〈a, b | aaa=a, abbba=b〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 19 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁹ ⇒ b
ba ⇒ ab³
a²b ⇒ b
a³ ⇒ a

Idempotents

3 elements

1
b⁸
a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
1	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
a	a	a²	ab	a	b	ab²	b²	ab³	b³	ab⁴	b⁴	ab⁵	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸
b	b	ab³	b²	b	ab⁴	b³	ab⁵	b⁴	ab⁶	b⁵	ab⁷	b⁶	ab⁸	b⁷	ab	b⁸	ab²	b	ab³
a²	a²	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
ab	ab	b³	ab²	ab	b⁴	ab³	b⁵	ab⁴	b⁶	ab⁵	b⁷	ab⁶	b⁸	ab⁷	b	ab⁸	b²	ab	b³
b²	b²	ab⁶	b³	b²	ab⁷	b⁴	ab⁸	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b	ab⁵	b²	ab⁶
ab²	ab²	b⁶	ab³	ab²	b⁷	ab⁴	b⁸	ab⁵	b	ab⁶	b²	ab⁷	b³	ab⁸	b⁴	ab	b⁵	ab²	b⁶
b³	b³	ab	b⁴	b³	ab²	b⁵	ab³	b⁶	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b	ab⁷	b²	ab⁸	b³	ab
ab³	ab³	b	ab⁴	ab³	b²	ab⁵	b³	ab⁶	b⁴	ab⁷	b⁵	ab⁸	b⁶	ab	b⁷	ab²	b⁸	ab³	b
b⁴	b⁴	ab⁴	b⁵	b⁴	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴
ab⁴	ab⁴	b⁴	ab⁵	ab⁴	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸	ab	b	ab²	b²	ab³	b³	ab⁴	b⁴
b⁵	b⁵	ab⁷	b⁶	b⁵	ab⁸	b⁷	ab	b⁸	ab²	b	ab³	b²	ab⁴	b³	ab⁵	b⁴	ab⁶	b⁵	ab⁷
ab⁵	ab⁵	b⁷	ab⁶	ab⁵	b⁸	ab⁷	b	ab⁸	b²	ab	b³	ab²	b⁴	ab³	b⁵	ab⁴	b⁶	ab⁵	b⁷
b⁶	b⁶	ab²	b⁷	b⁶	ab³	b⁸	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab	b⁶	ab²
ab⁶	ab⁶	b²	ab⁷	ab⁶	b³	ab⁸	b⁴	ab	b⁵	ab²	b⁶	ab³	b⁷	ab⁴	b⁸	ab⁵	b	ab⁶	b²
b⁷	b⁷	ab⁵	b⁸	b⁷	ab⁶	b	ab⁷	b²	ab⁸	b³	ab	b⁴	ab²	b⁵	ab³	b⁶	ab⁴	b⁷	ab⁵
ab⁷	ab⁷	b⁵	ab⁸	ab⁷	b⁶	ab	b⁷	ab²	b⁸	ab³	b	ab⁴	b²	ab⁵	b³	ab⁶	b⁴	ab⁷	b⁵
b⁸	b⁸	ab⁸	b	b⁸	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
ab⁸	ab⁸	b⁸	ab	ab⁸	b	ab²	b²	ab³	b³	ab⁴	b⁴	ab⁵	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 29 total

Length:	Presentation:	Description:	Related:
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 \| aaab=b, abbba=a〉	Finite non-commutative monoid with 19 elements	3 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