〈a, b | aaa=ab, bbab=bb〉

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

a⁸ ⇒ a⁵
ab ⇒ a³
b²a³ ⇒ b²
b³ ⇒ b²a²

Idempotents

4 elements

1
a⁶
ba⁷
b²a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
b	b	ba	b²	ba²	b²a	b²a²	ba³	b²a²	b²	ba⁴	b²	b²a	ba⁵	b²a	ba⁶	b²a²	ba⁷	b²	b²a
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
b²	b²	b²a	b²a²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²	b²a	b²a	b²a²	b²
a³	a³	a⁴	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
b²a	b²a	b²a²	b²	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a	b²a²	b²a²	b²	b²a
a⁴	a⁴	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
b²a²	b²a²	b²	b²a	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²a²	b²	b²	b²a	b²a²
a⁵	a⁵	a⁶	a⁷	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
a⁶	a⁶	a⁷	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba⁵	ba⁵	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
a⁷	a⁷	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba⁶	ba⁶	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
ba⁷	ba⁷	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 28 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=bb, aaba=ab〉	Finite non-commutative monoid with 19 elements

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aaa=ab, bbba=bb〉