The Collatz conjecture, also known as the $3n+1$ problem, has captivated mathematicians since its formulation by Lothar Collatz in 1937. Despite its simple statement; that iterated application of the Collatz function:

will eventually lead any positive integer to 1; the conjecture has remained unproven. Extensive research efforts, ranging from computational verifications to sophisticated analytical techniques [1, 15, 3, 17], have yet to yield a definitive proof. This enduring challenge highlights the need for novel perspectives and interdisciplinary approaches.

In this paper, we propose an algebraic approach to tackle the Collatz conjecture by studying the topological structures it induces. Specifically, we consider a topology $\tau$ on a set $X$ as a commutative semiring where union acts as addition and intersection as multiplication. This natural correspondence transforms the study of topological properties into an algebraic problem, utilizing the well-established theory of semirings.

Our work focuses on primal topologies, a specific class of topologies induced by functions $f:X\to X$, where open sets $\theta \subseteq X$ satisfy $f^{-1}(\theta )\subseteq \theta$ [10, 21]. When the Collatz function $f:\mathbb{N}\to \mathbb{N}$, as defined above, induces such a primal topology ${\tau }_f$ on the natural numbers $\mathbb{N}$, this topological space also forms an additively idempotent semiring.

This perspective allows us to reformulate the Collatz conjecture in purely algebraic terms. Our central result establishes a novel characterization: the Collatz conjecture holds if and only if the semiring ${\tau }_f$ possesses a unique maximal ideal. This reinterprets a long-standing problem in number theory as a question about the algebraic structure of a topological space, offering a new avenue for its investigation.

Beyond the Collatz conjecture, our research provides broader contributions to the understanding of semirings induced by topologies:

• •

  We show that topological spaces that are not $w$-$R_0$, that is, the intersection of the closures of all points is empty, as defined by Di Maio in [9] are precisely those local semirings whose only maximal ideal is an avoidance ideal of a closed set. This establishes a fundamental link between a separation axiom and the algebraic property of locality.

• •

  We show that when a primal topology is compact, its associated semiring decomposes as a finite direct sum of local semirings. Furthermore, we prove that primal compactness and connectedness together characterize the locality of these semirings.

This approach links algebraic properties of semirings with topological properties of spaces, and it has implications beyond the specific context of the Collatz conjecture. This framework could potentially be applied to analyze other discrete dynamical systems or algorithms where a function induces a topological structure, offering new insights into their behavior through the lens of semirings. This provides a versatile tool for exploring diverse mathematical structures.

This manuscript is organized as follows: Section 2 provides essential definitions and preliminary results on semirings and topology. Section 3 delves into the properties of local semirings induced by topologies. In Section 4, we examine compact primal topologies as direct sums of local semirings. Finally, Section 5 applies these concepts to the Collatz semiring, reformulating the conjecture, and posing open questions for future research.

This section introduces fundamental definitions and results from semiring theory and general topology that are essential for understanding the subsequent sections.

A semiring (commutative with nonzero identity) is an algebra $(R,+,\cdot ,0,1)$ where $R$ is a set with distinguished elements $0,1\in R$. The binary operations $+$ (sum) and $\cdot$ (multiplication) on $R$ satisfy the following axioms [12]:

1. (1)

   $(R,+,0)$ is a commutative monoid.

2. (2)

   $(R,\cdot ,1)$ is a commutative monoid, with $0\ne 1$.

3. (3)

   Multiplication distributes over addition: $a(b+c)=ab+ac$ for all $a,b,c\in R$.

4. (4)

   Zero is an annihilator for multiplication: $a\cdot 0=0$ for all $a\in R$.

A subset $I$ of a semiring $R$ is an ideal if it is closed under addition ($a,b\in I⟹a+b\in I$) and absorbs multiplication ($a\in I,r\in R⟹ra\in I$) . A prime ideal $P$ of $R$ is a proper ideal such that if $xy\in P$, then $x\in P$ or $y\in P$ [2]. A maximal ideal is a proper ideal that is not properly contained in any other ideal of $R$. A semiring $R$ is called a local semiring if it possesses a unique maximal ideal. (Such semirings are referred to as “quasi-local” in [12].)

