#16060 ⟨a, b | aaa=bb, abab=aa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 16 elements

Element profile

8 element center:
- 1, a³, ba², a⁴, ba³, a⁵, ba⁴, ba⁵
2 idempotent elements:
- 1, a³

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1

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

# ab:aaa=bb,abab=aa a/b
aaaaaa=aaa
abaa=baaa
aab=baaaaa
bb=aaa
abab=aa

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

20 unique, 175 total

Σ	#	Presentation	Description	Related
8	628	⟨a, b \| bb=aa, abab=1⟩	Finite non-Abelian group with 16 elements	58 iso
9	1331	⟨a, b \| aaaa=b, bbbb=1⟩	Isomorphic to ℤ₁₆	67 iso
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Finite non-commutative monoid with 16 elements	4 anti-iso
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Finite non-Abelian group with 16 elements	7 iso
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Isomorphic to ℕ_{(16 = 4)}	1 iso
10	4648	⟨a, b \| aaaa=b, bbbb=a⟩	Isomorphic to ℕ_{(16 = 1)}	5 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(16 = 5)}	2 iso
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Finite non-commutative monoid with 16 elements	6 iso
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	13259	⟨a, b \| bab=aba, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Finite non-commutative monoid with 16 elements
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Finite non-commutative monoid with 16 elements
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Finite non-commutative monoid with 16 elements
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Isomorphic to ℕ_{(16 = 3)}	2 iso
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Finite non-commutative monoid with 16 elements
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Finite non-commutative monoid with 16 elements
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Finite non-commutative monoid with 16 elements

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