#24660 ⟨a, b | aa=a, ababab=bb⟩

Properties

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

Element profile

1 element center:
- 1
4 idempotent elements:
- 1, a, b³, b³a
2 right zero elements:
- b³, b³a

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 shortlex with a < b

a² ⇒ a
ab² ⇒ b²
b²ab ⇒ b³
b⁴ ⇒ b³
(ab)³ ⇒ b²

# ab:aa=a,ababab=bb ab
aa=a
abb=bb
bbab=bbb
bbbb=bbb
ababab=bb

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	bab	b²a	b³	(ab)²	(ba)²	b³a	a(ba)²	b(ab)²	(ba)³
1	1	a	b	ab	ba	b²	aba	bab	b²a	b³	(ab)²	(ba)²	b³a	a(ba)²	b(ab)²	(ba)³
a	a	a	ab	ab	aba	b²	aba	(ab)²	b²a	b³	(ab)²	a(ba)²	b³a	a(ba)²	b²	b²a
b	b	ba	b²	bab	b²a	b³	(ba)²	b³	b³a	b³	b(ab)²	b³a	b³a	(ba)³	b³	b³a
ab	ab	aba	b²	(ab)²	b²a	b³	a(ba)²	b³	b³a	b³	b²	b³a	b³a	b²a	b³	b³a
ba	ba	ba	bab	bab	(ba)²	b³	(ba)²	b(ab)²	b³a	b³	b(ab)²	(ba)³	b³a	(ba)³	b³	b³a
b²	b²	b²a	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
aba	aba	aba	(ab)²	(ab)²	a(ba)²	b³	a(ba)²	b²	b³a	b³	b²	b²a	b³a	b²a	b³	b³a
bab	bab	(ba)²	b³	b(ab)²	b³a	b³	(ba)³	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
b²a	b²a	b²a	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
b³	b³	b³a	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
(ab)²	(ab)²	a(ba)²	b³	b²	b³a	b³	b²a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
(ba)²	(ba)²	(ba)²	b(ab)²	b(ab)²	(ba)³	b³	(ba)³	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
b³a	b³a	b³a	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
a(ba)²	a(ba)²	a(ba)²	b²	b²	b²a	b³	b²a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
b(ab)²	b(ab)²	(ba)³	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a
(ba)³	(ba)³	(ba)³	b³	b³	b³a	b³	b³a	b³	b³a	b³	b³	b³a	b³a	b³a	b³	b³a

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	16060	⟨a, b \| aaa=bb, abab=aa⟩	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

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