## Theorems (from clone theory) that can be stated only by using operations and their composition.

To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and relations in general) that can still be stated if we consider the operations as morphisms in an abstract category that has products (why we need the products will be explained below).

I believe that this can be best explained by using an example.

Take, for instance the well-known Swierczkowski-Lemma that says that if you obtain a projection from a given at least quaternary operation by identifying two of its arguments, then this projection must always be the same no matter which two variables one identifies. This lemma can be stated only be using operations and compositions between them. This can be done as follows.

Let $A$ be an object in a category with products. Let $f$ be a morphism from $A^n$ to $A$ ($n \geq 4$) such that $\langle \pi_{i_1}^n,\ldots,\pi_{i_n}^n \rangle \circ f$ is a projection whenever $i_1,\ldots,i_n$ are not pairwise distinct (of course, the projections are all defined from $A^n$ to $A$). Then, the projection is always the same.

In the category of sets, this Lemma coincides with the usual Swierczwoski-Lemma. The only things we needed for this statement are the operations (interpreted as morphisms from the $n$-th power of $A$ to $A$), the composition between them and the fact that we have products (such that the powers of $A$ and hence also the tuplings and the projections are available).

So my question is: What other theorem (or lemmas) from clone theory do you know that can be stated by only using these things? Note that I am not asking for the theorem to be still correct. This is something I will then figure out on my own (as a sidenote: the above stated "generalized" Swierczwoski-Lemma is still true). I just want to find candidates.

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Does "projection" mean: "projection from $A^n$ onto one of its factors"? What does the notation $\pi^n_i$ mean? In which order do you write the composition of maps? A Google search for "Swierczwoski-Lemma" gives precisely one hit which is this MathOverflow-Page. What would be a good reference for results like this? – Andreas Thom Sep 28 2010 at 20:19
You are right about the projections. The notation $\pi_i^n$ means the projection from $A^n$ to its $i$-th factor. I write composition from right to left, i.e. $f \circ g$ means "f after g". As for the "Swierczwoski"-Lemma, I have to apologize for the typo. It's called Swierczkowski Lemma (now a google-search will give you more hits). I have also corrected it above. The original reference is: S. Swierczkowski, Algebras which are independently generated by every n elements, Fun. Math., 49 (1960/61), pp. 93-104 – Sebastian Sep 28 2010 at 21:02
Furthermore, you can find the Swierczkowksi-Lemma in many works that deal with minimal clones (since it is particularly important for them). For instance, it is stated (and proved) in B. Csákány's wonderful survey "Minimal clones - a minicourse" that appeared in Algebra Universalis 54 (2005). – Sebastian Sep 28 2010 at 21:05
Thanks a lot for the answer. – Andreas Thom Sep 29 2010 at 8:15