# Information entropy re-defined in a category theory context using
preradicals

Sebastian Pardo G Department of Bioengineering
University of California, San Diego, USA.  Center for Engineered Natural
Intelligence
University of California, San Diego, USA.  Gabriel A. Silva  Department of
Bioengineering
University of California, San Diego, USA.  Department of Neurosciences
University of California, San Diego, USA.

###### Abstract

Algebraically, entropy can be defined for abelian groups and their
endomorphisms, and was latter extended to consider objects in a Flow category
derived from abelian categories, such as $R\textit{-}Mod$ with $R$ a ring.
Preradicals are endofunctors which can be realized as compatible choice
assignments in the category where they are defined. Here we present a formal
definition of entropy for preradicals on $R$-Mod and show that the concept of
entropy for preradicals respects their order as a big lattice. Also, due to
the connection between modules and complete bounded modular lattices, we
provide a definition of entropy for lattice preradicals, and show that this
notion is equivalent, from a functorial perspective, to the one defined for
module preradicals.

Corresponding author: GS<EMAIL_ADDRESS>

Keywords— Category of flows, module preradicals, lattice preradicals, entropy.

Mathematics Subject Classification. 16D90, 37A35, 18A99, 06C99.

##  1 Introduction

This work is the second of two related papers that aim to use Category Theory
as a natural way to describe information flows in networks and relating these
ideas to notions of Shannon information theory. In [17] we showed that
preradicals naturally describe the flow of information. To that end, we used
quiver representation theory to model a directed graph as a diagram with
objects and morphisms within the category $R$-Mod, for $R$ a principal ideal
domain. Particularly, we showed that preradicals generalize the concept of
persistence in filtrations, whose underlying structure has the form of a
directed acyclic graph.

Broadly, a preradical is an endofunctor 111An endofunctor is a functor with
the same domain and codomain which acts as a subfunctor of the identity
functor. Yet, as described in [11], preradicals can be considered as
compatible choice assignments in the category where they are defined. Here,
this property motivated us to study the dynamics that preradicals induce on
the objects and their corresponding endomorphisms. To approach this, we will
rely on concepts of entropy defined on abelian groups and their endomorphisms,
and on concepts of entropy defined on abelian categories.

The notion of entropy was first introduced in the field of thermodynamics as a
measure of the system’s disorder. It was later used in quantum mechanics where
it was endowed with a deep mathematical formalism. A third perspective that
came later, known as Shannon entropy, measures the efficiency of a system in
transmitting information. When it comes to a mathematical subject, entropy was
first defined in topology in [1] by Adler, Konheim and McAndrew, for a
continuous self-map of a compact space. However, from an algebraic
perspective, Weiss settles in [19] the notion of _algebraic entropy_ for an
endomorphism of an abelian group. On the other hand, in [12] Peter provides a
slightly different definition of entropy for an automorphism of an abelian
group, which was later extended in [5] to consider endomorphisms of abelian
groups. One substantial difference between Peter’s and Weiss’s definitions is
that, in the former one considers a supremum over all finite subsets of a
group $G$ while in the latter one takes the supremum over all finite
subgroups.

Afterwards, algebraic entropy was generalized in [15] to consider not only
abelian groups (or $\mathbb{Z}$-modules), but modules over a unitary ring $R$.
In this case, one considers _invariants_ on $R$-Mod to define an algebraic
entropy: invariants are extended real valued functions that are invariant
under isomorphic objects. Specially, we focus on invariants $i$ with at least
two minimal requirements with which one can associate an algebraic entropy to
it. These two conditions comprise what we normally call as subadditive
invariant. In this work, we will define the notion of entropy for a preradical
in terms of an algebraic $i$-entropy, where $i$ represents an invariant
defined on $R$-Mod.

Lastly, due to the relation between lattice preradical and module preradicals
described in [11], we use functors to provide a notion of entropy for lattice
preradicals, when restricted to a non-full subcategory $\mathcal{SL_{M}}$ of
the category $\mathcal{L_{M}}$ of complete bounded modular lattices and linear
morphisms. This functorial connection show that the definition of entropy for
lattice preradicals is order-respectful, and further, it is equivalent to the
entropy defined for module preradicals.

