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I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. The carrier of $A$ is a set (written $|A|$). The interpretation of $\omega\in \Omega$ is a function $a_{\omega}:|A|^{ar(\omega)}\longmapsto |A|$, where ar is the function associating to each operation symbol its arity.

Now, I can't understand the following "equivalent" definition: an $\Omega$-algebra is essentially a carrier $|A|$ and a single function $$a:\displaystyle\sum_{\omega\in\Omega}|A|^{ar(\omega)}\longmapsto |A|$$ in which $\sum$ is the disjoint union operator. The domain is the set of terms which consist of a single operation applied to elements of $|A|$ and the function $a$ evaluates those in $|A|$.

This sounds obscure to me: what actually this functions $a$ do? If the domain is the disjoint union, how can 'a single operation applied to....' be elements of that domain?

Danae Kissinger
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    You split up the big disjoint union into its summands. The operations are the restrictions. – Zhen Lin Feb 27 '15 at 13:45
  • Oh, and careful with your arrows. The bar opposite the arrowheads indicates element mapping, not functional mapping. (Think "maps to" when you just mean "to"). – Timtro Oct 12 '18 at 04:31
  • In programming terms, the $\sum$ makes for a bunch of if-clauses. Each clause is a separate function. – Nikolaj-K Oct 13 '19 at 10:52

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The function $a$ can receive arguments of several types (where each type is a term in the union). For consonance with the use of $\Sigma$, let's denote disjoint union as $+$.

Given a set $X$ and functions $a_1 \colon X \to X$ and $a_2 \colon X \times X \to X$, we may render the $a$s as a single function with signature $$ a \colon X + X \times X \to X, $$ where $$ a(y) = \left\{\begin{aligned} a_1(x) \quad &\text{iff}\ y = x \in X \\ a_2(x_1, x_2) \quad &\text{iff}\ y = (x_1, x_2) \in X \times X \end{aligned}\right. $$

So, given an $\Omega$-algebra $A$, with the operators $\omega$ and interpretations $$ a_\omega \colon \lvert A \rvert^{ar(\omega)} \longrightarrow \lvert A \rvert $$ we have the equivalent but more compact representation as the single function $$ a:\displaystyle\sum_{\omega\in\Omega}|A|^{ar(\omega)} \longrightarrow |A| $$ where $$ a(y) = \left\{\begin{aligned} a_{\omega_1}(…) \quad &\text{iff}\ …\\ a_{\omega_2}(…) \quad &\text{iff}\ …\\ &\vdots \end{aligned}\right. $$ for all $\omega \in \Omega$, where I hope you will forgive my abuse of notation in enumerating the $\omega$s.

Timtro
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