〈a, b | aaab=b, abbba=a〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 19 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹⁰ ⇒ b
ab ⇒ b⁷
b⁹a ⇒ a
a² ⇒ b⁶a

Idempotents

3 elements

1
b⁹
b³a

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
a	a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b
ba	ba	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²
b²a	b²a	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³
b³a	b³a	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴
b⁴a	b⁴a	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵
b⁵a	b⁵a	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶
b⁶a	b⁶a	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷
b⁷a	b⁷a	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴
b⁸	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸
b⁸a	b⁸a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵
b⁹	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 26 total

Length:	Presentation:	Description:	Related:
10	〈a, b \| aaa=a, abbba=b〉	Finite non-commutative monoid with 19 elements
10	〈a, b \| aba=b, bab=aaa〉	Finite non-commutative monoid with 19 elements
11	〈a, b \| aaaab=1, abbbbb=1〉	Isomorphic to ℤ₁₉	20 isomorphic
11	〈a, b \| aaa=ab, bbab=bb〉	Finite non-commutative monoid with 19 elements	1 isomorphic
11	〈a, b \| aaa=bb, aaba=ab〉	Finite non-commutative monoid with 19 elements

Other isomorphic instances

3 total

Length:	Presentation:
11	〈a, b \| aaab=b, babba=a〉
11	〈a, b \| aaab=b, bbaba=a〉
11	〈a, b \| aaab=b, bbbaa=a〉