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〈a, b | ab=aa, bbaaa=bb〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 15 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

ab ⇒ a²
b³ ⇒ b²a
b²a³ ⇒ b²
a⁶ ⇒ a³

Idempotents

4 elements

1
a³
b²a
ba⁵

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	ba⁵
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	ba⁵
a	a	a²	a²	a³	a³	a³	a⁴	a⁴	a⁴	a⁵	a⁵	a⁵	a³	a³	a⁴
b	b	ba	b²	ba²	b²a	b²a	ba³	b²a²	b²a²	ba⁴	b²	b²	ba⁵	b²a	b²a²
a²	a²	a³	a³	a⁴	a⁴	a⁴	a⁵	a⁵	a⁵	a³	a³	a³	a⁴	a⁴	a⁵
ba	ba	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁵	ba⁵	ba⁵	ba³	ba³	ba⁴
b²	b²	b²a	b²a	b²a²	b²a²	b²a²	b²	b²	b²	b²a	b²a	b²a	b²a²	b²a²	b²
a³	a³	a⁴	a⁴	a⁵	a⁵	a⁵	a³	a³	a³	a⁴	a⁴	a⁴	a⁵	a⁵	a³
ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁵	ba⁵	ba⁵	ba³	ba³	ba³	ba⁴	ba⁴	ba⁵
b²a	b²a	b²a²	b²a²	b²	b²	b²	b²a	b²a	b²a	b²a²	b²a²	b²a²	b²	b²	b²a
a⁴	a⁴	a⁵	a⁵	a³	a³	a³	a⁴	a⁴	a⁴	a⁵	a⁵	a⁵	a³	a³	a⁴
ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba⁵	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁵	ba⁵	ba³
b²a²	b²a²	b²	b²	b²a	b²a	b²a	b²a²	b²a²	b²a²	b²	b²	b²	b²a	b²a	b²a²
a⁵	a⁵	a³	a³	a⁴	a⁴	a⁴	a⁵	a⁵	a⁵	a³	a³	a³	a⁴	a⁴	a⁵
ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁵	ba⁵	ba⁵	ba³	ba³	ba⁴
ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁵	ba⁵	ba⁵	ba³	ba³	ba³	ba⁴	ba⁴	ba⁵

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

23 unique, 184 total

Length:	Presentation:	Description:	Related:
9	〈a, b \| aaa=bb, aba=a〉	Finite non-commutative monoid with 15 elements	2 isomorphic
9	〈a, b \| aaa=bb, aba=b〉	Finite non-commutative monoid with 15 elements	1 isomorphic
9	〈a, b \| aaab=1, bbbbb=1〉	Isomorphic to ℤ₁₅	138 isomorphic
9	〈a, b \| aaa=a, abbb=b〉	Finite non-commutative monoid with 15 elements	4 isomorphic
10	〈a, b \| aaa=bb, aba=aa〉	Finite non-commutative monoid with 15 elements	1 isomorphic
10	〈a, b \| aaa=bb, aba=bb〉	Finite non-commutative monoid with 15 elements	2 isomorphic
10	〈a, b \| aaa=a, bbbbb=a〉	Finite commutative monoid with 15 elements
10	〈a, b \| aaa=b, bbbbb=a〉	Finite commutative monoid with 15 elements
10	〈a, b \| aaa=b, bbbbb=b〉	Finite commutative monoid with 15 elements
11	〈a, b \| abb=aaa, baaa=b〉	Finite non-commutative monoid with 15 elements	3 isomorphic
11	〈a, b \| abb=aaa, baba=b〉	Finite non-commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aab=aaa, aba=bb〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| abb=aaa, bbb=ab〉	Finite non-commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| abb=aaa, bbb=ba〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| aaa=aa, bbbbb=a〉	Finite commutative monoid with 15 elements
11	〈a, b \| aaa=ab, baabb=b〉	Finite non-commutative monoid with 15 elements	5 isomorphic
11	〈a, b \| aaa=ab, bbbbb=a〉	Finite commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aaa=bb, bbbbb=a〉	Finite commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aaa=ab, bbbb=aa〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| aab=aa, bbbb=ab〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| aaa=b, bbbbb=ab〉	Finite commutative monoid with 15 elements
11	〈a, b \| aab=a, bbbbb=ba〉	Finite non-commutative monoid with 15 elements	1 anti-isomorphic
11	〈a, b \| aab=b, bbaa=aaa〉	Finite non-commutative monoid with 15 elements

Other isomorphic instances

6 total

Length:	Presentation:
11	〈a, b \| ab=aa, bbaab=bb〉
11	〈a, b \| ab=aa, bbaba=bb〉
11	〈a, b \| ab=aa, bbabb=bb〉
11	〈a, b \| ab=aa, bbbaa=bb〉
11	〈a, b \| ab=aa, bbbab=bb〉
11	〈a, b \| ab=aa, bbbba=bb〉