I am reading this paper.
Theorem 2.1 on page 3 says that every logic has a matrix semantics.
A matrix is a pair $\langle \mathcal{A}, X \rangle$ where $\mathcal{A}$ is an algebraic structure and $X$ is a subset of $A$ called the designated elements or designated truth values.
A matrix semantics is a whole class of these things where the algebras are all similar (have the same language).
A logic $\mathcal{S}$ in this case is a pair consisting of a language $\mathcal{L}$ and a consequence relation $\vdash_{\mathcal{S}}$.
I'm assuming that $\vdash_{\mathcal{S}}$ is a deductive system, subject to the following rules. These rules are mostly paraphrased from Blok and Pigozzi's Algebraizable Logics, but I don't have the book in front of me at the moment.
Let $\Gamma \vdash \Delta$ hold if and only if $\Gamma \vdash \varphi$ for all $\varphi$ in $\Delta$.
- substitution closure: For all substitutions $\sigma$, if $\Gamma \vdash \varphi$ then $\sigma(\Gamma) \vdash \sigma(\varphi)$.
- monotonicity of entailment: $\Gamma \vdash \varphi$ implies $\Delta \cup \Gamma \vdash \varphi$.
- identity: $\varphi \vdash \varphi$
- transitivity: If $\Gamma \vdash \Delta$ and $\Delta \vdash \varphi$ then $\Gamma \vdash \varphi$.
- compactness: If $\Gamma \vdash \varphi$ then there exists a finite $\Gamma_0 \subset \Gamma$ such that $\Gamma_0 \vdash \varphi$.
Anyway, the following explicit matrix semantics is given for an arbitrary deductive system.
$$ M := \{ \langle \textbf{Fm}, \Gamma \rangle : \Gamma \in \text{Th}(\vdash) \} $$
$\textbf{Fm}$ is the formula algebra, a completely free algebra on the language of the propositional logic. And $\text{Th}(\vdash)$ are the set of theories of $\vdash$. A theory is defined as a deductively closed set.
So, that makes sense and is actually quite clever. We can take an arbitrary deductive system and turn it into a whole mess of matrices by taking the formula algebra as our algebra and choosing each possible deductively closed set in it in turn as our designated truth values.
However, some propositional logics such as classical logic, intuitionistic logic, and minimal logic can have their semantics completely described by a single matrix.
Indeed matrix semantics-es where the class is a singleton don't seem to be mentioned in (the first few pages of) Moraschini's paper, but this seems like an interesting special case.
Anyway, this raises the question:
Are there propositional logics with a matrix semantics that can't be described with a single matrix?