#5340 ⟨a, b | aaa=bb, aab=ba⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 13 elements

Element profile

3 element center:
- 1, a³, a⁶
2 idempotent elements:
- 1, a⁶

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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a⁷ ⇒ a⁴
a⁶b ⇒ a³b
ba ⇒ a²b
b² ⇒ a³

# ab:aaa=bb,aab=ba reversed:a/b
aaaaaaa=aaaa
aaaaaab=aaab
ba=aab
bb=aaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b
1	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b
a	a	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b
b	b	a²b	a³	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴
a²	a²	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b
ab	ab	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵
a³	a³	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b
a²b	a²b	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b
a³b	a³b	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴
a⁵	a⁵	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b
a⁴b	a⁴b	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵
a⁶	a⁶	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b
a⁵b	a⁵b	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶	a⁴b	a⁵	a³b	a⁴	a⁵b	a⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 303 total

Σ	#	Presentation	Description	Related
9	1328	⟨a, b \| aaaa=b, abbb=1⟩	Isomorphic to ℤ₁₃	189 iso
9	2118	⟨a, b \| aba=b, baab=a⟩	Finite non-commutative monoid with 13 elements	4 iso
10	4642	⟨a, b \| aaaa=b, abbb=a⟩	Isomorphic to ℕ_{(13 = 1)}	10 iso
10	4643	⟨a, b \| aaaa=b, abbb=b⟩	Isomorphic to ℕ_{(13 = 4)}	6 iso
10	4683	⟨a, b \| aaab=b, abba=a⟩	Finite non-commutative monoid with 13 elements	15 iso, 1 anti-iso
10	5065	⟨a, b \| aaa=ab, babb=b⟩	Finite non-commutative monoid with 13 elements	7 iso
10	5334	⟨a, b \| aaa=ab, bba=bb⟩	Finite non-commutative monoid with 13 elements
10	5336	⟨a, b \| aaa=ab, bbb=ab⟩	Finite non-commutative monoid with 13 elements	1 iso
10	5337	⟨a, b \| aaa=ab, bbb=ba⟩	Finite non-commutative monoid with 13 elements
10	6305	⟨a, b \| aaa=b, abbbb=b⟩	Isomorphic to ℕ_{(13 = 3)}	2 iso
10	7143	⟨a, b \| bb=aa, aaab=ba⟩	Finite non-commutative monoid with 13 elements	1 iso
11	12240	⟨a, b \| aaab=aa, bbbb=a⟩	Isomorphic to ℕ_{(13 = 8)}	1 iso
11	12268	⟨a, b \| aaab=ab, bbbb=a⟩	Isomorphic to ℕ_{(13 = 5)}	3 iso
11	12324	⟨a, b \| aaab=bb, bbbb=a⟩	Isomorphic to ℕ_{(13 = 2)}	14 iso
11	14647	⟨a, b \| aaba=b, babbb=a⟩	Finite commutative monoid with 13 elements	2 iso
11	15520	⟨a, b \| aaa=bb, aabbb=b⟩	Finite commutative monoid with 13 elements	2 iso
11	16012	⟨a, b \| aaa=ab, abbb=bb⟩	Finite non-commutative monoid with 13 elements
11	16069	⟨a, b \| aaa=bb, abbb=ba⟩	Finite non-commutative monoid with 13 elements	1 anti-iso
11	16205	⟨a, b \| aab=ab, bbbb=aa⟩	Finite non-commutative monoid with 13 elements	1 iso, 1 anti-iso
11	16459	⟨a, b \| aba=bb, abbb=aa⟩	Finite non-commutative monoid with 13 elements	1 iso
11	16506	⟨a, b \| aab=ab, bbb=aaa⟩	Finite non-commutative monoid with 13 elements	1 iso
11	16515	⟨a, b \| aab=ba, bbb=aaa⟩	Finite non-commutative monoid with 13 elements
11	18958	⟨a, b \| aab=a, bbbbbb=a⟩	Isomorphic to ℕ_{(13 = 6)}	4 iso
11	20046	⟨a, b \| aab=a, bbbb=bba⟩	Finite non-commutative monoid with 13 elements
11	20927	⟨a, b \| ab=aa, aaaa=bbb⟩	Finite non-commutative monoid with 13 elements	7 iso
11	20991	⟨a, b \| ab=aa, baaa=bbb⟩	Finite non-commutative monoid with 13 elements	3 iso

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