The paper is organized as follows: in Section 2 we give some preliminary facts
about the structure of the category of flows $\textbf{Flow}_{\mathcal{C}}$ for
any category $\mathcal{C}$. We also give a brief introduction to preradicals
on $R$-Mod along with notation, terminology, and basic properties. Section 3
shows that any preradical defined on a category $\mathcal{C}$ induces a
preradical on the category of flows associated to $\mathcal{C}$. In
particular, we describe the $\alpha$ and $\omega$ preradicals in the category
of flows Flow${}_{R\textit{-}Mod}$ with respect to the category $R$-Mod. In
Section 4 we define entropy for preradicals on $R$-Mod and display some
examples. Right after, we describe the entropy for preradicals on the category
of flows Flow${}_{R\textit{-}Mod}$. Finally, we use some functorial properties
between the big lattice of lattice preradicals and the big lattice of module
preradicals to define entropy for lattice preradicals, and prove some of its
properties.

##  2 Preliminaries

###  2.1 The Category of Flows

Given a category $\mathcal{C}$, the identity functor on $\mathcal{C}$ defines
the special comma category $(\mathcal{C}\ \downarrow\ \mathcal{C})$. This
category, also known as the category of arrows, is usually denoted by
$\mathcal{C}^{2}$. The objects of the category $\mathcal{C}^{2}$ are triplets
$(C_{1},C_{2},\eta)$, where $C_{1},C_{2}$ are objects of $\mathcal{C}$ and
$\eta:C_{1}\longrightarrow C_{2}$ is a morphisms in $\mathcal{C}$. A morphism
between two objects $(C_{1},C_{2},\eta)$ and $(D_{1},D_{2},\psi)$ of
$\mathcal{C}^{2}$ is made of a pair $(f,g)$ where $f:C_{1}\longrightarrow
D_{1}$ and $g:C_{2}\longrightarrow D_{2}$ are morphisms in $\mathcal{C}$ such
that the diagram

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{D_{2}}$

commutes in $\mathcal{C}$. This arrow category is in fact isomorphic to the
functor category $\mathcal{C}^{I}$, where $I$ is the interval category
consisting of just two objects and only one nontrivial morphisms connecting
them $\\{0\longrightarrow 1\\}$.

We can impose two restrictions to the category $\mathcal{C}^{2}$ to obtain a
particular subcategory denoted as FlowC. The first restriction consists on
taking as triplets $(C_{1},C_{2},\eta)$ those that satisfies $C_{1}=C_{2}$.
This way, the objects of FlowC are objects in $\mathcal{C}$ equipped by
endomorphisms.

######  Definition 2.1.

A flow in $\mathcal{C}$ is a pair $(C,\varphi)$, where $C$ is an object in
$\mathcal{C}$ and $\varphi:C\longrightarrow C$ is an endomorphisms in
$\mathcal{C}$.

For the second restriction, we will take as morphisms those pairs $(f,g)$
satisfying $f=g$. Thus, a flow morphism between the flows $(C,\varphi)$ and
$(D,\psi)$ is a morphism $f:C\longrightarrow D$ in $\mathcal{C}$ such that the
diagram

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{D}$

commutes in $\mathcal{C}$. Therefore, FlowC is the category whose objects are
flows in $\mathcal{C}$ and whose morphisms are flow morphisms.

We notice that the above construction implies that the category FlowC will not
be a full subcategory of $\mathcal{C}^{2}$, although it can inherit some
properties of the category $\mathcal{C}$. For instance, as shown in [8], the
category FlowC is isomorphic to the functor category
$\mathcal{C}^{\mathbb{N}}$, where $\mathbb{N}$ is considered as a one object
category. Thus, the category FlowC will be abelian when so is $\mathcal{C}$.

There is a ’forgetful’ functor
$U:\textbf{Flow}_{\mathcal{C}}\longrightarrow{\mathcal{C}}$ given by the
assignment $(C,\eta)\longmapsto C$ on objects and the assignment $f\longmapsto
f$ on morphisms. This functor $U$ ignores the dynamic induced by the
endomorphisms, since:

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\overset{U}{\longmapsto}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{D}$$\textstyle{D.}$

There is also an embedding functor
$E:\mathcal{C}\longrightarrow\textbf{Flow}_{\mathcal{C}}$ given by the
assignment $C\longmapsto(C,Id_{C})$ on objects and the assignment
$f\longmapsto f$ on morphisms, that is,

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\overset{E}{\longmapsto}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$$\textstyle{D.}$

The next result will be useful for characterizing the subobjects in FlowC;
especially when inducing preradicals from $R$-Mod to
Flow${}_{R\textit{-}Mod}$.

