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This question is a follow-up to this one.

Say that a term reduct of a ring $\mathcal{R}=(R; +,\times,0,1)$ is a magma $\mathcal{M}$ whose domain is $R$ and whose magma operation is $(x,y)\mapsto t(x,y)$ for some (parameter-free, two-variable) term in the language of rings (denote this magma by "$\mathcal{R}_t$"). For example, if $\mathcal{S}=\mathbb{Z}/2\mathbb{Z}$ and $u(x,y)=(x+1)(y+1)$ then $\mathcal{S}_u$ is isomorphic the magma with domain $\{0,1\}$ and operation $(x,y)\mapsto \max\{1-x,1-y\}$.

I'm interested in the number of terms, or at least unary terms, of term reducts of rings. Specifically:

  • An algebra is primal iff every function on the algebra is represented by a term.

  • An algebra is unary-rich iff every unary function on the algebra is represented by a term.

For example, the $\mathcal{S}_u$ above is unary-rich (and I think primal, but I'm not certain about that).

My question is:

Which rings have primal (or at least unary-rich) term reducts?

Even the $\mathbb{Z}/p\mathbb{Z}$s seem nontrivial here. Note that no magma with more than one element which has an idempotent can be unary-rich, so we need to look at terms more complicated than just $+$ or $\times$.

Noah Schweber
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  • The only rings that are candidates for this are $\mathbb{Z}/p\mathbb{Z}$. If the original ring is not primal, then neither can any reduct be primal. Also, primal algebras must be simple and can have no proper nontrivial subalgebras. – Eran Jun 02 '21 at 15:49
  • @Eran Of course, silly me! I'm still digesting the previous answers on primal algebras so I forgot that. – Noah Schweber Jun 02 '21 at 15:50

1 Answers1

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Eran has noted that the only candidates are the rings of the form $\mathbb Z_p$. Each such ring has a primal magma reduct:

The operation $$ \begin{array}{|c||c|c|c|c|c|} \hline * & 0 & 1 & \cdots & n-2 & n-1\\ \hline \hline 0 & 1 & 0 & \cdots &0 & 0\\ \hline 1 & 0 & 2& &0 & 0\\ \hline \vdots & \vdots & & & & \\ \hline n-2& 0 & 0 & & n-1 & 0\\ \hline n-1& 0 & 0 & \cdots & 0 & 0\\ \hline \end{array} $$ defined by $x*y=0$ if $x\neq y$ and $x*x = x+1 \pmod{n}$ is a Sheffer operation on $\{0, 1, \ldots, n-1\}$. Using Lagrange interpolation, this can be shown to be a term function of $\mathbb Z_n$ if $n$ is prime. (The only thing you need for the Lagrange interpolation to work is that $\mathbb Z_n=\mathbb Z_p$ is a field and that every element of $\mathbb Z_p$ is the interpretation of a nullary term.)

Keith Kearnes
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