〈a, b | aaa=bb, aaba=ab〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 19 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a¹⁰ ⇒ a⁴
ba⁹ ⇒ ba³
ab ⇒ ba⁵
b² ⇒ a³

Idempotents

2 elements

1
a⁶

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	ba⁸
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
a	a	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
b	b	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a²	a²	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba	ba	ba²	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a³	a³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba²	ba²	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹
a⁴	a⁴	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴
ba³	ba³	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a⁵	a⁵	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³
ba⁴	ba⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
ba⁵	ba⁵	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶
a⁷	a⁷	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a⁸	a⁸	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba⁷	ba⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a⁹	a⁹	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba⁸	ba⁸	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹

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
10	〈a, b \| aba=b, bab=aaa〉	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