〈a, b | aab=bb, baba=aa〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁷ ⇒ a³
ba² ⇒ a⁴
a²ba ⇒ a⁵
b² ⇒ a²b
(ba)² ⇒ a²

Idempotents

3 elements

1
a⁴
a⁶b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	aba	bab	a⁴	a³b	(ab)²	a⁵	a⁴b	a⁶	a⁵b	a⁶b
1	1	a	b	a²	ab	ba	a³	a²b	aba	bab	a⁴	a³b	(ab)²	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a	a	a²	ab	a³	a²b	aba	a⁴	a³b	a⁵	(ab)²	a⁵	a⁴b	a⁵b	a⁶	a⁵b	a³	a⁶b	a³b
b	b	ba	a²b	a⁴	bab	a⁵	a⁵	a⁴b	a²	a⁵b	a⁶	a⁵b	a²b	a³	a⁶b	a⁴	a³b	a⁴b
a²	a²	a³	a²b	a⁴	a³b	a⁵	a⁵	a⁴b	a⁶	a⁵b	a⁶	a⁵b	a⁶b	a³	a⁶b	a⁴	a³b	a⁴b
ab	ab	aba	a³b	a⁵	(ab)²	a⁶	a⁶	a⁵b	a³	a⁶b	a³	a⁶b	a³b	a⁴	a³b	a⁵	a⁴b	a⁵b
ba	ba	a⁴	bab	a⁵	a⁴b	a²	a⁶	a⁵b	a³	a²b	a³	a⁶b	a³b	a⁴	a³b	a⁵	a⁴b	a⁵b
a³	a³	a⁴	a³b	a⁵	a⁴b	a⁶	a⁶	a⁵b	a³	a⁶b	a³	a⁶b	a³b	a⁴	a³b	a⁵	a⁴b	a⁵b
a²b	a²b	a⁵	a⁴b	a⁶	a⁵b	a³	a³	a⁶b	a⁴	a³b	a⁴	a³b	a⁴b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
aba	aba	a⁵	(ab)²	a⁶	a⁵b	a³	a³	a⁶b	a⁴	a³b	a⁴	a³b	a⁴b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
bab	bab	a²	a⁵b	a³	a²b	a⁴	a⁴	a³b	a⁵	a⁴b	a⁵	a⁴b	a⁵b	a⁶	a⁵b	a³	a⁶b	a³b
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a³	a³	a⁶b	a⁴	a³b	a⁴	a³b	a⁴b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a³b	a³b	a⁶	a⁵b	a³	a⁶b	a⁴	a⁴	a³b	a⁵	a⁴b	a⁵	a⁴b	a⁵b	a⁶	a⁵b	a³	a⁶b	a³b
(ab)²	(ab)²	a³	a⁶b	a⁴	a³b	a⁵	a⁵	a⁴b	a⁶	a⁵b	a⁶	a⁵b	a⁶b	a³	a⁶b	a⁴	a³b	a⁴b
a⁵	a⁵	a⁶	a⁵b	a³	a⁶b	a⁴	a⁴	a³b	a⁵	a⁴b	a⁵	a⁴b	a⁵b	a⁶	a⁵b	a³	a⁶b	a³b
a⁴b	a⁴b	a³	a⁶b	a⁴	a³b	a⁵	a⁵	a⁴b	a⁶	a⁵b	a⁶	a⁵b	a⁶b	a³	a⁶b	a⁴	a³b	a⁴b
a⁶	a⁶	a³	a⁶b	a⁴	a³b	a⁵	a⁵	a⁴b	a⁶	a⁵b	a⁶	a⁵b	a⁶b	a³	a⁶b	a⁴	a³b	a⁴b
a⁵b	a⁵b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁶	a⁵b	a³	a⁶b	a³	a⁶b	a³b	a⁴	a³b	a⁵	a⁴b	a⁵b
a⁶b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a³	a³	a⁶b	a⁴	a³b	a⁴	a³b	a⁴b	a⁵	a⁴b	a⁶	a⁵b	a⁶b

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 \| 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 \| 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