#60 ⟨a, b | aa=b, ab=b⟩

Properties

Presentation has sum-of-sides 6
Isomorphic to ℕ_{(3 = 2)}
Finite commutative monoid with 3 elements
Commutative Gröbner basis: ⟨a, b | a³=a², b=a²⟩
Not cancellative, because multiplication by a² is not injective:
- a² ≠ 1
- a² ⋅ a² = a²
- a² ⋅ 1 = a²
- Cancellative quotient is isomorphic to ℤ₁
Grothendieck group is isomorphic to ℤ₁
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

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

# ab:aa=b,ab=b a/b
aaa=aa
b=aa

Idempotents are shown in bold.

Idempotents are shown in bold.

4 unique, 2184 total

Σ	#	Presentation	Description	Related
5	6	⟨a, b \| aa=b, ab=1⟩	Isomorphic to ℤ₃	2029 iso
6	59	⟨a, b \| aa=b, ab=a⟩	Isomorphic to ℕ_{(3 = 1)}	61 iso
6	63	⟨a, b \| ab=a, ba=b⟩	Finite non-commutative monoid with 3 elements	61 iso, 17 anti-iso
8	891	⟨a, b \| aa=a, abba=b⟩	Finite commutative monoid with 3 elements	12 iso

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

23 total

Σ	#	Presentation	Mapping
7	248	⟨a, b \| aa=b, aaa=b⟩	φ(a) = a, φ(b) = aa
7	277	⟨a, b \| aa=b, ab=aa⟩	φ(a) = a, φ(b) = aa
7	281	⟨a, b \| ab=a, bb=ab⟩	φ(a) = aa, φ(b) = a
8	972	⟨a, b \| aa=b, aaa=aa⟩	φ(a) = a, φ(b) = aa
8	1007	⟨a, b \| ab=a, abb=bb⟩	φ(a) = aa, φ(b) = a
9	1613	⟨a, b \| aab=aa, abb=b⟩	φ(a) = a, φ(b) = aa
9	3103	⟨a, b \| ab=a, abbb=bb⟩	φ(a) = aa, φ(b) = a
10	5023	⟨a, b \| aaa=aa, aaaa=b⟩	φ(a) = a, φ(b) = aa
10	5109	⟨a, b \| aab=aa, abbb=b⟩	φ(a) = a, φ(b) = aa
10	9047	⟨a, b \| ab=a, abbbb=bb⟩	φ(a) = aa, φ(b) = a
11	12217	⟨a, b \| aaab=aa, aabb=b⟩	φ(a) = a, φ(b) = aa
11	12221	⟨a, b \| aaab=aa, abab=b⟩	φ(a) = a, φ(b) = aa
11	12223	⟨a, b \| aaab=aa, abba=b⟩	φ(a) = a, φ(b) = aa
11	12225	⟨a, b \| aaab=aa, abbb=b⟩	φ(a) = a, φ(b) = aa
11	12333	⟨a, b \| aaba=aa, abba=b⟩	φ(a) = a, φ(b) = aa
11	15402	⟨a, b \| aaa=aa, aaaaa=b⟩	φ(a) = a, φ(b) = aa
11	15576	⟨a, b \| aab=aa, abbbb=b⟩	φ(a) = a, φ(b) = aa
11	19407	⟨a, b \| aab=a, aabbb=bb⟩	φ(a) = aa, φ(b) = a
11	19423	⟨a, b \| aab=a, ababb=bb⟩	φ(a) = aa, φ(b) = a
11	19431	⟨a, b \| aab=a, abbab=bb⟩	φ(a) = aa, φ(b) = a
11	19435	⟨a, b \| aab=a, abbba=bb⟩	φ(a) = aa, φ(b) = a
11	19439	⟨a, b \| aab=a, abbbb=bb⟩	φ(a) = aa, φ(b) = a
11	24987	⟨a, b \| ab=a, abbbbb=bb⟩	φ(a) = aa, φ(b) = a

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