〈a, b | aa=a, bbbbb=ab〉

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 ⇒ b⁵
ab ⇒ b⁵
a² ⇒ a

Idempotents

4 elements

1
b⁸
a
b⁸a

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
a	a	a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
ba	ba	ba	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b²a	b²a	b²a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b³a	b³a	b³a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁴a	b⁴a	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b⁵a	b⁵a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b⁶a	b⁶a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁷a	b⁷a	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁸	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁸a	b⁸a	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 103 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=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

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aa=a, bbbbb=aab〉