An element $r\in R$ is a unit if there exists an element $r^{\prime }\in R$ such that $r{r}^{\prime }=1=r^{\prime }r$. The set of all units of $R$ is denoted by $U(R)$.

The following propositions regarding ideals and units in semirings are standard results [12]:

A semiring is said to be PM-Semiring if each of its prime ideals is contained in a unique maximal ideal.

Let ${\{R_j\}}_{j\in J}$ be a family of semirings. The direct product ${\prod }_{j\in J}{R}_j$ is defined as the set of all tuples ${(a_j)}_{j\in J}$ such that $a_j\in R_j$ for each $j\in J$.

This set forms a semiring with component-wise addition and multiplication: $(a_j)+(b_j):=(a_j+b_j)$ and $(a_j)\cdot (b_j):=(a_j\cdot b_j)$. In the finite case, the product is isomorphic to the direct sum ${⨁}_{j\in J}{R}_j$.

For completeness, we recall the definitions of compactness and connectedness in general topology.

A topological space $(X,\tau )$ is compact if every open cover of $X$ has a finite subcover. That is, if $X={\bigcup }_{i\in I}{U}_i$ for some family of open sets ${\{U_i\}}_{i\in I}$, then there exists a finite subset $F\subseteq I$ such that $X={\bigcup }_{j\in F}{U}_j$. A topological space $(X,\tau )$ is connected if it cannot be expressed as the union of two non-empty disjoint open sets. Equivalently, $X$ is connected if its only clopen subsets are $\mathrm{\varnothing }$ and $X$.

If $(X,\tau )$ is a topological space, $\tau$ can be seen as a commutative semiring, called the topology semiring, where the sum and multiplication are the union and intersection, respectively, and the identity of the sum is $\mathrm{\varnothing }$ and that of multiplication is $X$. This semiring is an additively idempotent ($x+x=x$) and multiplicatively idempotent ($x\ast x=x$) semiring, since the union of open sets is open and the intersection of two open sets is an open set. These types of semirings have been studied in [4, 25, 19, 14, 5, 6, 7].

An ideal of a semiring is considered finitely generated if it can be constructed from a finite set of elements within the semiring, in that case we write $⟨A_1,..,A_n⟩$. A special instance of a finitely generated ideal is a principal ideal, which is uniquely defined by being generated by exactly one element of the semiring. A commutative semiring $S$ is called a Bézout semiring if every finitely generated ideal of $S$ is principal. In a topology semiring every finite set of generators can be reduced to a set of one generator.

In fact, if $I$ is an ideal of $\tau$, then $I\subseteq ⟨\bigcup \{A:A\in I\}⟩$ since if $B\in I$, then, $B=⟨\bigcup \{A:A\in I\}⟩\cap B$. Therefore, $B\in ⟨\bigcup \{A:A\in I\}⟩$.

We can see that the set $\varphi (K)$ is an ideal of the semiring $\tau$ because it satisfies the conditions:

1. (1)

   Closure under addition: For any $A,B\in \varphi (K)$, their union also belongs to $\varphi (K)$, since $(A\cup B)\cap K=(A\cap K)\cup (B\cap K)=\mathrm{\varnothing }$.

2. (2)

   Absorption under multiplication: For any $A\in \varphi (K)$ and $C\in \tau$, their intersection remains within $\varphi (K)$, as $(A\cap C)\cap K=(A\cap K)\cap C=\mathrm{\varnothing }$.

In [9], Di Maio gives the definition of weakly-$R_0$ spaces for non trivial topologies. A topological space $(X,\tau )$ is said to be weakly-$R_0$ (briefly $w$-$R_0$) if ${\bigcap }_{x\in X}\overline{x}=\mathrm{\varnothing }$.

The following is the main theorem of the section.

The locality of the semiring topology is certainly a strong condition, as it causes the equivalence between compactness and being non $w$-$R_0$.

The next corollary follows from the two previous theorems.

Locality also forces the topological space to be connected.

