#1686 ⟨a, b | abb=aa, bbb=a⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(6 = 5)}
Finite commutative monoid with 6 elements
Commutative Gröbner basis: ⟨a, b | b⁶=b⁵, a=b³⟩
Not cancellative, because multiplication by b⁵ is not injective:
- b ≠ 1
- b⁵ ⋅ b = b⁵
- b⁵ ⋅ 1 = b⁵
- Cancellative quotient is isomorphic to ℤ₁
Grothendieck group is isomorphic to ℤ₁
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, b⁵
Zero element: b⁵

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(b) = 0; deg(a) = 1

b⁶ ⇒ b⁵
a ⇒ b³

# ab:abb=aa,bbb=a b/a
bbbbbb=bbbbb
a=bbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵
1	1	b	b²	b³	b⁴	b⁵
b	b	b²	b³	b⁴	b⁵	b⁵
b²	b²	b³	b⁴	b⁵	b⁵	b⁵
b³	b³	b⁴	b⁵	b⁵	b⁵	b⁵
b⁴	b⁴	b⁵	b⁵	b⁵	b⁵	b⁵
b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 1876 total

Σ	#	Presentation	Description	Related
6	22	⟨a, b \| ab=aa, bb=1⟩	Finite non-commutative monoid with 6 elements	44 iso, 63 anti-iso
6	26	⟨a, b \| aaa=1, abb=1⟩	Isomorphic to ℤ₆	1373 iso
7	160	⟨a, b \| ab=aa, bb=b⟩	Finite non-commutative monoid with 6 elements	4 anti-iso
7	245	⟨a, b \| aa=a, bbb=a⟩	Isomorphic to ℕ_{(6 = 3)}	49 iso
7	257	⟨a, b \| aa=b, bbb=a⟩	Isomorphic to ℕ_{(6 = 1)}	61 iso
7	258	⟨a, b \| aa=b, bbb=b⟩	Isomorphic to ℕ_{(6 = 2)}	55 iso
8	639	⟨a, b \| ab=aa, baa=b⟩	Finite non-commutative monoid with 6 elements	5 iso, 8 anti-iso
8	644	⟨a, b \| ab=aa, bbb=a⟩	Isomorphic to ℕ_{(6 = 4)}	46 iso
8	893	⟨a, b \| aa=a, abbb=b⟩	Finite non-commutative monoid with 6 elements	19 iso
8	1011	⟨a, b \| ab=a, baa=bb⟩	Finite non-commutative monoid with 6 elements	12 iso, 1 anti-iso
9	1427	⟨a, b \| abba=b, baba=1⟩	Finite non-Abelian group with 6 elements	66 iso
9	1581	⟨a, b \| aaa=aa, abb=b⟩	Finite non-commutative monoid with 6 elements	5 iso
9	1648	⟨a, b \| aab=bb, abb=a⟩	Finite commutative monoid with 6 elements	13 iso
9	3075	⟨a, b \| ab=a, aaaa=bb⟩	Finite commutative monoid with 6 elements	14 iso
9	3132	⟨a, b \| ab=a, bbbb=aa⟩	Finite commutative monoid with 6 elements	5 iso
9	3134	⟨a, b \| ab=a, bbbb=ba⟩	Finite non-commutative monoid with 6 elements	2 iso
9	3193	⟨a, b \| ab=a, bbb=aaa⟩	Finite commutative monoid with 6 elements	9 iso
9	3199	⟨a, b \| ab=a, bbb=bba⟩	Finite non-commutative monoid with 6 elements	3 iso
11	24145	⟨a, b \| aa=a, abbbbba=b⟩	Finite commutative monoid with 6 elements

