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〈a, b | aaa=bb, abab=aa〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 16 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁶ ⇒ a³
aba² ⇒ ba³
a²b ⇒ ba⁵
b² ⇒ a³
(ab)² ⇒ a²

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	aba	ba²	bab	a⁴	ba³	(ba)²	a⁵	ba⁴	ba⁵
1	1	a	b	a²	ab	ba	a³	aba	ba²	bab	a⁴	ba³	(ba)²	a⁵	ba⁴	ba⁵
a	a	a²	ab	a³	ba⁵	aba	a⁴	ba³	ba³	a²	a⁵	ba⁴	a³	a³	ba⁵	ba³
b	b	ba	a³	ba²	bab	a⁴	ba³	(ba)²	a⁵	ba⁴	ba⁴	a³	ba⁵	ba⁵	a⁴	a⁵
a²	a²	a³	ba⁵	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba⁴
ab	ab	aba	a⁴	ba³	a²	a⁵	ba⁴	a³	a³	ba⁵	ba⁵	a⁴	ba³	ba³	a⁵	a³
ba	ba	ba²	bab	ba³	a⁵	(ba)²	ba⁴	a³	a³	ba²	ba⁵	a⁴	ba³	ba³	a⁵	a³
a³	a³	a⁴	ba³	a⁵	ba⁴	ba⁴	a³	ba⁵	ba⁵	a⁴	a⁴	ba³	a⁵	a⁵	ba⁴	ba⁵
aba	aba	ba³	a²	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a⁴
ba²	ba²	ba³	a⁵	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a⁴
bab	bab	(ba)²	ba⁴	a³	ba²	ba⁵	a⁴	ba³	ba³	a⁵	a⁵	ba⁴	a³	a³	ba⁵	ba³
a⁴	a⁴	a⁵	ba⁴	a³	ba⁵	ba⁵	a⁴	ba³	ba³	a⁵	a⁵	ba⁴	a³	a³	ba⁵	ba³
ba³	ba³	ba⁴	a³	ba⁵	a⁴	a⁴	ba³	a⁵	a⁵	ba⁴	ba⁴	a³	ba⁵	ba⁵	a⁴	a⁵
(ba)²	(ba)²	a³	ba²	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba⁴
a⁵	a⁵	a³	ba⁵	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba⁴
ba⁴	ba⁴	ba⁵	a⁴	ba³	a⁵	a⁵	ba⁴	a³	a³	ba⁵	ba⁵	a⁴	ba³	ba³	a⁵	a³
ba⁵	ba⁵	ba³	a⁵	ba⁴	a³	a³	ba⁵	a⁴	a⁴	ba³	ba³	a⁵	ba⁴	ba⁴	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 175 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| bb=aa, abab=1〉	Finite non-Abelian group with 16 elements	4 isomorphic
8	〈a, b \| aa=1, ababbb=1〉	Finite non-Abelian group with 16 elements	40 isomorphic
9	〈a, b \| aaaa=b, bbbb=1〉	Isomorphic to ℤ₁₆	67 isomorphic
9	〈a, b \| aab=a, bbbb=b〉	Finite non-commutative monoid with 16 elements
10	〈a, b \| aaab=ba, abab=1〉	Finite non-Abelian group with 16 elements	7 isomorphic
10	〈a, b \| bab=aaa, aabb=1〉	Finite non-Abelian group with 16 elements	6 isomorphic
10	〈a, b \| aaaa=a, aabb=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aaaa=a, bbbb=a〉	Finite commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aaaa=b, bbbb=a〉	Finite commutative monoid with 16 elements	5 isomorphic
10	〈a, b \| aba=b, aaaabb=1〉	Finite non-Abelian group with 16 elements	3 isomorphic
10	〈a, b \| aba=b, aabbbb=1〉	Finite non-Abelian group with 16 elements	5 isomorphic
11	〈a, b \| aaaa=aa, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaaa=ab, bbbb=a〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aaaa=bb, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaab=bb, abba=a〉	Finite non-commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| bab=aab, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| bab=aba, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaaa=a, aabab=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
11	〈a, b \| aab=bb, abbba=a〉	Finite non-commutative monoid with 16 elements	3 isomorphic
11	〈a, b \| aaa=ab, babb=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=ab, bbaa=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aba=aa, bbbb=ab〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=b, abbbbb=b〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aba=b, aaaa=abb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| ab=aa, bbba=bbb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aa=a, ababab=bb〉	Finite non-commutative monoid with 16 elements