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〈a, b | aaaa=1, aabab=ab〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 20 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a²b ⇒ b²
bab ⇒ ab²
b³ ⇒ b
a⁴ ⇒ 1

Idempotents

5 elements

1
b²
aba
ba²
ab²a³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³
1	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³
a	a	a²	ab	a³	b²	aba	ab²	1	b²a	b	aba²	ab²a	b²a²	ba	aba³	ab²a²	b²a³	ba²	ab²a³	ba³
b	b	ba	b²	ba²	ab²	b²a	b	ba³	ab²a	ab	b²a²	ba	ab²a²	aba	b²a³	ba²	ab²a³	aba²	ba³	aba³
a²	a²	a³	b²	1	ab²	b²a	b	a	ab²a	ab	b²a²	ba	ab²a²	aba	b²a³	ba²	ab²a³	aba²	ba³	aba³
ab	ab	aba	ab²	aba²	b	ab²a	ab	aba³	ba	b²	ab²a²	aba	ba²	b²a	ab²a³	aba²	ba³	b²a²	aba³	b²a³
ba	ba	ba²	ab²	ba³	b	ab²a	ab	b	ba	b²	ab²a²	aba	ba²	b²a	ab²a³	aba²	ba³	b²a²	aba³	b²a³
b²	b²	b²a	b	b²a²	ab	ba	b²	b²a³	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³
a³	a³	1	ab²	a	b	ab²a	ab	a²	ba	b²	ab²a²	aba	ba²	b²a	ab²a³	aba²	ba³	b²a²	aba³	b²a³
aba	aba	aba²	b	aba³	ab	ba	b²	ab	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³
ab²	ab²	ab²a	ab	ab²a²	b²	aba	ab²	ab²a³	b²a	b	aba²	ab²a	b²a²	ba	aba³	ab²a²	b²a³	ba²	ab²a³	ba³
ba²	ba²	ba³	b	b	ab	ba	b²	ba	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³
b²a	b²a	b²a²	ab	b²a³	b²	aba	ab²	b²	b²a	b	aba²	ab²a	b²a²	ba	aba³	ab²a²	b²a³	ba²	ab²a³	ba³
aba²	aba²	aba³	ab	ab	b²	aba	ab²	aba	b²a	b	aba²	ab²a	b²a²	ba	aba³	ab²a²	b²a³	ba²	ab²a³	ba³
ab²a	ab²a	ab²a²	b²	ab²a³	ab²	b²a	b	ab²	ab²a	ab	b²a²	ba	ab²a²	aba	b²a³	ba²	ab²a³	aba²	ba³	aba³
ba³	ba³	b	ab	ba	b²	aba	ab²	ba²	b²a	b	aba²	ab²a	b²a²	ba	aba³	ab²a²	b²a³	ba²	ab²a³	ba³
b²a²	b²a²	b²a³	b²	b²	ab²	b²a	b	b²a	ab²a	ab	b²a²	ba	ab²a²	aba	b²a³	ba²	ab²a³	aba²	ba³	aba³
aba³	aba³	ab	b²	aba	ab²	b²a	b	aba²	ab²a	ab	b²a²	ba	ab²a²	aba	b²a³	ba²	ab²a³	aba²	ba³	aba³
ab²a²	ab²a²	ab²a³	ab²	ab²	b	ab²a	ab	ab²a	ba	b²	ab²a²	aba	ba²	b²a	ab²a³	aba²	ba³	b²a²	aba³	b²a³
b²a³	b²a³	b²	ab²	b²a	b	ab²a	ab	b²a²	ba	b²	ab²a²	aba	ba²	b²a	ab²a³	aba²	ba³	b²a²	aba³	b²a³
ab²a³	ab²a³	ab²	b	ab²a	ab	ba	b²	ab²a²	aba	ab²	ba²	b²a	aba²	ab²a	ba³	b²a²	aba³	ab²a²	b²a³	ab²a³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

28 unique, 64 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 \| aab=aa, baba=bb〉	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 \| 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

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aaaa=1, baaab=ab〉