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〈a, b | aaa=bb, aba=b〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 15 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁹ ⇒ a³
ba⁶ ⇒ b
ab ⇒ ba⁵
b² ⇒ a³

Idempotents

2 elements

1
a⁶

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

23 unique, 189 total

Length:	Presentation:	Description:	Related:
9	〈a, b \| aaa=bb, aba=a〉	Finite non-commutative monoid with 15 elements	2 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
11	〈a, b \| ab=aa, bbaaa=bb〉	Finite non-commutative monoid with 15 elements	6 isomorphic

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aba=b, abab=aaa〉