〈a, b | aba=b, bbbb=aa〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁹ ⇒ b
ba ⇒ ab⁵
a² ⇒ b⁴

Idempotents

2 elements

1
b⁸

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 104 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aa=1, abbba=b〉	Finite non-commutative monoid with 18 elements	14 isomorphic
9	〈a, b \| aa=1, abbbbb=b〉	Finite non-commutative monoid with 18 elements	22 isomorphic, 9 anti-isomorphic
10	〈a, b \| aaab=1, bbbbbb=1〉	Isomorphic to ℤ₁₈	33 isomorphic
10	〈a, b \| aba=a, aaab=bb〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| aba=b, baab=aa〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| bb=aa, aaaba=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
10	〈a, b \| aa=a, bbbbb=ab〉	Finite non-commutative monoid with 18 elements	1 isomorphic
10	〈a, b \| aa=1, abababa=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aaaa=ab, babb=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aab=bb, bbbba=a〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=ab, bbbb=ab〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=ab, bbbb=ba〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aab=bb, baba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=a, bbbbbb=a〉	Finite commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=a〉	Finite commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=b〉	Finite commutative monoid with 18 elements
11	〈a, b \| aab=b, bbbba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| bb=aa, ababa=aa〉	Finite non-commutative monoid with 18 elements	1 isomorphic