A theory for generalized morphisms and beyond

The notions of generalized morphisms introduced in this paper develop from very basic mathematical objects such as sets, functions, and partial functions. As a consequence, many familiar mathematical notions are covered by the generalized morphisms. Specifically, continuous functions between topological spaces, ring homomorphisms, module homomorphisms, group homomorphisms, and covariant functors between categories can be characterized in terms of the generalized morphisms. Indeed, in these examples, the generalized morphisms preserve the structures of the mathematical objects involved. Basically, this is because we define the generalized morphisms to be commutative with the intrinsic operations on the mathematical objects involved. Moreover, we show that the inverse of any bijective generalized morphism is also a generalized morphism (of the same kind), and hence a generalized isomorphism can be defined as a bijective generalized morphism.

The notion of generalized morphisms in our theory is not the same as the notion of morphisms in category theory. In category theory, a morphism between two objects does not necessarily preserve the structure of objects, while a generalized morphism in our theory does. Nevertheless, since covariant functors preserve the structures of categories, all covariant functors can be characterized in terms of the generalized morphisms.

One may have encountered concepts similar to the notions of our generalized morphisms, e.g. the notion of homomorphisms between structures in the field of mathematical logic. However, we have not found any research with any content similar to ours.

Besides introducing the new notions, we establish Galois correspondences for the Galois groups of the generalized automorphisms as well as for the “Galois monoids” of the generalized endomorphisms. As is well known, in the Galois theory for infinite algebraic field extensions, Krull topology is put on Galois groups, and in differential Galois theory, differential Galois groups are endowed with Zariski topology. Then the fundamental theorem in each of the two theories tells us that there exists a Galois correspondence between the set of all closed subgroups of the Galois group of a Galois field extension (or a Picard-Vessiot extension) and the set of all intermediate (differential) fields. Now for our theory, we first study the lattice structures related to Galois correspondences. Then, we can determine under what conditions we can characterize Galois correspondences with topology (as is done in the above two Galois theories).

Moreover, we study ways to construct the generalized morphisms and the generalized isomorphisms, which we think to be important because the morphisms and isomorphisms preserve the structures of the mathematical objects involved.

In Part I of this paper, which consists of Sections 2 to 7, we address the above issues for the case where only single-variable functions are involved, and in Part II, we address the above issues for the case where multivariable functions or partial functions are involved.

In Part III, new interpretations on solvability of polynomials and solvability of homogeneous linear differential equations are introduced. The key idea is to “decompose” an equation $Q$ into $n(\in{{\mathbb{Z}}^{+}})$ equations ${{Q}_{1}},\cdots,{{Q}_{n}}$ such that each solution of $Q$ can somehow be expressed in terms of the solutions of ${{Q}_{1}},\cdots,{{Q}_{n}}$, where for example in the case of a separable polynomial $p(x)$ in $Q:=(p(x)=0)$, the Galois group of each $p_{i}(x)$ in $Q_{i}:=(p_{i}(x)=0)$ is simple. Moreover, this idea is roughly generalized for “general” equation solving in terms of the theory developed in Parts I and II.

Some more results are presented in Part IV. For example, in Section 19, we obtain analogues of the well-known fact that the Galois group of an irreducible polynomial acts transitively on its roots, and in Section 21, we generalize the algebraic notions of transcendental elements over a field and purely transcendental field extensions. Moreover, in Section 22, we obtain an isomorphism theorem which generalizes the first isomorphism theorems (for groups, rings, and modules). And in Section LABEL:App_to_dyn, we show a close relation between a part of our theory and dynamical systems.

To help the reader better understand the structure of the paper, we simplify the table of contents as follows.