######  Lemma 2.2.

A flow morphism $f$ in FlowC is a monomorphisms if $U(f)=f$ is a monomorphism
in the category $\mathcal{C}$.

######  Proof.

For the sake of the proof, we will denote by $\bar{f}$ the morphism in FlowC
so that $U(\bar{f})=f$. Let $f$ be a monomorphism in $\mathcal{C}$ and let
$\bar{f}$ be the flow morphism such that $U(\bar{f})=f$. Suppose that
$\bar{f}:(C,\eta)\longrightarrow(D,\mu)$. If
$\bar{g}_{1},\bar{g}_{2}:(B,\xi)\longrightarrow(C,\eta)$ are two flow
morphisms such that $\bar{f}\circ\bar{g_{1}}=\bar{f}\circ\bar{g_{2}}$, then we
have the commutative diagrams

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{g_{1}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{g_{2}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{=}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{D}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{D}$

in $\mathcal{C}$ from which it follows that $(f\circ g_{1}\circ\xi)=(f\circ
g_{2}\circ\xi)$. Since $f$ is a monomorphism in $\mathcal{C}$, the last
equation simplifies to $g_{1}\circ\xi=g_{2}\circ\xi$. Now, as
$\bar{g}_{1},\bar{g}_{2}:(B,\xi)\longrightarrow(C,\eta)$ are both flow
morphisms, the condition $g_{1}\circ\xi=g_{2}\circ\xi$ implies that the
following diagrams define the same transformation:

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{g_{1}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{=}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{g_{2}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C,}$

Therefore, $\bar{g_{1}}=\bar{g_{2}}$. ∎

Formally, a subobject of the object $(C,\eta)$ in FlowC, is an equivalence
class of monomorphisms into $(C,\eta)$. Thus, we can write a subobject of
$(C,\eta)$ as a pair $((X,\alpha),f)$ where
$(X,\alpha)\overset{f}{\longrightarrow}(C,\eta)$ is a "representative"
monomorphism in FlowC. By Lemma 2.2, it suffices to consider a monomorphism
$X\overset{f}{\longrightarrow}C$ in the category $\mathcal{C}$. Hence, for the
case $\mathcal{C}=R\textit{-}Mod$, we might choose, as a representative of the
equivalence class of $(X,f)$, the pair $(f(X),\iota)$ where
$\iota:f(X)\hookrightarrow C$ denotes the inclusion map. Observe that, since
$f$ is a monomorphism in the category $R$-Mod, then $f$ is an injective
morphism and thus $X\cong f(X)$. Further, by the commutativity of the diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C,}$

induced by the flow morphism $(X,\alpha)\overset{f}{\longrightarrow}(C,\eta)$,
we have that $\eta(f(X))=f(\alpha(X))\subseteq f(X)$; in other words, the
endomorphism $\eta$ restricts to $f(X)$. Therefore, the diagram

$\textstyle{f(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{|}}$$\textstyle{f(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C}$

commutes in $R$-Mod, showing that the pair $((f(X),\eta_{|}),\iota)$ is also a
representative subobject of $(C,\eta)$ for the equivalence class containing
$((X,\alpha),f)$.

######  Remark 1.

The forgetful functor $U\colon\textbf{Flow}_{R\textit{-}Mod}\longrightarrow
R\textit{-}Mod$ maps each subobject in Flow${}_{R\textit{-}Mod}$ to a
subobject in $R$-Mod. In other words, if $((f(X),\eta_{|}),\iota)$ is a
representative subobject of $(C,\eta)$ for the class of equivalence containing
the flow monomorphism $((X,\alpha),f)$, then $((f(X),\iota)$ is a
representative subobject of $C$, for the class of equivalence that contains
the monomorphism $X\overset{f}{\longrightarrow}C$ in $R$-Mod.

