〈a, b | aaaa=ab, babb=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁸ ⇒ b
ba ⇒ b⁶
a²b ⇒ ab⁶
a⁴ ⇒ ab

Idempotents

3 elements

1
b⁷
ab²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a³	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷
1	1	a	b	a²	ab	b²	a³	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷
a	a	a²	ab	a³	ab⁶	ab²	ab	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵
b	b	b⁶	b²	b⁴	b⁷	b³	b²	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶
a²	a²	a³	ab⁶	ab	ab⁴	ab⁷	ab⁶	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³
ab	ab	ab⁶	ab²	ab⁴	ab⁷	ab³	ab²	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶
b²	b²	b⁷	b³	b⁵	b	b⁴	b³	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷
a³	a³	ab	ab⁴	ab⁶	ab²	ab⁵	ab⁴	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab
ab²	ab²	ab⁷	ab³	ab⁵	ab	ab⁴	ab³	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷
b³	b³	b	b⁴	b⁶	b²	b⁵	b⁴	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷	b³	b
ab³	ab³	ab	ab⁴	ab⁶	ab²	ab⁵	ab⁴	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab
b⁴	b⁴	b²	b⁵	b⁷	b³	b⁶	b⁵	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷	b³	b	b⁴	b²
ab⁴	ab⁴	ab²	ab⁵	ab⁷	ab³	ab⁶	ab⁵	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²
b⁵	b⁵	b³	b⁶	b	b⁴	b⁷	b⁶	b⁵	b	b⁶	b²	b⁷	b³	b	b⁴	b²	b⁵	b³
ab⁵	ab⁵	ab³	ab⁶	ab	ab⁴	ab⁷	ab⁶	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³
b⁶	b⁶	b⁴	b⁷	b²	b⁵	b	b⁷	b⁶	b²	b⁷	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴
ab⁶	ab⁶	ab⁴	ab⁷	ab²	ab⁵	ab	ab⁷	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴
b⁷	b⁷	b⁵	b	b³	b⁶	b²	b	b⁷	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵
ab⁷	ab⁷	ab⁵	ab	ab³	ab⁶	ab²	ab	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 102 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 \| 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

Other isomorphic instances

2 total

Length:	Presentation:
11	〈a, b \| aaaa=ab, bbab=b〉
11	〈a, b \| aaaa=ab, bbba=b〉