Part I. Theory for the case of unary functions

Section 2. Basic notions and properties

Section 3. Galois correspondences

Section 4. Lattice structures of objects arising in Galois correspondences on a $T$-space $S$

Section 5. Topologies employed to construct Galois correspondences

Section 6. Generalized morphisms and isomorphisms from a $T_{1}$-space to a $T_{2}$-space

Section 7. Constructions of the generalized morphisms and isomorphisms

Part II. Theory for the case of multivariable total or partial functions

Section 8. Basic notions for the case of multivariable (total) functions

Section 9. Basic notions for the case of (multivariable) partial functions

Section 10. Basic properties and more notions

Section 11. Galois correspondences

Section 12. Lattice structures of objects arising in Galois correspondences on a $T$-space $S$

Section 13. Topologies employed to construct Galois correspondences

Section 14. Constructions of the generalized morphisms and isomorphisms

Part III. Solvability of equations

Section 15. A solvability of polynomial equations

Section 16. A solvability of homogeneous linear differential equations

Section 17. A possible strategy for equation solving

Part IV. Other topics and future research

Section 18. Dualities of operator semigroups

Section 19. Fixed sets and transitive actions of $\operatorname{End}_{T}(S)$ and $\operatorname{Aut}_{T}(S)$

Section 20. Two questions on Galois $T$-extensions and normal subgroups of Galois $T$-groups

Section 21. $\mathcal{F}$-transcendental elements and $\mathcal{F}$-transcendental subsets

Section 22. A generalized first isomorphism theorem

Section 23. On topological spaces

Section LABEL:App_to_dyn. On dynamical systems

Section LABEL:Other_topics. Other topics for future research

The contents of Part I are roughly described section by section as follows.

Let $T$ be a set of functions from a set $D$ to $D$. If the composite of any two elements in $T$ still lies in $T$, then we call $T$ an operator semigroup on $D$ (cf. Definition 2.1.1). Let $U\subseteq D$. Then we call $\bigcup\nolimits_{f\in T}{\operatorname{Im}(f{{|}_{U}})}$ the $T$-space generated by $U$, where $\operatorname{Im}(f{{|}_{U}})$ denotes the image of the restriction of $f$ to $U$ (cf. Definition 2.2.1). $T$-spaces represent mathematical objects whose structures are preserved under generalized morphisms. A critical notion in this theory is $T$-morphism, which is defined as a map $\sigma$ from a $T$-space $S$ to a $T$-space such that $\sigma$ commutes with every element of $T$; that is, $\forall a\in S$ and $f\in T$, $\sigma(f(a))=f(\sigma(a))$ (cf. Definition 2.3.1). From these simple definitions much follows. For example, any ring homomorphism between $B[u]$ and $B[v]$ with field $B$ fixed pointwisely can be characterized in terms of a $T$-morphism (cf. Proposition 2.3.2), and a map from a topological space to itself is continuous if and only if the map induces a $T$-morphism (cf. Proposition 2.3.5). In Subsection 2.4, we show that the inverse of any bijective $T$-morphism from a $T$-space $S_{1}$ to a $T$-space $S_{2}$ is a $T$-morphism from $S_{2}$ to $S_{1}$, and hence a $T$-isomorphism can be defined as a bijective $T$-morphism.

In Section 3, we establish the Galois correspondence between the Galois groups of a $T$-space $S$ and the fixed subsets of $S$ under the actions of the $T$-automorphisms of $S$ (cf. Corollary 3.2.6). Moreover, we shall find the Galois correspondence between the “Galois monoids” of a $T$-space $S$ and the fixed subsets of $S$ under the actions of the $T$-endomorphisms of $S$ (cf. Corollary 3.2.5).

To better understand these Galois correspondences, in Section 4, we study the lattice structures of those objects which arise in Corollaries 3.2.5 and 3.2.6. Results in Section 4 will be employed in Section 5.

In the fundamental theorem for infinite algebraic field extensions (resp. for differential field extensions), the Galois correspondences are characterized with Krull topology (resp. Zariski topology) (see e.g. [1, 3, 4, 5, 9]). In Section 5, we shall see when and how we can characterize Galois correspondences with topology.