###  2.2 Preradicals

Broadly, a preradical on $\mathcal{C}$ can be defined in terms of the
2-category of all endofunctors222An endofunctor is a functor
$T:\mathcal{C}\longrightarrow\mathcal{C}$ whose domain is equal to its
codomain of $\mathcal{C}$. This 2-category coincides with the functor category
$\mathcal{C}^{\mathcal{C}}$, whose objects are all endofunctors
$\tau:\mathcal{C}\longrightarrow\mathcal{C}$, and whose morphisms are natural
transformations, usually denoted by $\tau_{1}\Longrightarrow\tau_{2}$. Thus, a
preradical $\sigma$ on $\mathcal{C}$ is the same as a co-pointed endofunctor
of $\mathcal{C}^{\mathcal{C}}$; this is, an endofunctor
$\sigma:\mathcal{C}\longrightarrow\mathcal{C}$ for which there is natural
transformation $\sigma\Longrightarrow Id_{\mathcal{C}}$, and where
$Id_{\mathcal{C}}$ denotes the identity functor. Observe that
$\sigma\Longrightarrow Id_{\mathcal{C}}$ means that one has a family of
morphisms $\\{\eta_{C}\\}_{C\in\mathcal{C}}$ -indexed by the objects of
$\mathcal{C}$\- such that, for each $C,D\in\mathcal{C}$ and each morphism
$C\overset{f}{\longrightarrow}D$ in $\mathcal{C}$, the following diagram
commutes

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma(f)}$$\scriptstyle{\eta_{C}}$$\textstyle{\sigma(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\eta_{D}}$

However, according to [13] one can also interpret a preradical on
$\mathcal{C}$ as a subfunctor of the identity functor $Id_{\mathcal{C}}$. In
other words, a preradical is a functor that assign to each object $A$ in
$\mathcal{C}$, a subobject $\sigma(A)$ in $\mathcal{C}$ such that, for any
morphism $A\overset{f}{\longrightarrow}B$ in $\mathcal{C}$, the restriction
and corestriction of $f$ to the respective subobjects defines a morphisms
$\sigma(f)\colon\sigma(A)\overset{f_{|}}{\longrightarrow}\sigma(B)$ in
$\mathcal{C}$, which makes the following diagram to commute

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$$\textstyle{\sigma(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma(f)}$$\scriptstyle{\iota_{A}}$$\textstyle{\sigma(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\iota_{B}}$

Notice that in the above diagram, $\iota_{A}$ and $\iota_{B}$ denote the
respective inclusion maps. In this way, as the authors describe in [11],
preradicals behave as compatible choice assignments, since these assign to
each object of the category $\mathcal{C}$ a subobject, in such a way that the
choices are compatible with respect to the morphism in $\mathcal{C}$. In fact,
this property is what makes preradicals useful for describing the flow of
information (see [17]).

In this work we will consider preradicals on a category $\mathcal{C}$ as
subfunctors of the identity functor $Id_{\mathcal{C}}$. Specially, we will
start considering preradicals on $R$-Mod for a unitary ring $R$. Here, objects
of $R$-Mod corresponds to $R$-modules and subobjects corresponds to
$R$-submodules. For a complete introduction to preradicals and its properties
on $R$-Mod see [7], [13] and [14].

######  Definition 2.3.

A preradical $\sigma$ on the category $R$-Mod is a functor $\sigma\colon
R\textit{-Mod}\longrightarrow R\textit{-Mod}$ that assigns to each module
$M\in R\textit{-Mod}$, a submodule $\sigma(M)$ such that for each morphism
$f\colon M\longrightarrow M^{\prime}$ in $R\textit{-Mod}$, we have the
commutative diagram

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M^{\prime}}$$\textstyle{\sigma(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma(f)}$$\scriptstyle{\iota}$$\textstyle{\sigma(M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\iota}$

Here, $\sigma(f)$ is the restriction and corestriction of $f$ to $\sigma(M)$
and $\sigma(M^{\prime})$ respectively, that is,

$\sigma(f)\colon=f\mid_{\sigma(M)}:\sigma(M)\longrightarrow\sigma(M^{\prime})$.

Also, $\iota$ represents the inclusion map.

######  Example.

Consider the category $\mathbb{Z}$-Mod of all $\mathbb{Z}$-modules. This
category is isomorphic to the category $Ab$ of all abelian groups. Now, given
$M\in\mathbb{Z}$-Mod, one can define

$\sigma(M)=\\{x\in M\mid 3x=0\\}$.

