Finite magmas representing all unary functions by terms

Question

Say that a magma $\mathcal{M}=(M;*)$ is unary-rich iff for every function $f:M\rightarrow M$ there is a (one-variable, parameter-free) term $t_f$ such that $t_f^\mathcal{M}=f$. For example:

• The one-element magma (and the empty magma if we permit such) is trivially unary-rich.

• A bit less trivially, there is a two-element unary-rich magma (in fact exactly 0ne up to isomorphism): namely $M=\{a,b\}$ and $*$ given by $$a*a=b, a*b=a, b*a=a, b*b=b$$ via the terms, $x, xx, (xx)x,$ and $(xx)(xx)$.

My question is:

Are there unary-rich magmas of every (or at least arbitrarily large) finite cardinality?

Of course every unary-rich magma is finite since there are only countably many terms. Note that if $\vert M\vert=n<\omega$ then $M$ is unary-rich iff $M$ has $n^n$-many distinct one-variable terms up to equality-as-functions.

Already the case of $3$-element magmas seems interesting and difficult to brute-force.

Answer 1

G. Rousseau, Completeness in finite algebras with a single operation, Proc. Amer. Math. Soc. 18 (1967) 1009-1013

This paper proves that a finite algebra with one $n$-ary operation, $n>1$, is primal iff it has no proper subalgebra, no non-identity automorphism, and no proper nontrivial congruence. Therefore the following are equivalent for a finite algebra with one binary operation:

$\bf A$ is unary-rich.

$\bf A$ is primal.

$\bf A$ has no proper subalgebra, no non-identity automorphism, and no proper nontrivial congruence.

Answer 2

From this answer by Eran, we know that the asymptotic proportion of primal magmas among all magmas is $1/e$.

In a primal algebra all operations defined on its carrier set are terms of the algebra, so certainly every primal magma is unary-rich, and it follows that the asymptotic proportion of unary-rich magmas is equal or greater than $1/e$.

In particular, there have to be arbitrarily large ones.
It may however be the case that is not one of every cardinality.

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Added. From the book "Universal Algebra and Applications in Theoretical Computer Science", by Klaus Denecke and Shelly Wismath, I've got the following results.

Proposition 10.5.7
(i) Let $\mathbf A$ be a primal algebra. Then

1. $\mathbf A$ has no proper subalgebras,
2. $\mathbf A$ has no non-identical automorphisms,
3. $\mathbf A$ is simple, and
4. $\mathbf A$ generates an arithmetical variety.

(ii) Every functional complete algebra is simple

and then

Theorem 10.5.8 For a finite algebra $\mathbf A$ the following propositions are equivalent:

1. $\mathbf A$ is primal.
2. $\mathbf A$ generates an arithmetical variety, has no non-identical automorphisms, has no proper subalgebras and is simple.
3. There is a ternary discriminator term which induces a term operation on $\mathbf A$, and the algebra $\mathbf A$ has no proper subalgebras and no non-identical automorphisms.

Here a ternary discriminator term is an operation $t:A^3\to A$ defined by $$t(x,y,z) = \begin{cases} z &\text{ if } x=y,\\ x &\text{otherwise,} \end{cases} $$

and a variety is arithmetical iff it is both congruence-distributive and congruence-permutable.

Answer 3

The following is an example of a unary rich magma on $3$ elements I think, the multiplication table is the following:

$\begin{pmatrix} 1 0 0\\ 0 2 0 \\ 1 0 0 \end{pmatrix}$

The terms found for each function are as follows:

$ 0\rightarrow 0 \quad 1\rightarrow 0 \quad 2\rightarrow 0 $

$((xx)x)$

$ 0\rightarrow 0 \quad 1\rightarrow 0 \quad 2\rightarrow 1 $

$(x(xx))$

$ 0\rightarrow 0 \quad 1\rightarrow 0 \quad 2\rightarrow 2 $

$((x(xx))(((xx)x)((xx)x)))$

$ 0\rightarrow 0 \quad 1\rightarrow 1 \quad 2\rightarrow 0 $

$((xx)(x(xx)))$

$ 0\rightarrow 0 \quad 1\rightarrow 1 \quad 2\rightarrow 1 $

$((xx)((xx)x))$

$ 0\rightarrow 0 \quad 1\rightarrow 1 \quad 2\rightarrow 2 $

$x$

$ 0\rightarrow 0 \quad 1\rightarrow 2 \quad 2\rightarrow 0 $

$(x((x(xx))(x(xx))))$

$ 0\rightarrow 0 \quad 1\rightarrow 2 \quad 2\rightarrow 1 $

$(x((x(xx))((xx)x)))$

$ 0\rightarrow 0 \quad 1\rightarrow 2 \quad 2\rightarrow 2 $

$(((xx)((xx)x))(((xx)x)((xx)x)))$

$ 0\rightarrow 1 \quad 1\rightarrow 0 \quad 2\rightarrow 0 $

$(x(x(xx)))$

$ 0\rightarrow 1 \quad 1\rightarrow 0 \quad 2\rightarrow 1 $

$(x((xx)x))$

$ 0\rightarrow 1 \quad 1\rightarrow 0 \quad 2\rightarrow 2 $

$((x(xx))((xx)((xx)x)))$

$ 0\rightarrow 1 \quad 1\rightarrow 1 \quad 2\rightarrow 0 $

$((x(xx))((xx)x))$

$ 0\rightarrow 1 \quad 1\rightarrow 1 \quad 2\rightarrow 1 $

$(((xx)x)((xx)x))$

$ 0\rightarrow 1 \quad 1\rightarrow 1 \quad 2\rightarrow 2 $

$((x(xx))(x(xx)))$

$ 0\rightarrow 1 \quad 1\rightarrow 2 \quad 2\rightarrow 0 $

$(xx)$

$ 0\rightarrow 1 \quad 1\rightarrow 2 \quad 2\rightarrow 1 $

$(x((xx)(x(xx))))$

$ 0\rightarrow 1 \quad 1\rightarrow 2 \quad 2\rightarrow 2 $

$(((xx)((xx)x))((xx)((xx)x)))$

$ 0\rightarrow 2 \quad 1\rightarrow 0 \quad 2\rightarrow 0 $

$((xx)((x(xx))(x(xx))))$

$ 0\rightarrow 2 \quad 1\rightarrow 0 \quad 2\rightarrow 1 $

$((xx)(xx))$

$ 0\rightarrow 2 \quad 1\rightarrow 0 \quad 2\rightarrow 2 $

$((x((xx)x))(((xx)x)((xx)x)))$

$ 0\rightarrow 2 \quad 1\rightarrow 1 \quad 2\rightarrow 0 $

$((xx)(x((xx)x)))$

$ 0\rightarrow 2 \quad 1\rightarrow 1 \quad 2\rightarrow 1 $

$((xx)(x(x(xx))))$

$ 0\rightarrow 2 \quad 1\rightarrow 1 \quad 2\rightarrow 2 $

$((x((xx)x))(x((xx)x)))$

$ 0\rightarrow 2 \quad 1\rightarrow 2 \quad 2\rightarrow 0 $

$(((x(xx))(x(xx)))((x(xx))(x(xx))))$

$ 0\rightarrow 2 \quad 1\rightarrow 2 \quad 2\rightarrow 1 $

$(((x(xx))(x(xx)))((x(xx))((xx)x)))$

$ 0\rightarrow 2 \quad 1\rightarrow 2 \quad 2\rightarrow 2 $

$((((xx)x)((xx)x))(((xx)x)((xx)x)))$

I found $1596$ different ones by only checking a small set of terms and going over all the $3^9$ magmas on $0,1,2$.