Section 6 introduces the notions of $\theta$-morphism and $\theta$-isomorphism. As a generalization of the notion of $T$-morphism, a $\theta$-morphism (cf. Definitions 6.1.1 and 6.1.8) is from a ${{T}_{1}}$-space to a ${{T}_{2}}$-space, where both ${{T}_{1}}$ and ${{T}_{2}}$ are operator semigroups and $\theta\subseteq{{T}_{1}}\times{{T}_{2}}$. We shall find that some more familiar mathematical notions can be characterized by $\theta$-morphisms (cf. Propositions 6.1.2, 6.1.3 and 6.1.10). In Subsection 6.2, we show that the inverse of any bijective $\theta$-morphism from a $T_{1}$-space $S_{1}$ to a $T_{2}$-space $S_{2}$ is a $\theta^{-1}$-morphism from $S_{2}$ to $S_{1}$, and thus a $\theta$-isomorphism can be defined as a bijective $\theta$-morphism.

Main results in Section 7 are about constructions of $T$-morphisms and $\theta$-morphisms. Roughly speaking, for a $T$-morphism or $\theta$-morphism from a $T$-space $S$, Subsections 7.1, 7.2 and 7.3 discuss how to construct the morphism by a map from a set $U$ which generates $S$ (i.e. $S=\bigcup\nolimits_{f\in T}{\operatorname{Im}(f{{|}_{U}})}$). Moreover, in Subsection 7.4, we shall introduce a construction of $T$-morphisms in terms of topology.

The contents of Part II are described as follows.

To make our theory more general, in Section 8, we generalize the notion of operator semigroup to incorporate functions of more than one variable (cf. Definition 8.1.2). And correspondingly, the concepts of $T$-spaces, $T$-morphisms and $\theta$-morphisms are generalized (cf. Definitions 8.2.1, 8.3.1, and 8.5.4). Then we shall find that ring homomorphisms, module homomorphisms, and group homomorphisms can be characterized in terms of $T$-morphisms (cf. Propositions 8.3.4 to 8.3.7) or $\theta$-morphisms (cf. Propositions 8.5.7 to 8.5.11).

To further generalize our theory, in Section 9, we allow operators in $T$ to be partial functions (cf. Definition 9.1.4). And correspondingly, the notions of $T$-spaces, $T$-morphisms and $\theta$-morphisms are generalized (cf. Definitions 9.2.1, 9.3.1, and 9.5.5). Because operators in $T$ are now allowed to be partial functions, we can generate fields, differential fields, and categories as $T$-spaces in a natural way (cf. Examples 9.2.2 to 9.2.4). Then we shall find that ring homomorphisms between fields, differential ring homomorphisms between differential fields, and covariant functors between categories can be characterized in terms of $T$-morphisms (cf. Propositions 9.3.4 and 9.3.6) or $\theta$-morphisms (cf. Propositions 9.5.8 and 9.5.9).

For the case of (partial) functions of more than one variable, results obtained in Sections 2, 3, 4, 5 and 7 are generalized in Sections 10, 11, 12, 13 and 14, respectively. (Note that results obtained in Section 6 are generalized in Sections 8 and 9.)

Part III addresses the following issues.

A main goal of the classical Galois theory is to study the solvability by radicals of polynomial equations. In Section 15, however, we shall introduce another understanding of solvability of polynomial equations, which somehow corresponds to a composition series of the Galois group of the polynomial. And in Subsection 15.2 we shall explain why we introduce this notion of solvability. Roughly speaking, the central idea is that a separable polynomial $p(x)$ can be “decomposed” into $n$ polynomials ${{p}_{1}}(x),\cdots,{{p}_{n}}(x)$ such that the Galois group of each ${{p}_{i}}(x)$ is simple and any root of $p(x)$ can be expressed in terms of the roots of ${{p}_{1}}(x),\cdots,{{p}_{n}}(x)$ (cf. Corollary 15.2.5).

