〈a, b | aba=b, bab=aaa〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 19 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a¹¹ ⇒ a³
ba⁸ ⇒ b
ab ⇒ ba⁷
b² ⇒ a⁴

Idempotents

2 elements

1
a⁸

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰
a	a	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a³
b	b	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a³	ba	ba²
a²	a²	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	a⁴
ba	ba	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	ba³
a³	a³	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a³	ba⁴	a⁴	a⁵
ba²	ba²	ba³	a¹⁰	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	ba⁴
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	b	a¹⁰	ba	a³	ba²	a⁴	ba³	a⁵	a⁶
ba³	ba³	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a³	ba⁷	a⁴	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	ba⁵
a⁵	a⁵	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	a⁷
ba⁴	ba⁴	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a³	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	ba⁶
a⁶	a⁶	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	a⁸
ba⁵	ba⁵	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	ba⁷
a⁷	a⁷	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	a⁹
ba⁶	ba⁶	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷	b
a⁸	a⁸	a⁹	b	a¹⁰	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰
ba⁷	ba⁷	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a³	ba⁷	a⁴	b	ba
a⁹	a⁹	a¹⁰	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a³
a¹⁰	a¹⁰	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 29 total

Length:	Presentation:	Description:	Related:
10	〈a, b \| aaa=a, abbba=b〉	Finite non-commutative monoid with 19 elements
11	〈a, b \| aaaab=1, abbbbb=1〉	Isomorphic to ℤ₁₉	20 isomorphic
11	〈a, b \| aaab=b, abbba=a〉	Finite non-commutative monoid with 19 elements	3 isomorphic
11	〈a, b \| aaa=ab, bbab=bb〉	Finite non-commutative monoid with 19 elements	1 isomorphic
11	〈a, b \| aaa=bb, aaba=ab〉	Finite non-commutative monoid with 19 elements