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〈a, b | aaaa=1, aaabab=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 20 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

b² ⇒ b
bab ⇒ ab
a⁴ ⇒ 1
ba²b ⇒ a²b
ba³b ⇒ a³b

Idempotents

5 elements

1
b
a³ba
a²ba²
aba³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
1	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
a	a	a²	ab	a³	a²b	aba	1	a³b	a²ba	aba²	b	a³ba	a²ba²	aba³	ba	a³ba²	a²ba³	ba²	a³ba³	ba³
b	b	ba	b	ba²	ab	ba	ba³	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
a²	a²	a³	a²b	1	a³b	a²ba	a	b	a³ba	a²ba²	ab	ba	a³ba²	a²ba³	aba	ba²	a³ba³	aba²	ba³	aba³
ab	ab	aba	ab	aba²	a²b	aba	aba³	a³b	a²ba	aba²	b	a³ba	a²ba²	aba³	ba	a³ba²	a²ba³	ba²	a³ba³	ba³
ba	ba	ba²	ab	ba³	a²b	aba	b	a³b	a²ba	aba²	b	a³ba	a²ba²	aba³	ba	a³ba²	a²ba³	ba²	a³ba³	ba³
a³	a³	1	a³b	a	b	a³ba	a²	ab	ba	a³ba²	a²b	aba	ba²	a³ba³	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a²b	a²b	a²ba	a²b	a²ba²	a³b	a²ba	a²ba³	b	a³ba	a²ba²	ab	ba	a³ba²	a²ba³	aba	ba²	a³ba³	aba²	ba³	aba³
aba	aba	aba²	a²b	aba³	a³b	a²ba	ab	b	a³ba	a²ba²	ab	ba	a³ba²	a²ba³	aba	ba²	a³ba³	aba²	ba³	aba³
ba²	ba²	ba³	a²b	b	a³b	a²ba	ba	b	a³ba	a²ba²	ab	ba	a³ba²	a²ba³	aba	ba²	a³ba³	aba²	ba³	aba³
a³b	a³b	a³ba	a³b	a³ba²	b	a³ba	a³ba³	ab	ba	a³ba²	a²b	aba	ba²	a³ba³	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a²ba	a²ba	a²ba²	a³b	a²ba³	b	a³ba	a²b	ab	ba	a³ba²	a²b	aba	ba²	a³ba³	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
aba²	aba²	aba³	a³b	ab	b	a³ba	aba	ab	ba	a³ba²	a²b	aba	ba²	a³ba³	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
ba³	ba³	b	a³b	ba	b	a³ba	ba²	ab	ba	a³ba²	a²b	aba	ba²	a³ba³	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a³ba	a³ba	a³ba²	b	a³ba³	ab	ba	a³b	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
a²ba²	a²ba²	a²ba³	b	a²b	ab	ba	a²ba	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
aba³	aba³	ab	b	aba	ab	ba	aba²	a²b	aba	ba²	a³b	a²ba	aba²	ba³	a³ba	a²ba²	aba³	a³ba²	a²ba³	a³ba³
a³ba²	a³ba²	a³ba³	ab	a³b	a²b	aba	a³ba	a³b	a²ba	aba²	b	a³ba	a²ba²	aba³	ba	a³ba²	a²ba³	ba²	a³ba³	ba³
a²ba³	a²ba³	a²b	ab	a²ba	a²b	aba	a²ba²	a³b	a²ba	aba²	b	a³ba	a²ba²	aba³	ba	a³ba²	a²ba³	ba²	a³ba³	ba³
a³ba³	a³ba³	a³b	a²b	a³ba	a³b	a²ba	a³ba²	b	a³ba	a²ba²	ab	ba	a³ba²	a²ba³	aba	ba²	a³ba³	aba²	ba³	aba³

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, 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 \| aaaa=1, abaaab=b〉