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〈a, b | aba=ab, bbbb=1〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 20 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a² ⇒ a
aba ⇒ ab
ab²a ⇒ ab²
b⁴ ⇒ 1
ab³a ⇒ ab³

Idempotents

5 elements

1
a
bab³
b²ab²
b³ab

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab³	bab²	b²ab	b³a	bab³	b²ab²	b³ab	b²ab³	b³ab²	b³ab³
1	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab³	bab²	b²ab	b³a	bab³	b²ab²	b³ab	b²ab³	b³ab²	b³ab³
a	a	a	ab	ab	ab	ab²	ab²	ab²	ab²	ab³	ab³	ab³	ab³	ab³	a	a	a	ab	ab	ab²
b	b	ba	b²	bab	b²a	b³	bab²	b²ab	b³a	1	bab³	b²ab²	b³ab	a	b²ab³	b³ab²	ab	b³ab³	ab²	ab³
ab	ab	ab	ab²	ab²	ab²	ab³	ab³	ab³	ab³	a	a	a	a	a	ab	ab	ab	ab²	ab²	ab³
ba	ba	ba	bab	bab	bab	bab²	bab²	bab²	bab²	bab³	bab³	bab³	bab³	bab³	ba	ba	ba	bab	bab	bab²
b²	b²	b²a	b³	b²ab	b³a	1	b²ab²	b³ab	a	b	b²ab³	b³ab²	ab	ba	b³ab³	ab²	bab	ab³	bab²	bab³
ab²	ab²	ab²	ab³	ab³	ab³	a	a	a	a	ab	ab	ab	ab	ab	ab²	ab²	ab²	ab³	ab³	a
bab	bab	bab	bab²	bab²	bab²	bab³	bab³	bab³	bab³	ba	ba	ba	ba	ba	bab	bab	bab	bab²	bab²	bab³
b²a	b²a	b²a	b²ab	b²ab	b²ab	b²ab²	b²ab²	b²ab²	b²ab²	b²ab³	b²ab³	b²ab³	b²ab³	b²ab³	b²a	b²a	b²a	b²ab	b²ab	b²ab²
b³	b³	b³a	1	b³ab	a	b	b³ab²	ab	ba	b²	b³ab³	ab²	bab	b²a	ab³	bab²	b²ab	bab³	b²ab²	b²ab³
ab³	ab³	ab³	a	a	a	ab	ab	ab	ab	ab²	ab²	ab²	ab²	ab²	ab³	ab³	ab³	a	a	ab
bab²	bab²	bab²	bab³	bab³	bab³	ba	ba	ba	ba	bab	bab	bab	bab	bab	bab²	bab²	bab²	bab³	bab³	ba
b²ab	b²ab	b²ab	b²ab²	b²ab²	b²ab²	b²ab³	b²ab³	b²ab³	b²ab³	b²a	b²a	b²a	b²a	b²a	b²ab	b²ab	b²ab	b²ab²	b²ab²	b²ab³
b³a	b³a	b³a	b³ab	b³ab	b³ab	b³ab²	b³ab²	b³ab²	b³ab²	b³ab³	b³ab³	b³ab³	b³ab³	b³ab³	b³a	b³a	b³a	b³ab	b³ab	b³ab²
bab³	bab³	bab³	ba	ba	ba	bab	bab	bab	bab	bab²	bab²	bab²	bab²	bab²	bab³	bab³	bab³	ba	ba	bab
b²ab²	b²ab²	b²ab²	b²ab³	b²ab³	b²ab³	b²a	b²a	b²a	b²a	b²ab	b²ab	b²ab	b²ab	b²ab	b²ab²	b²ab²	b²ab²	b²ab³	b²ab³	b²a
b³ab	b³ab	b³ab	b³ab²	b³ab²	b³ab²	b³ab³	b³ab³	b³ab³	b³ab³	b³a	b³a	b³a	b³a	b³a	b³ab	b³ab	b³ab	b³ab²	b³ab²	b³ab³
b²ab³	b²ab³	b²ab³	b²a	b²a	b²a	b²ab	b²ab	b²ab	b²ab	b²ab²	b²ab²	b²ab²	b²ab²	b²ab²	b²ab³	b²ab³	b²ab³	b²a	b²a	b²ab
b³ab²	b³ab²	b³ab²	b³ab³	b³ab³	b³ab³	b³a	b³a	b³a	b³a	b³ab	b³ab	b³ab	b³ab	b³ab	b³ab²	b³ab²	b³ab²	b³ab³	b³ab³	b³a
b³ab³	b³ab³	b³ab³	b³a	b³a	b³a	b³ab	b³ab	b³ab	b³ab	b³ab²	b³ab²	b³ab²	b³ab²	b³ab²	b³ab³	b³ab³	b³ab³	b³a	b³a	b³ab

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

28 unique, 63 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 \| 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 \| 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

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| abab=abb, bbbb=1〉

Other anti-isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| abab=bab, bbbb=1〉