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〈a, b | aaa=1, abbb=b〉

Monoid presentation of length 8

Properties

Finite non-commutative monoid with 21 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁷ ⇒ b
ab ⇒ b⁵
a³ ⇒ 1

Idempotents

4 elements

1
b⁶
b²a
b⁴a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
1	1	a	b	a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
a	a	a²	b⁵	1	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b	b	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
a²	a²	1	b³	a	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
ba	ba	ba²	b⁶	b	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b²	b²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
ba²	ba²	b	b⁴	ba	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b²a	b²a	b²a²	b	b²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
b³	b³	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b²a²	b²a²	b²	b⁵	b²a	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b³a	b³a	b³a²	b²	b³	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
b⁴	b⁴	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b³a²	b³a²	b³	b⁶	b³a	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b⁴a	b⁴a	b⁴a²	b³	b⁴	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
b⁵	b⁵	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b⁴a²	b⁴a²	b⁴	b	b⁴a	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
b⁵a	b⁵a	b⁵a²	b⁴	b⁵	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b⁶	b⁶	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
b⁵a²	b⁵a²	b⁵	b²	b⁵a	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
b⁶a	b⁶a	b⁶a²	b⁵	b⁶	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b⁶a²	b⁶a²	b⁶	b³	b⁶a	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²

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=ab, bbb=1〉	Finite non-commutative monoid with 21 elements	6 isomorphic, 9 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

10 total

Length:	Presentation:
10	〈a, b \| aaa=1, aabbb=ab〉
10	〈a, b \| aaa=1, babab=ab〉
11	〈a, b \| aaa=1, aaaabbb=b〉
11	〈a, b \| aaa=1, aababab=b〉
11	〈a, b \| aaa=1, aabbaab=b〉
11	〈a, b \| aaa=1, abaaabb=b〉
11	〈a, b \| aaa=1, ababaab=b〉
11	〈a, b \| aaa=1, abbaaab=b〉
11	〈a, b \| aaa=1, abbb=aaab〉
11	〈a, b \| aaa=1, baaa=abbb〉

Other anti-isomorphic instances

5 total

Length:	Presentation:
9	〈a, b \| abb=aaa, bbb=1〉
10	〈a, b \| aaa=1, abbba=ab〉
10	〈a, b \| aaa=1, baabb=ba〉
11	〈a, b \| aaa=1, aaabbba=b〉
11	〈a, b \| aaa=1, abbb=abaa〉