Note that $\sigma(M)$ is a submodule of $M$. Also, for any morphism
$f:M\longrightarrow M^{\prime}$ in $\mathbb{Z}$-Mod and for any $y\in M$, one
has that $f(3y)=3f(y)$. Thus, if $x\in\sigma(M)$ then $3f(x)=f(3x)=0$, which
implies that $f(x)\in\sigma(M^{\prime})$. Hence,
$f(\sigma(M))\subseteq\sigma(M^{\prime})$ which in turns implies that

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M^{\prime}}$$\textstyle{\sigma(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{|}}$$\scriptstyle{\iota}$$\textstyle{\sigma(M^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

commutes in $\mathbb{Z}\texttt{-}Mod$.

In the category $R$-Mod one can define four principal operations between
preradicals: if $M\in R$-Mod and $\sigma,\tau$ are two preradicals, then

  * i)

$(\sigma\wedge\tau)(M)=\sigma(M)\cap\sigma(M)$;

  * ii)

$(\sigma\vee\tau)(M)=\sigma(M)+\tau(M)$;

  * iii)

$(\sigma\cdot\tau)(M)=\sigma(\tau(M))$;

  * (iv)

$(\sigma:\tau)(M)$ is the submodule of $M$ such that

$(\sigma:\tau)(M)/\sigma(M)=\tau(M/\sigma(M))$.

The above operations are called the meet, the join, the product and the
coproduct respectively. We note that the product $(\sigma\cdot\tau)$
corresponds to the composition between functors $\tau$ and $\sigma$; while the
coproduct $(\sigma:\tau)$ involves taking a quotient module (induced by
$\sigma$), apply functor $\tau$, and then use the Correspondence Theorem for
Modules to obtain the submodule $(\sigma:\tau)(M)$ of $M$. As regards the join
and the meet operations, these can be extended to consider an arbitrary family
of preradicals. In other words, for any family of preradicals
$\\{\tau_{\alpha}\\}_{\alpha\in I}$ for $M$ an $R$-module,

  * i)

$\big{(}\bigvee_{\alpha\in I}\ \tau_{\alpha}\big{)}(M)=\Sigma_{\alpha\in I}\
\tau_{\alpha}(M)$,

  * ii)

$\big{(}\bigwedge_{\alpha\in I}\ \tau_{\alpha}\big{)}(M)=\cap_{\alpha\in I}\
\tau_{\alpha}(M)$.

We denote by $R$-pr the collection of all preradicals on $R$-Mod. There is a
natural partial ordering in $R$-pr given by $\sigma\leq\tau$ if and only if
$\sigma(M)\leq\tau(M)$ for each $M\in R$-Mod. This partial ordering together
with the meet and join operations make of $R$-pr a big lattice. 333A big
lattice is a class (not necessarily a set) having joins and meets for
arbitrary families (indexed by a class) of elements

We close this section with the description of two particular preradicals on
$R$-Mod. Let $M$ be an $R$-module and let $N$ be a submodule of $M$. On the
one hand, we define the preradical $\alpha_{N}^{M}$ which evaluated in a $K\in
R$-Mod is

$\alpha_{N}^{M}(K)=\sum\\{f(N)\mid f\colon M\longrightarrow K\\}$.

Observe that $f(N)$ is a submodule of $K$ for each
$M\overset{f}{\longrightarrow}K$. On the other hand, we define the preradical
$\omega_{N}^{M}$ which evaluated in $K\in R$-Mod is

$\omega_{N}^{M}(K)=\cap\\{f^{-1}(N)\mid f\colon K\longrightarrow M\\}$.

Notice here that $f^{-1}(N)$ is a submodule of $K$, for every $f\colon
K\longrightarrow M$.

##  3 Preradicals in the Category of Flows

In this section we will show that each preradical on $R$-Mod induces a
preradical on the category of flows Flow${}_{R\textit{-}Mod}$ and viceversa.
We start by noticing that, given a flow $C\overset{\eta}{\longrightarrow}C$
and a preradical $\sigma$ on $R$-Mod, the diagram

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta|_{\sigma(C)}}$$\scriptstyle{\iota}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

commutes in $R$-Mod. This in turn implies that
$\eta\mid_{\sigma(C)}:\sigma(C)\longrightarrow\sigma(C)$ is an endomorphism in
$R$-Mod, and thus the pair $(\sigma(C),\eta\mid_{\sigma(C)})$ is an object in
Flow${}_{R\textit{-}Mod}$. Furthermore, we have that
$\big{(}\sigma(C),\eta\mid_{\sigma(C)}\big{)}$ together with the inclusion map
$\iota$ defines a subobject of the flow object $(C,\eta)$ in
Flow${}_{R\textit{-}Mod}$.

