〈a, b | aba=b, baab=aa〉

Monoid presentation of length 10

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a¹⁰ ⇒ a²
ba⁸ ⇒ b
ab ⇒ ba⁷
b² ⇒ a⁴

Idempotents

2 elements

1
a⁸

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹
a	a	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a²
b	b	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a²	b	a³	ba
a²	a²	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a²	ba⁵	a³
ba	ba	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a²	ba²
a³	a³	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	ba	a⁹	ba²	a²	ba³	a³	ba⁴	a⁴
ba²	ba²	ba³	a²	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	b	a²	ba	a³	ba²	a⁴	ba³	a⁵
ba³	ba³	ba⁴	a⁹	ba⁵	a²	ba⁶	a³	ba⁷	a⁴	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴
a⁵	a⁵	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶
ba⁴	ba⁴	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a²	b	a³	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵
a⁶	a⁶	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷
ba⁵	ba⁵	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶
a⁷	a⁷	a⁸	ba	a⁹	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸
ba⁶	ba⁶	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a²	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷
a⁸	a⁸	a⁹	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹
ba⁷	ba⁷	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	a⁹	ba⁵	a²	ba⁶	a³	ba⁷	a⁴	b
a⁹	a⁹	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a²

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, bbbb=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