#253 ⟨a, b | aa=b, abb=a⟩

Properties

Presentation has sum-of-sides 7
Isomorphic to ℕ_{(5 = 1)}
Finite commutative monoid with 5 elements
Commutative Gröbner basis: ⟨a, b | a⁵=a, b=a²⟩
Not cancellative, because multiplication by a³ is not injective:
- a⁴ ≠ 1
- a³ ⋅ a⁴ = a³
- a³ ⋅ 1 = a³
- Cancellative quotient is isomorphic to ℤ₄
Grothendieck group is isomorphic to ℤ₄
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁴

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

a⁵ ⇒ a
b ⇒ a²

# ab:aa=b,abb=a a/b
aaaaa=a
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴
1	1	a	a²	a³	a⁴
a	a	a²	a³	a⁴	a
a²	a²	a³	a⁴	a	a²
a³	a³	a⁴	a	a²	a³
a⁴	a⁴	a	a²	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

11 unique, 1391 total

Σ	#	Presentation	Description	Related
6	44	⟨a, b \| aa=b, abb=1⟩	Isomorphic to ℤ₅	1132 iso
7	254	⟨a, b \| aa=b, abb=b⟩	Isomorphic to ℕ_{(5 = 2)}	43 iso
7	268	⟨a, b \| ab=a, baa=b⟩	Finite non-commutative monoid with 5 elements	63 iso, 23 anti-iso
8	574	⟨a, b \| aaa=b, aab=b⟩	Isomorphic to ℕ_{(5 = 3)}	27 iso
8	950	⟨a, b \| ab=a, bbbb=a⟩	Isomorphic to ℕ_{(5 = 4)}	32 iso
8	995	⟨a, b \| ab=a, aaa=bb⟩	Finite commutative monoid with 5 elements	19 iso
8	1019	⟨a, b \| ab=a, bba=bb⟩	Finite non-commutative monoid with 5 elements	25 iso
8	1020	⟨a, b \| ab=a, bbb=aa⟩	Finite commutative monoid with 5 elements	9 iso
8	1022	⟨a, b \| ab=a, bbb=ba⟩	Finite non-commutative monoid with 5 elements	4 iso
10	8617	⟨a, b \| aa=a, abbbba=b⟩	Finite commutative monoid with 5 elements	3 iso
11	15426	⟨a, b \| aaa=aa, abbba=b⟩	Finite commutative monoid with 5 elements

