〈a, b | bb=aa, abab=ab〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 17 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁸ ⇒ b⁴
ab⁴ ⇒ b⁷
bab³ ⇒ b³
b²a ⇒ ab²
a² ⇒ b²
(ab)² ⇒ ab

Idempotents

4 elements

1
b⁴
ab
(ba)²

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	bab	b³	ab³	(ba)²	bab²	b⁴	b⁵	b⁶	b⁷
1	1	a	b	ab	ba	b²	aba	ab²	bab	b³	ab³	(ba)²	bab²	b⁴	b⁵	b⁶	b⁷
a	a	b²	ab	b³	aba	ab²	bab²	b⁴	ab	ab³	b⁵	aba	ab²	b⁷	b⁴	b⁵	b⁶
b	b	ba	b²	bab	ab²	b³	(ba)²	bab²	ab³	b⁴	b³	ab²	b⁷	b⁵	b⁶	b⁷	b⁴
ab	ab	aba	ab²	ab	b⁴	ab³	aba	ab²	b⁵	b⁷	ab³	b⁴	b⁶	b⁴	b⁵	b⁶	b⁷
ba	ba	b³	bab	b⁴	(ba)²	bab²	b⁷	b⁵	bab	b³	b⁶	(ba)²	bab²	b⁴	b⁵	b⁶	b⁷
b²	b²	ab²	b³	ab³	bab²	b⁴	ab²	b⁷	b³	b⁵	b⁴	bab²	b⁴	b⁶	b⁷	b⁴	b⁵
aba	aba	ab³	ab	b⁷	aba	ab²	b⁶	b⁴	ab	ab³	b⁵	aba	ab²	b⁷	b⁴	b⁵	b⁶
ab²	ab²	b⁴	ab³	b⁵	ab²	b⁷	b⁴	b⁶	ab³	b⁴	b⁷	ab²	b⁷	b⁵	b⁶	b⁷	b⁴
bab	bab	(ba)²	bab²	bab	b⁵	b³	(ba)²	bab²	b⁶	b⁴	b³	b⁵	b⁷	b⁵	b⁶	b⁷	b⁴
b³	b³	bab²	b⁴	b³	b⁷	b⁵	bab²	b⁴	b⁴	b⁶	b⁵	b⁷	b⁵	b⁷	b⁴	b⁵	b⁶
ab³	ab³	ab²	b⁷	ab³	b⁶	b⁴	ab²	b⁷	b⁷	b⁵	b⁴	b⁶	b⁴	b⁶	b⁷	b⁴	b⁵
(ba)²	(ba)²	b³	bab	b⁴	(ba)²	bab²	b⁷	b⁵	bab	b³	b⁶	(ba)²	bab²	b⁴	b⁵	b⁶	b⁷
bab²	bab²	b⁵	b³	b⁶	bab²	b⁴	b⁵	b⁷	b³	b⁵	b⁴	bab²	b⁴	b⁶	b⁷	b⁴	b⁵
b⁴	b⁴	b⁷	b⁵	b⁴	b⁴	b⁶	b⁷	b⁵	b⁵	b⁷	b⁶	b⁴	b⁶	b⁴	b⁵	b⁶	b⁷
b⁵	b⁵	b⁴	b⁶	b⁵	b⁵	b⁷	b⁴	b⁶	b⁶	b⁴	b⁷	b⁵	b⁷	b⁵	b⁶	b⁷	b⁴
b⁶	b⁶	b⁵	b⁷	b⁶	b⁶	b⁴	b⁵	b⁷	b⁷	b⁵	b⁴	b⁶	b⁴	b⁶	b⁷	b⁴	b⁵
b⁷	b⁷	b⁶	b⁴	b⁷	b⁷	b⁵	b⁶	b⁴	b⁴	b⁶	b⁵	b⁷	b⁵	b⁷	b⁴	b⁵	b⁶

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=aa〉	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