Analogously, in Section 16, we shall introduce a new interpretation of solvability of homogeneous linear differential equations, which somehow corresponds to a “composition Zariski-closed series” of the differential Galois group of the equation. In Subsection 16.3 we shall explain why we introduce this notion of solvability. As what we did in Section 15, our main approach is to “decompose” a homogeneous linear differential equation $L(Y)=0$ into $n$ homogeneous linear differential equations ${{L}_{1}}(Y)=0,\cdots,{{L}_{n}}(Y)=0$ so that any solution of $L(Y)=0$ can be expressed in terms of the solutions of ${{L}_{1}}(Y)=0,\cdots,{{L}_{n}}(Y)=0$ (cf. Corollary 16.3.6).

Section 17 generalizes some results in Sections 15 and 16, and it roughly describes in terms of our theory a possible strategy for equation solving.

In Part IV we give some more results which we think to be important or deserve deeper research.

Section 18 is about dualities of operator semigroups. Let $T$ be an operator semigroup and let $S$ be a $T$-space. We denote by $\operatorname{End}_{T}(S)$ (resp. $\operatorname{Aut}_{T}(S)$) the set of all $T$-endomorphisms of $S$ (resp. the set of all $T$-automorphisms of $S$). Then, roughly speaking, there is a duality between $\operatorname{End}_{T}(S)$ and the maximum operator semigroup which “accommodates” $\operatorname{End}_{T}(S)$ (i.e. $\forall\sigma\in\operatorname{End}_{T}(S)$, $\sigma$ would still be a $T$-endomorphism of $S$ if $T$ were extended to the maximum). Analogously, there is a duality between $\operatorname{Aut}_{T}(S)$ and the maximum operator semigroup which “accommodates” $\operatorname{Aut}_{T}(S)$.

In Section 19, we shall talk about fixed subsets of a $T$-space $S$ under the actions of $\operatorname{End}_{T}(S)$ and $\operatorname{Aut}_{T}(S)$. And we shall obtain some transitive properties of the actions of $\operatorname{End}_{T}(S)$ and $\operatorname{Aut}_{T}(S)$ on $S$, which are analogues of the well-known fact that the Galois group of an irreducible polynomial acts transitively on its roots.

Section 20 addresses an analogue of the issue which is normally the last part of the fundamental theorem of a Galois theory: the correspondence between the normal subgroups of the Galois group and the Galois extensions of the base field.

In Section 21, the algebraic notions of transcendental elements over a field and purely transcendental field extensions are generalized.

In Section 22, we shall obtain an isomorphism theorem which generalizes the first isomorphism theorems (for groups, rings, and modules).

In Section 23, we shall introduce for topological spaces some notions and properties related to our theory. And we shall employ Corollaries 3.2.5 and 3.2.6 to obtain Galois correspondences on topological spaces.

Let $(M,S,\Phi)$ be a dynamical system, where $M$ is a monoid, $S$ is the phase space and $\Phi$ is the evolution function. In Section LABEL:App_to_dyn, it is shown that $(M,S,\Phi)$ induces in a natural way an operator semigroup $T$ on $S$ such that $T$ contains the identity function and $S$ is a $T$-space. Conversely, for any operator semigroup $T$ which contains the identity function, any $T$-space $S$ induces a dynamical system $(T,S,\Phi)$ in a natural way. Hence we can apply our theory for operator semigroups to dynamical systems. In particular, we shall obtain Galois correspondences on dynamical systems.

Section LABEL:Other_topics gives some more suggestions on future research.

Part I. THEORY FOR THE CASE OF UNARY FUNCTIONS

In Part I of the article, we introduce the generalized morphisms where all functions involved are unary. The generalized morphisms preserve structures of mathematical objects which we call $T$-spaces, where $T$ stands for a semigroup of functions with composition of functions as the binary operation.