〈a, b | aaa=ab, bbbb=ba〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁹ ⇒ a⁴
ba⁷ ⇒ ba²
ab ⇒ a³
b²a ⇒ ba³
b⁴ ⇒ ba

Idempotents

3 elements

1
a⁵
ba³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
1	1	a	b	a²	ba	b²	a³	ba²	b³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁷	a⁵	a⁶	a⁶	a⁷	a⁷	a⁸	a⁸	a⁴	a⁴
b	b	ba	b²	ba²	ba³	b³	ba³	ba⁴	ba	ba⁴	ba⁵	ba⁵	ba⁶	ba⁶	ba²	ba²	ba³	ba³
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a⁸	a⁶	a⁷	a⁷	a⁸	a⁸	a⁴	a⁴	a⁵	a⁵
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba²	ba⁵	ba⁶	ba⁶	ba²	ba²	ba³	ba³	ba⁴	ba⁴
b²	b²	ba³	b³	ba⁴	ba⁵	ba	ba⁵	ba⁶	ba³	ba⁶	ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵
a³	a³	a⁴	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁴	a⁷	a⁸	a⁸	a⁴	a⁴	a⁵	a⁵	a⁶	a⁶
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵
b³	b³	ba⁵	ba	ba⁶	ba²	ba³	ba²	ba³	ba⁵	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba⁶	ba⁶	ba²	ba²
a⁴	a⁴	a⁵	a⁶	a⁶	a⁷	a⁸	a⁷	a⁸	a⁵	a⁸	a⁴	a⁴	a⁵	a⁵	a⁶	a⁶	a⁷	a⁷
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba²	ba⁶	ba²	ba⁴	ba²	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba⁶	ba⁶
a⁵	a⁵	a⁶	a⁷	a⁷	a⁸	a⁴	a⁸	a⁴	a⁶	a⁴	a⁵	a⁵	a⁶	a⁶	a⁷	a⁷	a⁸	a⁸
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba²	ba³	ba²	ba³	ba⁵	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba⁶	ba⁶	ba²	ba²
a⁶	a⁶	a⁷	a⁸	a⁸	a⁴	a⁵	a⁴	a⁵	a⁷	a⁵	a⁶	a⁶	a⁷	a⁷	a⁸	a⁸	a⁴	a⁴
ba⁵	ba⁵	ba⁶	ba²	ba²	ba³	ba⁴	ba³	ba⁴	ba⁶	ba⁴	ba⁵	ba⁵	ba⁶	ba⁶	ba²	ba²	ba³	ba³
a⁷	a⁷	a⁸	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a⁸	a⁶	a⁷	a⁷	a⁸	a⁸	a⁴	a⁴	a⁵	a⁵
ba⁶	ba⁶	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba²	ba⁵	ba⁶	ba⁶	ba²	ba²	ba³	ba³	ba⁴	ba⁴
a⁸	a⁸	a⁴	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁴	a⁷	a⁸	a⁸	a⁴	a⁴	a⁵	a⁵	a⁶	a⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 104 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aa=1, abbba=b〉	Finite non-commutative monoid with 18 elements	14 isomorphic
9	〈a, b \| aa=1, abbbbb=b〉	Finite non-commutative monoid with 18 elements	22 isomorphic, 9 anti-isomorphic
10	〈a, b \| aaab=1, bbbbbb=1〉	Isomorphic to ℤ₁₈	33 isomorphic
10	〈a, b \| aba=a, aaab=bb〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| aba=b, baab=aa〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| aba=b, bbbb=aa〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| bb=aa, aaaba=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
10	〈a, b \| aa=a, bbbbb=ab〉	Finite non-commutative monoid with 18 elements	1 isomorphic
10	〈a, b \| aa=1, abababa=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aaaa=ab, babb=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aab=bb, bbbba=a〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=ab, bbbb=ab〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aab=bb, baba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=a, bbbbbb=a〉	Finite commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=a〉	Finite commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=b〉	Finite commutative monoid with 18 elements
11	〈a, b \| aab=b, bbbba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| bb=aa, ababa=aa〉	Finite non-commutative monoid with 18 elements	1 isomorphic