=aa, aabba=b〉
11	〈a, b \| aaa=aa, ababa=b〉