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For an object $X$ in a category with finite products, define its endomorphism Lawvere theory to be the Lawvere theory generated by $X$: its $n$-ary operations are given by $\text{Hom}(X^n, X)$, and so accordingly it describes the most general algebraic structure that $X$ possesses. Some quick examples:

  1. The endomorphism Lawvere theory of the abelian group $\mathbb{Z}$ is the Lawvere theory of abelian groups. (This is specific to abelian groups.)

  2. The endomorphism Lawvere theory of the set $2 = \{ 0, 1 \}$ is the Lawvere theory of Boolean algebras, or equivalently Boolean rings. This recently came up here.

  3. The endomorphism Lawvere theory of Sierpinski space is, I believe, the Lawvere theory of (bounded) distributive lattices, although I haven't checked this.

  4. As an interesting generalization of the second example, the endomorphism Lawvere theory of the set $3 = \{ 0, 1, 2 \}$ is the Lawvere theory of a ternary generalization of Boolean rings: $\mathbb{F}_3$-algebras such that every element $x$ satisfies $x^3 = x$. This follows from the fact that every ternary function $3^n \to 3$ can be represented as a polynomial over $\mathbb{F}_3$ which is unique if we require that the degree of the polynomial in each variable is at most $2$, or equivalently if we impose the relation $x^3 = x$ on each variable.

Unfortunately, I'm not sure how to continue this pattern: that is,

What is the endomorphism Lawvere theory of the finite set $4 = \{ 0, 1, 2, 3 \}$? Of any finite set?

(An answer should be a description in terms of generators and relations - some operations that generate all the others under composition and product, and the axioms they satisfy - which is ideally related to familiar algebraic structures such as rings.)

I am only confident I know the answer for a finite set of prime size $p$, which again generalizes the Boolean case: it should be $\mathbb{F}_p$-algebras such that every element satisfies $x^p = x$.

Qiaochu Yuan
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    It seems your description using $\mathbb{F}_p$-algebras works when $p$ is a prime power, not just a prime. – Eric Wofsey Jan 12 '18 at 22:44
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    Hmm, looks like you're right; we're really just using properties of finite fields. Okay, cool. So now the smallest set I don't know the answer for is $6$. – Qiaochu Yuan Jan 12 '18 at 22:48
  • And for sets whose cardinality is not a prime power, you can't just use the ring operations for any ring structure, since the ring splits as a direct product by the Chinese remainder theorem and all polynomials will respect this splitting (for instance, the image of any polynomial must be a "rectangle" for the splitting). – Eric Wofsey Jan 12 '18 at 22:57
  • Just to make sure I follow correctly: Are these algebras unital? – Lukas Heger Jan 12 '18 at 23:03
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    @MatheinBoulomenos: yes. Otherwise we would not be able to express constant functions. – Qiaochu Yuan Jan 12 '18 at 23:10
  • If $A$ is a finite commutative ring, then the $A$-algebra structure on $A$ together with the characteristic function of ${0}$ will generate the endomorphism Lawvere theory of the set $A$. Describing the relations seems tricky though. – Eric Wofsey Jan 12 '18 at 23:20
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    I think Todd Trimble's answer here is relevant. – Eran Jan 13 '18 at 00:04
  • To be more clear, I think you are just looking for a primal algebra of order $n$, and the $n$-valued Post algebra is an example of this. – Eran Jan 13 '18 at 00:09
  • @Eran: thanks. Apparently a primal algebra is an algebra $A$ such that every function $A^n \to A$ is a term. I want something stronger than this, namely I want every function to be a unique term. Todd's answer is definitely relevant, and it looks like there's a reasonably nice description by generators and relations in it. – Qiaochu Yuan Jan 13 '18 at 00:47
  • @QiaochuYuan : why a unique term ? Since you're looking at the Lawvere theory, essentially you're interested in the clone, and for that the specific presentation doesn't matter (that's one of the points of Lawvere theories) – Maxime Ramzi Jan 13 '18 at 10:46
  • @Max: maybe I've misunderstood the definition of a primal algebra, or maybe we are using the word "unique" differently. I took the definition to mean the following, in Lawvere theory terms: a primal algebra is a pair consisting of a Lawvere theory and a model $A$ of that Lawvere theory such that the map from $n$-ary operations in the Lawvere theory to functions $A^n \to A$ is surjective. I want the Lawvere theory to be exactly the endomorphism Lawvere theory of $A$, which means I want this map to be bijective, not surjective. Consider explicitly $A = 2$; I want the Lawvere... – Qiaochu Yuan Jan 13 '18 at 11:49
  • theory to be Boolean rings and not, say, $\mathbb{F}_2$-algebras, which would satisfy surjectivity but not bijectivity. – Qiaochu Yuan Jan 13 '18 at 11:50
  • Oh ! Ok I think I get it; I misunderstood you because I thought you meant "terms" in the syntactic sense, which didn't make much sense in this setting – Maxime Ramzi Jan 13 '18 at 12:03
  • Yes, I meant terms up to equivalence (I don't know if this is the right word; I mean I consider two terms the same if they define the same function on all models, or equivalently if the axioms prove they are equal). Have I correctly understood the definition of a primal algebra then? – Qiaochu Yuan Jan 13 '18 at 12:07

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