######  Proposition 3.1.

Let $\sigma$ be a preradical on $R$-Mod, and let Flow${}_{R\textit{-}Mod}$ be
the category of flows associated to $R$-Mod. Then $\sigma$ induces a
preradical $\bar{\sigma}$ on Flow${}_{R\textit{-}Mod}$ in the following way:
$\bar{\sigma}$ assigns to each object $(C,\eta)$ the subobject
$(\sigma(C),\eta_{|_{\sigma(C)}})$, and assigns to each flow morphism
$(C,\eta)\overset{f}{\longrightarrow}(D,\mu)$ the flow morphism $\sigma(f)$
given by

$\sigma(f)\colon(\sigma(C),\eta_{|_{\sigma(C)}})\longrightarrow(\sigma(D),\mu_{|_{\sigma(D)}})$

where $\sigma(f)=f_{|_{\sigma(C)}}$.

######  Proof.

Let $\sigma$ be a preradical on $R$-Mod, and let $(C,\eta)$ and $(D,\mu)$ be
objects in Flow${}_{R\textit{-}Mod}$ with
$(C,\eta)\overset{f}{\longrightarrow}(D,\mu)$ a flow morphism. For the sake of
this prove, we will denote $(\sigma(C),\eta_{|_{\sigma(C)}})$ by
$(\sigma(C),\eta_{|})$ when the context allows no confusion. Hence, we will
show that

$\textstyle{(C,\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(D,\mu)}$$\textstyle{(\sigma(C),\eta_{|})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{\sigma(C)}}$$\scriptstyle{\iota}$$\textstyle{(\sigma(D),\mu_{|})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

is a commutative diagram in FlowC.

As we noticed previously, as $\eta:C\longrightarrow C$ and
$\mu:D\longrightarrow D$ are both endomorphisms in $R$-Mod, then we have the
following commutative diagrams

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta|_{\sigma(C)}}$$\scriptstyle{\iota}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$
and
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{D}$$\textstyle{\sigma(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu|_{\sigma(D)}}$$\scriptstyle{\iota}$$\textstyle{\sigma(D).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

From each of these diagrams we infer that $(\sigma(C),\eta\mid_{\sigma(C)})$
and $(\sigma(D),\mu|_{\sigma(D)})$ are subobjects of $(C,\eta)$ and $(D,\mu)$
respectively. Thus, in order to show that $\sigma$ induces a preradical in
FlowC, it suffices to prove that $f|_{\sigma(C)}$ is a flow morphisms from
$(\sigma(C),\eta\mid_{\sigma(C)})$ to $(\sigma(D),\mu|_{\sigma(D)})$.

On the one hand, since $\sigma$ is a preradical on $R$-Mod and
$f:C\longrightarrow D$ is a morphism in $R$-Mod, then
$f|_{\sigma(C)}\colon\sigma(C)\longrightarrow\sigma(D)$ is also a morphism in
$R$-Mod. On the other hand, for the flow morphisms
$f:(C,\eta)\longrightarrow(D,\mu)$ we have the commutative diagram

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{D}$

from which we obtain that $f\circ\eta=\mu\circ f$. Therefore, when restricted
to $\sigma(C)$ we get the following commutative diagram

$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta|_{\sigma(C)}}$$\scriptstyle{f|_{\sigma(C)}}$$\textstyle{\sigma(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{\sigma(C)}}$$\textstyle{\sigma(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu|_{\sigma(D)}}$$\textstyle{\sigma(D),}$

which shows that $f|_{\sigma(C)}$ is a flow morphism between the objects
$(\sigma(C),\eta\mid_{\sigma(C)})$ and $(\sigma(D),\mu|_{\sigma(D)})$ in
Flow${}_{R\textit{-}Mod}$. ∎

We will now describe the induced preradicals on Flow${}_{R\textit{-}Mod}$ that
correspond to the $\alpha$ and $\omega$ preradicals on $R$-Mod.

