In the category of sets and partial functions, $\mathsf{PAR}$, while the disjoint union $\sqcup$ is the usual categorical coproduct, the Cartesian product $\times$ becomes a restriction categorical analogue of the categorical product: a restriction product. Nevertheless, $\mathsf{PAR}$ does have a usual categorical product as well in the form $A \& B := A \sqcup B \sqcup (A \times B)$. Surprisingly, asking that a distributive restriction category (a restriction category with restriction products $\times$ and coproducts $\oplus$) has $A \& B$ a categorical product is enough to imply that the category is a classical restriction category. This is a restriction category which has joins and relative complements and, thus, supports classical Boolean reasoning. The first and main observation of the paper is that a distributive restriction category is classical if and only if $A \& B := A \oplus B \oplus (A \times B)$ is a categorical product in which case we call $\&$ the ''classical'' product. In fact, a distributive restriction category has a categorical product if and only if it is a classified restriction category. This is in the sense that every map $A \to B$ factors uniquely through a total map $A \to B \oplus \mathsf{1}$, where $\mathsf{1}$ is the restriction terminal object. This implies the second significant observation of the paper, namely, that a distributive restriction category has a classical product if and only if it is the Kleisli category of the exception monad $\_ \oplus \mathsf{1}$ for an ordinary distributive category. Thus having a classical product has a significant structural effect on a distributive restriction category. In particular, the classical product not only provides an alternative axiomatization for being classical but also for being the Kleisli category of the exception monad on an ordinary distributive category.

翻译：在集合与偏函数构成的范畴$\mathsf{PAR}$中，虽无交并$\sqcup$是通常的范畴余积，但笛卡尔积$\times$成为范畴积的限制范畴类比物：限制积。尽管如此，$\mathsf{PAR}$确实也具有通常的范畴积，其形式为$A \& B := A \sqcup B \sqcup (A \times B)$。令人惊讶的是，要求一个分配限制范畴（具有限制积$\times$和余积$\oplus$的限制范畴）将$A \& B$作为范畴积，足以推出该范畴是经典限制范畴。这是一种具有连接和相对补元的限制范畴，因此支持经典的布尔推理。本文首先且主要的观察是：分配限制范畴是经典范畴当且仅当$A \& B := A \oplus B \oplus (A \times B)$是范畴积，此时我们将$\&$称为"经典"积。事实上，分配限制范畴具有范畴积当且仅当它是分类限制范畴。这意味着每个映射$A \to B$唯一地通过一个全映射$A \to B \oplus \mathsf{1}$进行分解，其中$\mathsf{1}$是限制终对象。这引出了本文第二个重要观察：分配限制范畴具有经典积当且仅当它是关于普通分配范畴的异常单子$\_ \oplus \mathsf{1}$的Kleisli范畴。因此，具有经典积对分配限制范畴具有显著的结构性影响。特别地，经典积不仅为经典性提供了另一种公理化框架，也为普通分配范畴上异常单子的Kleisli范畴提供了另一种刻画方式。