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〈a, b | aa=1, abbabbbb=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 22 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a² ⇒ 1
bab ⇒ ab²
b⁶ ⇒ b

Idempotents

3 elements

1
b⁵
ab⁵a

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
a	a	1	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a
b	b	ba	b²	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	aba
ab	ab	aba	ab²	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ba
ba	ba	b	ab²	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ba
b²	b²	b²a	b³	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	ab²a
aba	aba	ab	b²	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	aba
ab²	ab²	ab²a	ab³	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	b²a
b²a	b²a	b²	ab³	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	b²a
b³	b³	b³a	b⁴	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	ab³a
ab²a	ab²a	ab²	b³	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	ab²a
ab³	ab³	ab³a	ab⁴	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	b³a
b³a	b³a	b³	ab⁴	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	b³a
b⁴	b⁴	b⁴a	b⁵	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
ab³a	ab³a	ab³	b⁴	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	ab³a
ab⁴	ab⁴	ab⁴a	ab⁵	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	b⁴a
b⁴a	b⁴a	b⁴	ab⁵	b⁵	ab⁵a	ab	b⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	b⁴a
b⁵	b⁵	b⁵a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
ab⁴a	ab⁴a	ab⁴	b⁵	ab⁵	b⁵a	b	ab⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
ab⁵	ab⁵	ab⁵a	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a
b⁵a	b⁵a	b⁵	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a
ab⁵a	ab⁵a	ab⁵	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 32 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=a, bbbbbb=ab〉	Finite non-commutative monoid with 22 elements

Other isomorphic instances

3 total

Length:	Presentation:
11	〈a, b \| aa=1, abbbabbb=b〉
11	〈a, b \| aa=1, bbabbbb=ab〉
11	〈a, b \| aa=1, bbbabbb=ab〉