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〈a, b | aab=aaa, aba=bb〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 15 elements

Completion parameters

Reduction order: right-to-left a/b

Complete rewriting system

a⁵ ⇒ a⁴
a²b ⇒ a³
ba⁴ ⇒ a⁴
b² ⇒ aba
(ba)² ⇒ (ab)²

Idempotents

2 elements

1
a⁴

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	aba	bab	a⁴	a³b	a²ba	(ba)²	a³ba
1	1	a	b	a²	ab	ba	a³	a²b	aba	bab	a⁴	a³b	a²ba	(ba)²	a³ba
a	a	a²	ab	a³	a²b	aba	a⁴	a³b	a²ba	(ba)²	a⁴	a⁴	a³ba	a⁴	a⁴
b	b	ba	aba	a³	bab	a⁴	a⁴	a³b	(ba)²	a⁴	a⁴	a⁴	a³ba	a⁴	a⁴
a²	a²	a³	a²b	a⁴	a³b	a²ba	a⁴	a⁴	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
ab	ab	aba	a²ba	a⁴	(ba)²	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
ba	ba	a³	bab	a⁴	a³b	(ba)²	a⁴	a⁴	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a³	a³	a⁴	a³b	a⁴	a⁴	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a²b	a²b	a²ba	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
aba	aba	a⁴	(ba)²	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
bab	bab	(ba)²	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a³b	a³b	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a²ba	a²ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
(ba)²	(ba)²	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
a³ba	a³ba	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

23 unique, 190 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 \| 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