Other isomorphic instances

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

49 total

Σ	#	Presentation	Mapping
9	1694	⟨a, b \| bab=aa, bbb=a⟩	φ(a) = bbb, φ(b) = b
9	2998	⟨a, b \| ab=a, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
9	3170	⟨a, b \| aa=b, bbb=abb⟩	φ(a) = b, φ(b) = bb
9	3171	⟨a, b \| aa=b, bbb=bab⟩	φ(a) = b, φ(b) = bb
10	5127	⟨a, b \| aab=ab, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	5159	⟨a, b \| aab=ba, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	5243	⟨a, b \| aba=ab, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	9109	⟨a, b \| ab=a, bbbbb=ab⟩	φ(a) = bbbbb, φ(b) = b
10	9197	⟨a, b \| aa=b, aaab=bbb⟩	φ(a) = b, φ(b) = bb
10	9205	⟨a, b \| aa=b, aaba=bbb⟩	φ(a) = b, φ(b) = bb
10	9209	⟨a, b \| aa=b, aabb=abb⟩	φ(a) = b, φ(b) = bb
10	9211	⟨a, b \| aa=b, aabb=bab⟩	φ(a) = b, φ(b) = bb
10	9212	⟨a, b \| aa=b, aabb=bba⟩	φ(a) = b, φ(b) = bb
10	9217	⟨a, b \| aa=b, abab=abb⟩	φ(a) = b, φ(b) = bb
10	9219	⟨a, b \| aa=b, abab=bab⟩	φ(a) = b, φ(b) = bb
10	9220	⟨a, b \| aa=b, abab=bba⟩	φ(a) = b, φ(b) = bb
10	9225	⟨a, b \| aa=b, abba=abb⟩	φ(a) = b, φ(b) = bb
10	9226	⟨a, b \| aa=b, abba=bab⟩	φ(a) = b, φ(b) = bb
10	9239	⟨a, b \| aa=b, baab=abb⟩	φ(a) = b, φ(b) = bb
10	9240	⟨a, b \| aa=b, baab=bab⟩	φ(a) = b, φ(b) = bb
11	19306	⟨a, b \| aaa=b, aaaaa=bb⟩	φ(a) = b, φ(b) = bbb
11	19855	⟨a, b \| aaa=b, aaab=aab⟩	φ(a) = b, φ(b) = bbb
11	19856	⟨a, b \| aaa=b, aaab=aba⟩	φ(a) = b, φ(b) = bbb
11	19858	⟨a, b \| aaa=b, aaab=baa⟩	φ(a) = b, φ(b) = bbb
11	19863	⟨a, b \| aaa=b, aaba=aab⟩	φ(a) = b, φ(b) = bbb
11	19864	⟨a, b \| aaa=b, aaba=aba⟩	φ(a) = b, φ(b) = bbb
11	19866	⟨a, b \| aaa=b, aaba=baa⟩	φ(a) = b, φ(b) = bbb
11	25265	⟨a, b \| aa=b, aaaaa=bbb⟩	φ(a) = b, φ(b) = bb
11	25269	⟨a, b \| aa=b, aaaab=abb⟩	φ(a) = b, φ(b) = bb
11	25271	⟨a, b \| aa=b, aaaab=bab⟩	φ(a) = b, φ(b) = bb
11	25272	⟨a, b \| aa=b, aaaab=bba⟩	φ(a) = b, φ(b) = bb
11	25277	⟨a, b \| aa=b, aaaba=abb⟩	φ(a) = b, φ(b) = bb
11	25279	⟨a, b \| aa=b, aaaba=bab⟩	φ(a) = b, φ(b) = bb
11	25280	⟨a, b \| aa=b, aaaba=bba⟩	φ(a) = b, φ(b) = bb
11	25293	⟨a, b \| aa=b, aabaa=abb⟩	φ(a) = b, φ(b) = bb
11	25294	⟨a, b \| aa=b, aabaa=bab⟩	φ(a) = b, φ(b) = bb
11	25655	⟨a, b \| ab=a, bbbbb=abb⟩	φ(a) = bbbbb, φ(b) = b
11	25730	⟨a, b \| aa=b, aabb=aaab⟩	φ(a) = b, φ(b) = bb
11	25731	⟨a, b \| aa=b, aabb=aaba⟩	φ(a) = b, φ(b) = bb
11	25734	⟨a, b \| aa=b, abaa=aabb⟩	φ(a) = b, φ(b) = bb
11	25736	⟨a, b \| aa=b, abab=aaab⟩	φ(a) = b, φ(b) = bb
11	25737	⟨a, b \| aa=b, abab=aaba⟩	φ(a) = b, φ(b) = bb
11	25739	⟨a, b \| aa=b, abab=abaa⟩	φ(a) = b, φ(b) = bb
11	25741	⟨a, b \| aa=b, abba=aaab⟩	φ(a) = b, φ(b) = bb
11	25742	⟨a, b \| aa=b, abba=aaba⟩	φ(a) = b, φ(b) = bb
11	25753	⟨a, b \| aa=b, baaa=aabb⟩	φ(a) = b, φ(b) = bb
11	25754	⟨a, b \| aa=b, baaa=abab⟩	φ(a) = b, φ(b) = bb
11	25757	⟨a, b \| aa=b, baab=aaab⟩	φ(a) = b, φ(b) = bb
11	25758	⟨a, b \| aa=b, baab=aaba⟩	φ(a) = b, φ(b) = bb

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