Back

〈a, b | aab=aa, baba=bb〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 20 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁴ ⇒ b³
bab³ ⇒ bab²
b²a ⇒ bab²
a²b ⇒ a²
(ba)² ⇒ b²
a⁴ ⇒ a²

Idempotents

6 elements

1
b³
a²
abab²
ba²
aba³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	bab	b³	aba²	(ab)²	ab³	ba³	bab²	aba³	abab²
1	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	bab	b³	aba²	(ab)²	ab³	ba³	bab²	aba³	abab²
a	a	a²	ab	a³	a²	aba	ab²	a²	a³	a²	aba²	(ab)²	ab³	a²	a³	a²	aba³	abab²	a³	a³
b	b	ba	b²	ba²	bab	bab²	b³	ba³	b²	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
a²	a²	a³	a²	a²	a³	a³	a²	a³	a²	a³	a²	a³	a²	a³	a²	a³	a³	a³	a²	a²
ab	ab	aba	ab²	aba²	(ab)²	abab²	ab³	aba³	ab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba	ba	ba²	bab	ba³	ba²	b²	bab²	ba²	ba³	ba²	bab²	b³	bab²	ba²	ba³	ba²	b³	b³	ba³	ba³
b²	b²	bab²	b³	b³	bab²	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
a³	a³	a²	a³	a³	a²	a²	a³	a²	a³	a²	a³	a²	a³	a²	a³	a²	a²	a²	a³	a³
aba	aba	aba²	(ab)²	aba³	aba²	ab²	abab²	aba²	aba³	aba²	abab²	ab³	abab²	aba²	aba³	aba²	ab³	ab³	aba³	aba³
ab²	ab²	abab²	ab³	ab³	abab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba²	ba²	ba³	ba²	ba²	ba³	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba³	ba³	ba²	ba²
bab	bab	b²	bab²	bab²	b³	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	b³	b³	bab²	bab²
b³	b³	bab²	b³	b³	bab²	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
aba²	aba²	aba³	aba²	aba²	aba³	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba³	aba³	aba²	aba²
(ab)²	(ab)²	ab²	abab²	abab²	ab³	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	ab³	ab³	abab²	abab²
ab³	ab³	abab²	ab³	ab³	abab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba³	ba³	ba²	ba³	ba³	ba²	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba²	ba²	ba³	ba³
bab²	bab²	b³	bab²	bab²	b³	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	b³	b³	bab²	bab²
aba³	aba³	aba²	aba³	aba³	aba²	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba²	aba²	aba³	aba³
abab²	abab²	ab³	abab²	abab²	ab³	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	ab³	ab³	abab²	abab²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

28 unique, 65 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aab=a, bbbb=1〉	Finite non-commutative monoid with 20 elements	3 isomorphic, 1 anti-isomorphic
8	〈a, b \| ab=aa, bbbb=1〉	Finite non-commutative monoid with 20 elements	1 isomorphic
9	〈a, b \| abab=a, bbbb=1〉	Finite non-commutative monoid with 20 elements	2 isomorphic
9	〈a, b \| aba=ab, bbbb=1〉	Finite non-commutative monoid with 20 elements	1 isomorphic, 1 anti-isomorphic
9	〈a, b \| ab=aa, bbbb=b〉	Finite non-commutative monoid with 20 elements
10	〈a, b \| abb=aab, aaaa=1〉	Finite non-commutative monoid with 20 elements	2 isomorphic
10	〈a, b \| baa=abb, abab=1〉	Finite non-Abelian group with 20 elements	1 isomorphic, 1 anti-isomorphic
10	〈a, b \| aaaaa=1, abbbb=1〉	Isomorphic to ℤ₂₀	16 isomorphic
10	〈a, b \| aab=bb, aaaa=a〉	Finite non-commutative monoid with 20 elements
10	〈a, b \| aaa=bb, bab=aa〉	Finite non-commutative monoid with 20 elements	1 isomorphic
10	〈a, b \| aba=a, aaaa=bb〉	Finite non-commutative monoid with 20 elements
10	〈a, b \| aba=b, aaaa=bb〉	Finite non-commutative monoid with 20 elements
10	〈a, b \| ab=aa, bbbb=bb〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aaaa=aa, abbb=b〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aaab=bb, bbba=a〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aaaa=a, bbbbb=a〉	Finite commutative monoid with 20 elements
11	〈a, b \| aaaa=b, bbbbb=a〉	Finite commutative monoid with 20 elements
11	〈a, b \| aaaa=b, bbbbb=b〉	Finite commutative monoid with 20 elements
11	〈a, b \| aab=ba, abbabb=1〉	Finite non-Abelian group with 20 elements	2 isomorphic
11	〈a, b \| aba=bb, baaab=a〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aba=aa, aaaa=bb〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aba=aa, aaab=bb〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aba=bb, abb=aaa〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| aaaa=1, aaabab=b〉	Finite non-commutative monoid with 20 elements	1 isomorphic
11	〈a, b \| aaaa=1, aabab=ab〉	Finite non-commutative monoid with 20 elements	1 isomorphic
11	〈a, b \| aba=b, aaaaabb=1〉	Finite non-Abelian group with 20 elements	3 isomorphic
11	〈a, b \| aab=a, bbbbb=bb〉	Finite non-commutative monoid with 20 elements
11	〈a, b \| ab=aa, bbbb=bbb〉	Finite non-commutative monoid with 20 elements