〈a, b | aa=a, abbba=b〉

Monoid presentation of length 9

Properties

Finite commutative monoid with 4 elements
Isomorphic to commutative presentation: 〈a, b | 2b=b, a+b=a, 3a=a〉
Grothendieck group is ℤ₂

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a² ⇒ a
ab ⇒ b
ba ⇒ b
b³ ⇒ b

Idempotents

3 elements

1
a
b²

Cayley table

Idempotents are shown in bold.

	1	a	b	b²
1	1	a	b	b²
a	a	a	b	b²
b	b	b	b²	b
b²	b²	b²	b	b²

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1599 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
10	〈a, b \| aaa=aa, abba=b〉	Finite commutative monoid with 4 elements	2 isomorphic

Other isomorphic instances

8 total

Length:	Presentation:
10	〈a, b \| aa=a, aabbba=b〉
10	〈a, b \| aa=a, ababba=b〉
11	〈a, b \| aa=a, aaabbba=b〉
11	〈a, b \| aa=a, aababba=b〉
11	〈a, b \| aa=a, aabbaba=b〉
11	〈a, b \| aa=a, aabbbaa=b〉
11	〈a, b \| aa=a, abaabba=b〉
11	〈a, b \| aa=a, abababa=b〉