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〈a, b | aa=a, bbbbbb=ab〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 22 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹¹ ⇒ b⁶
ab ⇒ b⁶
a² ⇒ a

Idempotents

4 elements

1
b¹⁰
a
b¹⁰a

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	b⁹a	b¹⁰	b¹⁰a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
a	a	a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
ba	ba	ba	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b²a	b²a	b²a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b³a	b³a	b³a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁴a	b⁴a	b⁴a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁵a	b⁵a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁶a	b⁶a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁷a	b⁷a	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁸	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁸a	b⁸a	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁹	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁹a	b⁹a	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b¹⁰	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b¹⁰a	b¹⁰a	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 35 total

Length:	Presentation:	Description:	Related:
9	〈a, b \| aa=1, ababba=b〉	Finite non-commutative monoid with 22 elements	15 isomorphic, 12 anti-isomorphic
10	〈a, b \| aa=1, abbbbbb=b〉	Finite non-commutative monoid with 22 elements	1 isomorphic
11	〈a, b \| aba=b, baaab=aa〉	Finite non-commutative monoid with 22 elements
11	〈a, b \| aa=1, abbabbbb=b〉	Finite non-commutative monoid with 22 elements	3 isomorphic