Given a non-empty set $X$ and a function $f:X\to X$ we say that ${\tau }_f$ is the primal topology generated by $f$ if the open sets in ${\tau }_f$ is the family $\{A\subseteq X:f^{-1}(A)\subseteq A\}$. It is clear that ${\tau }_f$ is an Alexandroff topology (the arbitrary intersection of open sets is open). These topologies were introduced in [21] with the name of Functional Alexandroff spaces and independently in [10], where they are called primal topologies. Further results on primal topological spaces appear in recent works such as [8, 18, 16]

The orbit of a point $x\in X$ is defined as the set $O_f(x)=\{f^n(x):n\ge 0\}$. The closure $\overline{x}:=\overline{\{x\}}$ of the set $\{x\}$ coincides with the orbit $O_f(x)$. We say that $x$ is a periodic point if for some $n\in \mathbb{N}$, $f^n(x)=x$, and for any periodic point $x$, we say that $O_f(x)$ is a periodic orbit. The smallest open neighbourhood of $x$ is the set $\mathrm{\Gamma }(x)=\{f^{-n}(x):n\ge 0\}$.

In an Alexandroff space, a quasi-order is defined by $x{\le }_{f}y$ if and only if $y\in \overline{x}$. The sets $(\uparrow x)=\{y\in X:x{\le }_{f}y\}$ and $(\downarrow x)=\{y\in X:y{\le }_{f}x\}$ are precisely the sets $\overline{x}$ and $\mathrm{\Gamma }(x)$, respectively.

Recall that a subset W of X is said to be invariant or $f$-invariant if $f(W)\subseteq W$. A subset W of X is called a minimal set of $f$ if W is a minimal element in the set of all non-empty closed invariant sets of $f$ equipped with the inclusion ordering. In [10] it is proven that the minimal sets of a primal space $(X,{\tau }_f)$ are exactly the orbits of periodic points.

From Theorem 3.8, it follows that if $(X,{\tau }_f)$ is not a $w$-$R_0$ primal space, then ${\tau }_f$ is a local semiring. However, in [23] it is shown that every primal topological space with a single periodic orbit contained in the closure of every point is not $w$-$R_0$. In fact, [13] proves that in primal spaces, this condition is equivalent to being compact and connected. This yields the following corollary.

On the other hand, [11] proves that a primal topological space is compact if and only if it has finitely many periodic points (and hence finitely many periodic orbits) and all other points are eventually periodic.

From this, the following result is deduced:

Let us provide a concrete example that illustrates the theorem.

The $3n+1$ problem or Collatz conjecture has been around since 1937 and no one has been able to prove or disprove it. It seems that another approach is needed, which is why we present an algebraic way of attacking the problem.

In [23], the set $\mathbb{N}$ and the Collatz function $f:\mathbb{N}\to \mathbb{N}$ are considered to study the primal Collatz space $(\mathbb{N},{\tau }_f)$. It has been proven that the $3n+1$ problem is true if and only if the primal space $(\mathbb{N},{\tau }_f)$ is not $w$-$R_0$, which allows us to write the following theorem in terms of semirings.

If $R$ is a commutative semiring, we denote by $\mathrm{Spec}(R)$ the set of all prime ideals of $R$. For any ideal $I$ we define $V(I)=\{P\in \mathrm{Spec}(R):I\subseteq P\}$ and $D(I):=\mathrm{Spec}(R)\setminus V(I)$. The sets $V(I)$ with $I$ an ideal of $R$, are the closed sets for a topology on $\mathrm{Spec}(R)$, called the Zariski topology, which will be denoted by ${\zeta }_R$.

For any topology $\tau$, let $\overline{\tau }$ be the closure of $\tau$ as a subset of the Cantor cube $2^X$ with the product topology [22]. Furthermore, $\overline{\tau }$ can be seen as the Alexandroffication of the topology $\tau$, that is, the smallest Alexandroff topology containing $\tau$. Ruza and Vielma [20] prove the following theorem for PM-semirings, using the terminology "Gelfand semirings".

Finally, we leave the following interesting open questions: