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〈a, b | abab=a, 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

a³ ⇒ a
aba ⇒ a²b
ab² ⇒ a²
b⁴ ⇒ 1

Idempotents

5 elements

1
a²
bab
b²a
b³a²b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a²b	ba²	bab	b²a	b³	ba²b	b²a²	b²ab	b³a	b²a²b	b³a²	b³ab	b³a²b
1	1	a	b	a²	ab	ba	b²	a²b	ba²	bab	b²a	b³	ba²b	b²a²	b²ab	b³a	b²a²b	b³a²	b³ab	b³a²b
a	a	a²	ab	a	a²b	a²b	a²	ab	ab	a	a	a²b	a²	a²	ab	ab	a²b	a²b	a²	a
b	b	ba	b²	ba²	bab	b²a	b³	ba²b	b²a²	b²ab	b³a	1	b²a²b	b³a²	b³ab	a	b³a²b	a²	ab	a²b
a²	a²	a	a²b	a²	ab	ab	a	a²b	a²b	a²	a²	ab	a	a	a²b	a²b	ab	ab	a	a²
ab	ab	a²b	a²	ab	a	a	a²b	a²	a²	ab	ab	a	a²b	a²b	a²	a²	a	a	a²b	ab
ba	ba	ba²	bab	ba	ba²b	ba²b	ba²	bab	bab	ba	ba	ba²b	ba²	ba²	bab	bab	ba²b	ba²b	ba²	ba
b²	b²	b²a	b³	b²a²	b²ab	b³a	1	b²a²b	b³a²	b³ab	a	b	b³a²b	a²	ab	ba	a²b	ba²	bab	ba²b
a²b	a²b	ab	a	a²b	a²	a²	ab	a	a	a²b	a²b	a²	ab	ab	a	a	a²	a²	ab	a²b
ba²	ba²	ba	ba²b	ba²	bab	bab	ba	ba²b	ba²b	ba²	ba²	bab	ba	ba	ba²b	ba²b	bab	bab	ba	ba²
bab	bab	ba²b	ba²	bab	ba	ba	ba²b	ba²	ba²	bab	bab	ba	ba²b	ba²b	ba²	ba²	ba	ba	ba²b	bab
b²a	b²a	b²a²	b²ab	b²a	b²a²b	b²a²b	b²a²	b²ab	b²ab	b²a	b²a	b²a²b	b²a²	b²a²	b²ab	b²ab	b²a²b	b²a²b	b²a²	b²a
b³	b³	b³a	1	b³a²	b³ab	a	b	b³a²b	a²	ab	ba	b²	a²b	ba²	bab	b²a	ba²b	b²a²	b²ab	b²a²b
ba²b	ba²b	bab	ba	ba²b	ba²	ba²	bab	ba	ba	ba²b	ba²b	ba²	bab	bab	ba	ba	ba²	ba²	bab	ba²b
b²a²	b²a²	b²a	b²a²b	b²a²	b²ab	b²ab	b²a	b²a²b	b²a²b	b²a²	b²a²	b²ab	b²a	b²a	b²a²b	b²a²b	b²ab	b²ab	b²a	b²a²
b²ab	b²ab	b²a²b	b²a²	b²ab	b²a	b²a	b²a²b	b²a²	b²a²	b²ab	b²ab	b²a	b²a²b	b²a²b	b²a²	b²a²	b²a	b²a	b²a²b	b²ab
b³a	b³a	b³a²	b³ab	b³a	b³a²b	b³a²b	b³a²	b³ab	b³ab	b³a	b³a	b³a²b	b³a²	b³a²	b³ab	b³ab	b³a²b	b³a²b	b³a²	b³a
b²a²b	b²a²b	b²ab	b²a	b²a²b	b²a²	b²a²	b²ab	b²a	b²a	b²a²b	b²a²b	b²a²	b²ab	b²ab	b²a	b²a	b²a²	b²a²	b²ab	b²a²b
b³a²	b³a²	b³a	b³a²b	b³a²	b³ab	b³ab	b³a	b³a²b	b³a²b	b³a²	b³a²	b³ab	b³a	b³a	b³a²b	b³a²b	b³ab	b³ab	b³a	b³a²
b³ab	b³ab	b³a²b	b³a²	b³ab	b³a	b³a	b³a²b	b³a²	b³a²	b³ab	b³ab	b³a	b³a²b	b³a²b	b³a²	b³a²	b³a	b³a	b³a²b	b³ab
b³a²b	b³a²b	b³ab	b³a	b³a²b	b³a²	b³a²	b³ab	b³a	b³a	b³a²b	b³a²b	b³a²	b³ab	b³ab	b³a	b³a	b³a²	b³a²	b³ab	b³a²b

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 \| 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 \| 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

2 total

Length:	Presentation:
11	〈a, b \| abbb=aba, bbbb=1〉
11	〈a, b \| aaaa=1, ababa=ab〉