〈a, b | aa=1, abababa=b〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁵ ⇒ b
bab⁴ ⇒ ba
b²a ⇒ ab²
a² ⇒ 1
(ba)² ⇒ abab³

Idempotents

2 elements

1
b⁴

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	bab	b³	(ab)²	ab³	bab²	b⁴	abab²	ab⁴	bab³	abab³
1	1	a	b	ab	ba	b²	aba	ab²	bab	b³	(ab)²	ab³	bab²	b⁴	abab²	ab⁴	bab³	abab³
a	a	1	ab	b	aba	ab²	ba	b²	(ab)²	ab³	bab	b³	abab²	ab⁴	bab²	b⁴	abab³	bab³
b	b	ba	b²	bab	ab²	b³	abab³	bab²	ab³	b⁴	aba	bab³	ab⁴	b	(ab)²	ba	ab	abab²
ab	ab	aba	ab²	(ab)²	b²	ab³	bab³	abab²	b³	ab⁴	ba	abab³	b⁴	ab	bab	aba	b	bab²
ba	ba	b	bab	b²	abab³	bab²	ab²	b³	aba	bab³	ab³	b⁴	(ab)²	ba	ab⁴	b	abab²	ab
b²	b²	ab²	b³	ab³	bab²	b⁴	abab²	ab⁴	bab³	b	abab³	ab	ba	b²	aba	ab²	bab	(ab)²
aba	aba	ab	(ab)²	ab²	bab³	abab²	b²	ab³	ba	abab³	b³	ab⁴	bab	aba	b⁴	ab	bab²	b
ab²	ab²	b²	ab³	b³	abab²	ab⁴	bab²	b⁴	abab³	ab	bab³	b	aba	ab²	ba	b²	(ab)²	bab
bab	bab	abab³	bab²	aba	b³	bab³	ab	(ab)²	b⁴	ba	ab²	abab²	b	bab	ab³	abab³	b²	ab⁴
b³	b³	bab²	b⁴	bab³	ab⁴	b	(ab)²	ba	ab	b²	abab²	bab	ab²	b³	abab³	bab²	ab³	aba
(ab)²	(ab)²	bab³	abab²	ba	ab³	abab³	b	bab	ab⁴	aba	b²	bab²	ab	(ab)²	b³	bab³	ab²	b⁴
ab³	ab³	abab²	ab⁴	abab³	b⁴	ab	bab	aba	b	ab²	bab²	(ab)²	b²	ab³	bab³	abab²	b³	ba
bab²	bab²	b³	bab³	b⁴	(ab)²	ba	ab⁴	b	abab²	bab	ab	b²	abab³	bab²	ab²	b³	aba	ab³
b⁴	b⁴	ab⁴	b	ab	ba	b²	aba	ab²	bab	b³	(ab)²	ab³	bab²	b⁴	abab²	ab⁴	bab³	abab³
abab²	abab²	ab³	abab³	ab⁴	bab	aba	b⁴	ab	bab²	(ab)²	b	ab²	bab³	abab²	b²	ab³	ba	b³
ab⁴	ab⁴	b⁴	ab	b	aba	ab²	ba	b²	(ab)²	ab³	bab	b³	abab²	ab⁴	bab²	b⁴	abab³	bab³
bab³	bab³	(ab)²	ba	abab²	b	bab	ab³	abab³	b²	bab²	ab⁴	aba	b³	bab³	ab	(ab)²	b⁴	ab²
abab³	abab³	bab	aba	bab²	ab	(ab)²	b³	bab³	ab²	abab²	b⁴	ba	ab³	abab³	b	bab	ab⁴	b²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 102 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
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 \| aaa=ab, bbbb=ba〉	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

Other isomorphic instances

2 total

Length:	Presentation:
10	〈a, b \| aa=1, ababab=ba〉
10	〈a, b \| aa=1, babab=aba〉