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〈a, b | aaa=ab, bbb=1〉

Monoid presentation of length 8

Properties

Finite non-commutative monoid with 21 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁷ ⇒ a
ab ⇒ a³
b³ ⇒ 1

Idempotents

4 elements

1
a⁶
ba⁴
b²a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a	a⁶	a	a²	a	a²	a³	a³	a⁴	a⁵
b	b	ba	b²	ba²	b²a	1	ba³	b²a²	a	ba⁴	b²a³	a²	ba⁵	b²a⁴	a³	ba⁶	b²a⁵	a⁴	b²a⁶	a⁵	a⁶
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a	a⁶	a	a²	a	a²	a³	a²	a³	a⁴	a⁴	a⁵	a⁶
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba	ba⁶	ba	ba²	ba	ba²	ba³	ba³	ba⁴	ba⁵
b²	b²	b²a	1	b²a²	a	b	b²a³	a²	ba	b²a⁴	a³	ba²	b²a⁵	a⁴	ba³	b²a⁶	a⁵	ba⁴	a⁶	ba⁵	ba⁶
a³	a³	a⁴	a⁵	a⁵	a⁶	a	a⁶	a	a²	a	a²	a³	a²	a³	a⁴	a³	a⁴	a⁵	a⁵	a⁶	a
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba	ba⁶	ba	ba²	ba	ba²	ba³	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶
b²a	b²a	b²a²	b²a³	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a	b²a⁶	b²a	b²a²	b²a	b²a²	b²a³	b²a³	b²a⁴	b²a⁵
a⁴	a⁴	a⁵	a⁶	a⁶	a	a²	a	a²	a³	a²	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁶	a	a²
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba	ba⁶	ba	ba²	ba	ba²	ba³	ba²	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba
b²a²	b²a²	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a	b²a⁶	b²a	b²a²	b²a	b²a²	b²a³	b²a²	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶
a⁵	a⁵	a⁶	a	a	a²	a³	a²	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a	a	a²	a³
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba	ba²	ba	ba²	ba³	ba²	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁶	ba	ba²
b²a³	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a	b²a⁶	b²a	b²a²	b²a	b²a²	b²a³	b²a²	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a
a⁶	a⁶	a	a²	a²	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a	a⁶	a	a²	a²	a³	a⁴
ba⁵	ba⁵	ba⁶	ba	ba	ba²	ba³	ba²	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba	ba	ba²	ba³
b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a	b²a²	b²a	b²a²	b²a³	b²a²	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a	b²a²
ba⁶	ba⁶	ba	ba²	ba²	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba	ba⁶	ba	ba²	ba²	ba³	ba⁴
b²a⁵	b²a⁵	b²a⁶	b²a	b²a	b²a²	b²a³	b²a²	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a	b²a	b²a²	b²a³
b²a⁶	b²a⁶	b²a	b²a²	b²a²	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a	b²a⁶	b²a	b²a²	b²a²	b²a³	b²a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

12 unique, 49 total

Length:	Presentation:	Description:	Related:
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 \| bab=aaa, bbb=aa〉	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

Other isomorphic instances

6 total

Length:	Presentation:
10	〈a, b \| aaa=1, abbb=aba〉
10	〈a, b \| aaa=1, babb=baa〉
11	〈a, b \| aaa=1, aaabbb=ba〉
11	〈a, b \| aaa=1, ababba=ab〉
11	〈a, b \| aaa=1, baaabb=ba〉
11	〈a, b \| aaa=1, aaaba=bbb〉

Other anti-isomorphic instances

9 total

Length:	Presentation:
9	〈a, b \| aaa=1, aabbb=b〉
9	〈a, b \| aaa=1, abbab=b〉
10	〈a, b \| aaa=1, abbb=aab〉
11	〈a, b \| aaa=1, aaabbb=ab〉
11	〈a, b \| aaa=1, aabbab=ab〉
11	〈a, b \| aaa=1, aabbba=ba〉
11	〈a, b \| aaa=1, baaabb=ab〉
11	〈a, b \| aaa=1, baabab=ab〉
11	〈a, b \| aaa=1, aaaab=bbb〉