The title has it all. I'm looking for a reference to the following:



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> **Q.** Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the category of topological spaces)?
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This is, at the end of the day, a special instance of a famous problem (concerning the cancellativity, up to iso, of direct products or other "binary operations" in all sorts of categories), originated from a question of M.S. Ulam that first appeared as Problème 56 on p. 285 of Fund. Math **20** (1933), No. 1; see also footnote 2 in R.H. Fox, *On a problem of S. Ulam concerning cartesian products*, Fund. Math. **27** (1947), No. 1, 278-287. But the literature is vast (even sticking to topological spaces), and I couldn't find a reference to the case in which I'm presently interested. 


Let me mention that I don't have an example showing that $X \times Y \cong X \times Z$ need *not* imply $Y \cong Z$ in my question (to be honest, I haven't even tried to find one), though I don't believe the contrary.




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*Background.* The point of the question is simple, so let me make it a bit more complicate, but only in the hope of also making it a little more interesting (at least from certain points of view). 


Fix a non-empty universe $\mathscr U$. Let $\mathscr T$ be the class of all $\mathscr U$-small, *non-empty* (topological) spaces, and $\sim$ the equivalence relation on $\mathscr T$ that identifies two spaces iff they are homeomorphic. Then the quotient of $\mathscr T$ by $\sim$ is naturally made into a commutative, reduced monoid, $\mathcal M$, by endowing it with the obvious binary operation induced by the direct product of spaces. 


Let $\mathcal M\_{\rm fin}$ be the submonoid of $\mathcal M$ consisting of all and only the equivalence classes containing finite spaces: This is a divisor-closed, unit-cancellative submonoid of $\mathcal M$ (for terminology, see the notes at the bottom of this post), and it is straigthfoward that there is a function $\lambda: \mathcal M\_{\rm fin} \to \mathbf N$ such that $\lambda(\mathsf t\_1) < \lambda(\mathsf t\_2)$ for all *distinct* classes $\mathsf t\_1, \mathsf t\_2 \in \mathcal M\_{\rm fin}$ with $\mathsf t\_1 \mid \mathsf t\_2$ in $\mathcal M\_{\rm fin}$ (just map each isomorphism class $\mathsf t \in \mathcal M\_{\rm fin}$ to the number of elements of any space $T \in \mathsf t$). So, it follows from [elsewhere](https://mathoverflow.net/questions/264379/) that $\mathcal M\_{\rm fin}$ is a BF-monoid, i.e., (i) every class is a finite product of atoms of $\mathcal M\_{\rm fin}$ and (ii) the factorizations (into atoms) of a fixed class cannot be arbitrarily large. Accordingly, it's natural to ask if $\mathcal M\_{\rm fin}$ is cancellative, as this would make life much easier (at least from the perspective of factorization theory, which is the motivation behind my question).


*Notes.* Let $H$ be a multiplicatively written monoid $H$. We say that $H$ is *unit-cancellative* if $xy = x$ or $yx = x$, for some $x, y \in H$, implies $y \in H^\times$, where $H^\times$ is the group of units of $H$. An element $a \in H$ is referred to as an *atom* if (i') $a \notin H^\times$ and (ii') $a \ne xy$ for all $x, y \in H \setminus H^\times$. Finally, a submonoid $M$ of $H$ is called *divisor-closed* if $x \in M$ whenever $x \mid\_H y$ and $y \in M$.



When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)
For simplicity of presentation, I follow Lovasz in his paper Operations with Structures and work with structures with one finitary relation which I will choose to be binary. The extension to multiple finitary relations and algebraic structures is straightforward, and the paper of Lovasz and the book *Algebras, Lattices, Varieties* both give hints on the extension.


First, let us note a case where a relational structure $A$ is not isomorphic to $C$, yet $A\times C \simeq C \times C$. Indeed, let both structures have as universe the two element set with elements $a$ and $b$, and for $A$ the binary relation is identity, while for $C$ the binary relation is the complement, or non identity. The Cartesian product of the structure $C$ with itself has the ordered pair $(a,a)$ related to $(b,b)$ , $(a,b)$ related to $(b,a)$, and vice versa being a symmetric relation. For $A \times C$ we have $(a,a)$ related to $(a,b)$ , $(b,b)$ related to $(b,a)$, and again vice versa. The map that swaps the elements $(b,b)$ and $(a,b)$ gives the desired isomorphism between the two products. This example is noted in both the book and the paper. Thus, any class which contains $A, C$, and the two products fails to have cancellation as well as unique factorization. Fortunately, Salvo Tringali is not asking about such classes yet.


If we take Eric Wofsey's map $h$, we have a map from isomorphism types of finite structures (of the same finite type) into the monoid of a countable power of the nonnegative integers with multiplication. Because of this, we get uniqueness of kth roots in the monoid of interest. In order to get cancellation, we need a little more, namely that the cancelled algebra has a one element subuniverse, or in relational terms that the cancelled structure has an $x$ with $R(x,x,...,x)$ .


For the class of finite topological spaces, we need a way to produce a finitary relation or operation so that we can apply either Lovasz's theorem 4.1 or the book's theorem 5.23. Given a space $X$, and a point $x$, let $O(x)$ be the intersection of all open sets containing $x$, and let $R(x,y)$ be the relation that $O(x) \subseteq O(y)$. When $X$ is finite, this is saying that $x$ is in every open set containing $y$. I leave as an exercise that finite topological spaces $X$ and $Y$ are isomorphic if and only if the associated relational structures are isomorphic. It might help to observe that $R$ is a pre-order on $X$.


Now all these relational structures satisfy $R(x,x)$, not just for one $x$, but for every $x \in X$. Thus we can use Lovasz's Theorem 4.1 on the finite relational structures to get the result for the topologies.


I am still looking for a clean way to transform a topology into an algebra with at least one one element subuniverse, so that I can recover the topology from the algebra. I may need to use a pointed set to accomplish this. Until then, we have to wait before using Theorem 5.23.


I am not well acquainted with category theory to generalize or reframe this result in those terms. It is important for cancellation to have "enough maps", as Eric has noted. I still believe a category-theoretic version exists. I had not read Lovasz's paper before this question appeared and initially thought it might be category-theoretic in nature. It isn't.


Lovasz's paper has not only Cartesian product and disjoint union of relational structures, but also a form of exponentiation of one structure by another. He then expresses polynomials by replacing an integer exponent n by the n element structure with diagonal relation, and otherwise using arbitrary structures as coefficients, and then noting that if "the sum of the coefficients" has at least one diagonal element, then the polynomial as a function of the structure X is injective. He notes with the example above that exponentiation using arbitrary structures as exponents is not injective.


Gerhard "Demurs Upon Considering Structural Tetration" Paseman, 2017.05.15.


