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〈a, b | bab=aaa, bbb=aa〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 21 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹⁰ ⇒ b⁷
ab⁷ ⇒ ab⁴
bab ⇒ ab³
b³a ⇒ ab³
a² ⇒ b³

Idempotents

2 elements

1
b⁹

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
a	a	b³	ab	b⁴	aba	ab²	ab⁵	b⁵	ab²a	ab³	ab⁴	b⁶	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b	b	ba	b²	ab³	b²a	b³	b⁶	ab⁴	ab³	b⁴	b⁸	ab⁵	b⁵	ab⁶	b⁶	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab	ab	aba	ab²	b⁶	ab²a	ab³	ab⁶	b⁷	b⁶	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
ba	ba	b⁴	ab³	b⁵	b⁶	ab⁴	ab⁴	b⁶	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²	b²	b²a	b³	ab⁵	ab³	b⁴	b⁷	ab⁶	ab⁵	b⁵	b⁹	ab⁴	b⁶	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
aba	aba	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab²	ab²	ab²a	ab³	b⁸	b⁶	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²a	b²a	b⁵	ab⁵	b⁶	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b³	b³	ab³	b⁴	ab⁴	ab⁵	b⁵	b⁸	ab⁵	ab⁴	b⁶	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab²a	ab²a	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab³	ab³	b⁶	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁴	b⁴	ab⁵	b⁵	ab⁶	ab⁴	b⁶	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab⁴	ab⁴	b⁸	ab⁵	b⁹	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b⁵	b⁵	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab⁵	ab⁵	b⁷	ab⁶	b⁸	b⁹	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b⁶	b⁶	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab⁶	ab⁶	b⁹	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁷	b⁷	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
b⁸	b⁸	ab⁴	b⁹	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
b⁹	b⁹	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

12 unique, 64 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aaa=ab, bbb=1〉	Finite non-commutative monoid with 21 elements	6 isomorphic, 9 anti-isomorphic
8	〈a, b \| aaa=1, abbb=b〉	Finite non-commutative monoid with 21 elements	10 isomorphic, 5 anti-isomorphic
9	〈a, b \| aaa=ab, bbb=b〉	Finite non-commutative monoid with 21 elements	1 anti-isomorphic
9	〈a, b \| aaa=1, babb=ab〉	Finite non-commutative monoid with 21 elements	6 isomorphic
10	〈a, b \| aaa=aa, bbb=ab〉	Finite non-commutative monoid with 21 elements
10	〈a, b \| aaa=a, abbbb=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aaaaa=b, abbbb=1〉	Isomorphic to ℤ₂₁	13 isomorphic
11	〈a, b \| bab=aaa, abba=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aab=ba, ababab=1〉	Finite non-Abelian group with 21 elements	1 isomorphic
11	〈a, b \| aaa=a, ababbb=b〉	Finite non-commutative monoid with 21 elements	1 isomorphic
11	〈a, b \| aba=b, baaaab=a〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aaa=1, abbab=aba〉	Finite non-commutative monoid with 21 elements