Other isomorphic instances

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

71 total

Σ	#	Presentation	Mapping
7	255	⟨a, b \| aa=b, bab=a⟩	φ(a) = a, φ(b) = aa
8	573	⟨a, b \| aaa=b, aab=a⟩	φ(a) = aaa, φ(b) = a
8	575	⟨a, b \| aaa=b, aba=a⟩	φ(a) = aaa, φ(b) = a
8	902	⟨a, b \| aa=b, aaab=a⟩	φ(a) = a, φ(b) = aa
8	904	⟨a, b \| aa=b, aaba=a⟩	φ(a) = a, φ(b) = aa
8	921	⟨a, b \| ab=a, aaaa=b⟩	φ(a) = a, φ(b) = aaaa
9	2058	⟨a, b \| aab=b, aabb=a⟩	φ(a) = aa, φ(b) = a
9	2062	⟨a, b \| aab=b, abab=a⟩	φ(a) = aa, φ(b) = a
9	2070	⟨a, b \| aab=b, baab=a⟩	φ(a) = aa, φ(b) = a
9	2110	⟨a, b \| aba=b, aabb=a⟩	φ(a) = aa, φ(b) = a
9	2112	⟨a, b \| aba=b, abab=a⟩	φ(a) = aa, φ(b) = a
9	2114	⟨a, b \| aba=b, abba=a⟩	φ(a) = aa, φ(b) = a
9	2896	⟨a, b \| aa=b, aaaaa=a⟩	φ(a) = a, φ(b) = aa
9	2939	⟨a, b \| ab=a, aaaab=b⟩	φ(a) = a, φ(b) = aaaa
9	2941	⟨a, b \| ab=a, aaaba=b⟩	φ(a) = a, φ(b) = aaaa
9	2945	⟨a, b \| ab=a, aabaa=b⟩	φ(a) = a, φ(b) = aaaa
9	2953	⟨a, b \| ab=a, abaaa=b⟩	φ(a) = a, φ(b) = aaaa
10	6278	⟨a, b \| aaa=b, aaaaa=a⟩	φ(a) = aaa, φ(b) = a
10	6321	⟨a, b \| aab=a, aaaab=b⟩	φ(a) = aaa, φ(b) = a
10	6323	⟨a, b \| aab=a, aaaba=b⟩	φ(a) = aaa, φ(b) = a
10	6327	⟨a, b \| aab=a, aabaa=b⟩	φ(a) = aaa, φ(b) = a
10	6335	⟨a, b \| aab=a, abaaa=b⟩	φ(a) = aaa, φ(b) = a
10	6451	⟨a, b \| aba=a, aaaba=b⟩	φ(a) = aaa, φ(b) = a
10	6455	⟨a, b \| aba=a, aabaa=b⟩	φ(a) = aaa, φ(b) = a
10	8719	⟨a, b \| ab=a, aaaabb=b⟩	φ(a) = a, φ(b) = aaaa
10	8723	⟨a, b \| ab=a, aaabab=b⟩	φ(a) = a, φ(b) = aaaa
10	8725	⟨a, b \| ab=a, aaabba=b⟩	φ(a) = a, φ(b) = aaaa
10	8731	⟨a, b \| ab=a, aabaab=b⟩	φ(a) = a, φ(b) = aaaa
10	8733	⟨a, b \| ab=a, aababa=b⟩	φ(a) = a, φ(b) = aaaa
10	8737	⟨a, b \| ab=a, aabbaa=b⟩	φ(a) = a, φ(b) = aaaa
10	8747	⟨a, b \| ab=a, abaaab=b⟩	φ(a) = a, φ(b) = aaaa
10	8749	⟨a, b \| ab=a, abaaba=b⟩	φ(a) = a, φ(b) = aaaa
10	8753	⟨a, b \| ab=a, ababaa=b⟩	φ(a) = a, φ(b) = aaaa
10	8761	⟨a, b \| ab=a, abbaaa=b⟩	φ(a) = a, φ(b) = aaaa
11	14369	⟨a, b \| aaaa=b, aaaaa=a⟩	φ(a) = a, φ(b) = aaaa
11	14412	⟨a, b \| aaab=a, aaaab=b⟩	φ(a) = a, φ(b) = aa
11	14414	⟨a, b \| aaab=a, aaaba=b⟩	φ(a) = a, φ(b) = aa
11	14418	⟨a, b \| aaab=a, aabaa=b⟩	φ(a) = a, φ(b) = aa
11	14426	⟨a, b \| aaab=a, abaaa=b⟩	φ(a) = a, φ(b) = aa
11	14542	⟨a, b \| aaba=a, aaaba=b⟩	φ(a) = a, φ(b) = aa
11	14546	⟨a, b \| aaba=a, aabaa=b⟩	φ(a) = a, φ(b) = aa
11	14554	⟨a, b \| aaba=a, abaaa=b⟩	φ(a) = a, φ(b) = aa
11	18966	⟨a, b \| aab=b, aaaabb=a⟩	φ(a) = aa, φ(b) = a
11	18970	⟨a, b \| aab=b, aaabab=a⟩	φ(a) = aa, φ(b) = a
11	18978	⟨a, b \| aab=b, aabaab=a⟩	φ(a) = aa, φ(b) = a
11	18994	⟨a, b \| aab=b, abaaab=a⟩	φ(a) = aa, φ(b) = a
11	19026	⟨a, b \| aab=b, baaaab=a⟩	φ(a) = aa, φ(b) = a
11	19170	⟨a, b \| aba=b, aaabab=a⟩	φ(a) = aa, φ(b) = a
11	19176	⟨a, b \| aba=b, aabaab=a⟩	φ(a) = aa, φ(b) = a
11	19178	⟨a, b \| aba=b, aababa=a⟩	φ(a) = aa, φ(b) = a
11	19192	⟨a, b \| aba=b, abaaba=a⟩	φ(a) = aa, φ(b) = a
11	24347	⟨a, b \| ab=a, aaaabbb=b⟩	φ(a) = a, φ(b) = aaaa
11	24355	⟨a, b \| ab=a, aaababb=b⟩	φ(a) = a, φ(b) = aaaa
11	24359	⟨a, b \| ab=a, aaabbab=b⟩	φ(a) = a, φ(b) = aaaa
11	24361	⟨a, b \| ab=a, aaabbba=b⟩	φ(a) = a, φ(b) = aaaa
11	24371	⟨a, b \| ab=a, aabaabb=b⟩	φ(a) = a, φ(b) = aaaa
11	24375	⟨a, b \| ab=a, aababab=b⟩	φ(a) = a, φ(b) = aaaa
11	24377	⟨a, b \| ab=a, aababba=b⟩	φ(a) = a, φ(b) = aaaa
11	24383	⟨a, b \| ab=a, aabbaab=b⟩	φ(a) = a, φ(b) = aaaa
11	24385	⟨a, b \| ab=a, aabbaba=b⟩	φ(a) = a, φ(b) = aaaa
11	24389	⟨a, b \| ab=a, aabbbaa=b⟩	φ(a) = a, φ(b) = aaaa
11	24403	⟨a, b \| ab=a, abaaabb=b⟩	φ(a) = a, φ(b) = aaaa
11	24407	⟨a, b \| ab=a, abaabab=b⟩	φ(a) = a, φ(b) = aaaa
11	24409	⟨a, b \| ab=a, abaabba=b⟩	φ(a) = a, φ(b) = aaaa
11	24415	⟨a, b \| ab=a, ababaab=b⟩	φ(a) = a, φ(b) = aaaa
11	24417	⟨a, b \| ab=a, abababa=b⟩	φ(a) = a, φ(b) = aaaa
11	24421	⟨a, b \| ab=a, ababbaa=b⟩	φ(a) = a, φ(b) = aaaa
11	24431	⟨a, b \| ab=a, abbaaab=b⟩	φ(a) = a, φ(b) = aaaa
11	24433	⟨a, b \| ab=a, abbaaba=b⟩	φ(a) = a, φ(b) = aaaa
11	24437	⟨a, b \| ab=a, abbabaa=b⟩	φ(a) = a, φ(b) = aaaa
11	24445	⟨a, b \| ab=a, abbbaaa=b⟩	φ(a) = a, φ(b) = aaaa

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