〈a, b | bb=aa, abab=aa〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 17 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁷ ⇒ a³
ba⁶ ⇒ ba²
aba² ⇒ ba⁵
a²b ⇒ ba²
b² ⇒ a²
(ab)² ⇒ a²

Idempotents

2 elements

1
a⁴

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	aba	ba²	bab	a⁴	ba³	(ba)²	a⁵	ba⁴	a⁶	ba⁵
1	1	a	b	a²	ab	ba	a³	aba	ba²	bab	a⁴	ba³	(ba)²	a⁵	ba⁴	a⁶	ba⁵
a	a	a²	ab	a³	ba²	aba	a⁴	ba³	ba⁵	a²	a⁵	ba²	a³	a⁶	ba³	a³	ba⁴
b	b	ba	a²	ba²	bab	a³	ba³	(ba)²	a⁴	ba⁵	ba⁴	a⁵	ba²	ba⁵	a⁶	ba²	a³
a²	a²	a³	ba²	a⁴	ba⁵	ba³	a⁵	ba²	ba⁴	a³	a⁶	ba⁵	a⁴	a³	ba²	a⁴	ba³
ab	ab	aba	a³	ba⁵	a²	a⁴	ba²	a³	a⁵	ba⁴	ba³	a⁶	ba⁵	ba⁴	a³	ba⁵	a⁴
ba	ba	ba²	bab	ba³	a⁴	(ba)²	ba⁴	a⁵	a³	ba²	ba⁵	a⁴	ba³	ba²	a⁵	ba³	a⁶
a³	a³	a⁴	ba⁵	a⁵	ba⁴	ba²	a⁶	ba⁵	ba³	a⁴	a³	ba⁴	a⁵	a⁴	ba⁵	a⁵	ba²
aba	aba	ba⁵	a²	ba²	a⁵	a³	ba³	a⁶	a⁴	ba⁵	ba⁴	a⁵	ba²	ba⁵	a⁶	ba²	a³
ba²	ba²	ba³	a⁴	ba⁴	a³	a⁵	ba⁵	a⁴	a⁶	ba³	ba²	a³	ba⁴	ba³	a⁴	ba⁴	a⁵
bab	bab	(ba)²	ba³	a³	ba²	ba⁴	a⁴	ba³	ba⁵	a⁶	a⁵	ba²	a³	a⁶	ba³	a³	ba⁴
a⁴	a⁴	a⁵	ba⁴	a⁶	ba³	ba⁵	a³	ba⁴	ba²	a⁵	a⁴	ba³	a⁶	a⁵	ba⁴	a⁶	ba⁵
ba³	ba³	ba⁴	a³	ba⁵	a⁶	a⁴	ba²	a³	a⁵	ba⁴	ba³	a⁶	ba⁵	ba⁴	a³	ba⁵	a⁴
(ba)²	(ba)²	a³	ba²	a⁴	ba⁵	ba³	a⁵	ba²	ba⁴	a³	a⁶	ba⁵	a⁴	a³	ba²	a⁴	ba³
a⁵	a⁵	a⁶	ba³	a³	ba²	ba⁴	a⁴	ba³	ba⁵	a⁶	a⁵	ba²	a³	a⁶	ba³	a³	ba⁴
ba⁴	ba⁴	ba⁵	a⁶	ba²	a⁵	a³	ba³	a⁶	a⁴	ba⁵	ba⁴	a⁵	ba²	ba⁵	a⁶	ba²	a³
a⁶	a⁶	a³	ba²	a⁴	ba⁵	ba³	a⁵	ba²	ba⁴	a³	a⁶	ba⁵	a⁴	a³	ba²	a⁴	ba³
ba⁵	ba⁵	ba²	a⁵	ba³	a⁴	a⁶	ba⁴	a⁵	a³	ba²	ba⁵	a⁴	ba³	ba²	a⁵	ba³	a⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 71 total

Length:	Presentation:	Description:	Related:
10	〈a, b \| aaaa=b, abbbb=1〉	Isomorphic to ℤ₁₇	39 isomorphic
10	〈a, b \| aab=b, babbb=a〉	Finite non-commutative monoid with 17 elements	1 isomorphic
10	〈a, b \| aba=b, baaab=a〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| aba=b, babbb=a〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| bb=aa, abab=ab〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| bb=aa, abab=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=aaa, bbb=ab〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=aaa, bbb=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aba=aaa, baa=bb〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aba=aaa, bbb=ab〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| abb=aba, bbb=aa〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aaaa=b, abbbb=a〉	Finite commutative monoid with 17 elements	2 isomorphic
11	〈a, b \| aaaa=b, abbbb=b〉	Finite commutative monoid with 17 elements	2 isomorphic
11	〈a, b \| aabb=a, baaaa=b〉	Finite non-commutative monoid with 17 elements	1 isomorphic
11	〈a, b \| abba=a, aaaab=b〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aaa=ab, babbb=b〉	Finite non-commutative monoid with 17 elements	3 isomorphic
11	〈a, b \| aab=aa, bbbb=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=bb, bab=aaa〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=b, abbbba=a〉	Finite non-commutative monoid with 17 elements	4 isomorphic