# Kuratowski's 14 theorem and universal algebra

For a tuple of functions $$\overline{p}$$ on a set $$Y$$, let $$cl_{\overline{p}}$$ be the associated closure operation: $$cl_{\overline{p}}(Z)$$ is the smallest subset of $$Y$$ containing $$Z$$ and closed under each of the operations in $$\overline{p}$$. For $$\overline{p}, Y$$ as above and $$A\subseteq Y$$, let $$\sharp_{\overline{p},Y}(A)$$ be the size of the smallest set $$\subseteq\mathcal{P}(Y)$$ contianing $$A$$ as an element and closed under complementation and $$cl_{\overline{p}}$$.

Given an algebra $$\mathcal{A}$$ (in the sense of universal algebra), let the Kuratowski spectrum of $$\mathcal{A}$$ be the set of cardinals $$n$$ with the following property:

For some $$\mathcal{B}\in\mathsf{HSP}(\mathcal{A})$$ and some finite set $$\overline{p}=p_1,...,p_k$$ of polynomials (= terms with parameters) in $$\mathcal{B}$$, there is an $$X\subseteq\mathcal{B}$$ with $$\sharp_{\overline{p}, \mathcal{B}}(X)=n$$.

Hammer proved (see the introduction to Shallit/Willard) a general result that implies that we always have the Kuratowski spectrum is a subset of $$[14]=\{1,2,...,14\}$$. Meanwhile, $$2$$ is always in the Kuratowski spectrum (consider the trivial algebra), and it's easy to whip up examples of algebras with Kuratowski spectrum either of the extremes $$[14]$$ or $$\{2\}$$.

($$1$$ is impossible for the silly reason that the empty set isn't allowed as a carrier set of an algebra. If we do allow empty algebras, then every Kuratowski spectrum contains $$\{1,2\}$$, and the interesting question is what happens between $$\{1,2\}$$ and $$[14]$$.)

I'm generally curious about what the Kuratowski spectrum tells us about an algebra. One natural question here is which subsets of $$[14]$$ occur as Kuratowski spectra, but that seems difficult. I'm hoping the following will be easier to answer:

Question 1: Are there nontrivial algebras with Kuratowski spectrum $$\not=[14]$$?

In case the answer is yes, the natural follow-up question is:

Question 2: Are there algebras with incomparable (with respect to $$\subseteq$$) Kuratowski spectra?

This is an affirmative answer to the first question.

Let $$\mathcal{A}$$ be any algebra whose fundamental operations are constant. Let $$\mathcal{B}\in\mathsf{HSP}(\mathcal{A})$$ be any algebra. The fundamental operations of $$\mathcal{B}$$ will also be constant, so the polynomial operations of $$\mathcal{B}$$ will be constant operations or projection operations.

Fix a sequence $$\overline{p}=(p_1,\ldots,p_k)$$ of polynomial functions of $$\mathcal{B}$$. Let $$C$$ be the set of images of the constant functions in the sequence $$\overline{p}$$. For any subset $$Z\subseteq B$$, one verifies that $$cl_{\overline{p}}(Z)=Z\cup C$$.

If we start with any subset $$X\subseteq B$$ and close under $$cl_{\overline{p}}$$ and complementation ($$Z\mapsto Z^c := B-Z$$), we can only generate:

$$\{X, X^c, X\cup C, X^c\cup C, X\cap C^c, X^c\cap C^c\}.$$

Thus, at most 6 sets can be generated from $$X$$ by $$cl_{\overline{p}}$$ and complementation, not 14.