######  Proposition 3.2.

Let $(M,\eta)$ be an object in Flow${}_{R\textit{-}Mod}$ and let
$(N,\eta_{|_{N}})$ be a subobject of $(M,\eta)$. Then, for any $(K,\mu)$ in
Flow${}_{R\textit{-}Mod}$

$\bar{\alpha}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K,\mu)=\big{(}\underset{f}{\sum}\
f(N),\mu_{|}\big{)}$

defines a preradical on Flow${}_{R\textit{-}Mod}$.
Here the sum is taken over all flow morphisms
$(M,\eta)\overset{f}{\longrightarrow}(K,\mu)$.

######  Proof.

Let $(M,\eta)$ be an object in Flow${}_{R\textit{-}Mod}$ and let
$(N,\eta_{|_{N}})$ be a subobject of $(M,\eta)$. Before we start with the
proof, we will notice that $\big{(}\underset{f}{\sum}\ f(N),\mu_{|}\big{)}$
defines a subobject of $(K,\mu)$. For any flow morphism
$f\colon(M,\eta)\longrightarrow(K,\mu)$ we have a commutative diagram

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{K}$

in $R$-Mod, from which it follows that

$\mu(f(N))=(\mu\circ f)(N)=(f\circ\eta)(N)\subseteq f(N)$.

Considering the above, we have a commutative diagram

$\textstyle{f(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{|}}$$\scriptstyle{\iota}$$\textstyle{f(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{K,}$

which implies that $(f(N),\mu_{|_{f(N)}})$ is a subobject of $(K,\mu)$. When
considering all flow morphisms $f\colon(M,\eta)\longrightarrow(K,\mu)$, the
latter argument can also be apply to $\underset{f}{\sum}\ f(N)$, as one also
has that $\mu\big{(}\underset{f}{\sum}\
f(N)\big{)}\subseteq\underset{f}{\sum}\ f(N)$. Therefore, we can say that
$\big{(}\underset{f}{\sum}\ f(N),\mu_{|_{\Sigma}}\big{)}$ is a subobject of
$(K,\mu)$ in Flow${}_{R\textit{-}Mod}$.

Now, let $(K,\mu)\overset{h}{\longrightarrow}(K^{\prime},\mu^{\prime})$ be a
flow morphism. On the one hand, observe that for any flow morphisms
$f\colon(M,\eta)\longrightarrow(K,\mu)$, the composition $(h\circ
f)\colon(M,\eta)\longrightarrow(K^{\prime},\mu^{\prime})$ is also a flow
morphisms. Hence,

$h\Big{(}\Big{\\{}\underset{f}{\sum}\ f(N)\mid f\colon M\longrightarrow
K\mbox{ in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}\Big{)}$
$=\Big{\\{}\underset{f}{\sum}\ (h\circ f)(N)\mid f\colon M\longrightarrow
K\mbox{ in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}$
$\subseteq\Big{\\{}\underset{f^{\prime}}{\sum}\ f^{\prime}(N)\mid
f^{\prime}\colon M\longrightarrow K^{\prime}\mbox{ in
}\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}.$

On the other hand, since $h$ is a flow morphisms, we have the commutative
diagram

$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{h}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}}$$\textstyle{K^{\prime},}$

which evaluated in $\underset{f}{\sum}\ f(N)$ induces the commutative diagram

$\textstyle{\underset{f}{\sum}\
f(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{|}}$$\scriptstyle{h_{|}}$$\textstyle{\underset{f}{\sum}\
f(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{|}}$$\textstyle{\underset{f^{\prime}}{\sum}\
f^{\prime}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}_{|}}$$\textstyle{\underset{f^{\prime}}{\sum}\
f^{\prime}(N)}$

in $R$-Mod. This shows that $h_{|}:\Big{(}\underset{f}{\sum}\
f(N),\mu_{|}\Big{)}\longrightarrow\Big{(}\underset{f}{\sum}\
f^{\prime}(N),\mu^{\prime}_{|}\Big{)}$ is a flow morphism which makes

$\textstyle{(K,\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(K^{\prime},\mu^{\prime})}$$\textstyle{\bar{\alpha}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K,\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{h_{|}}$$\textstyle{\bar{\alpha}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K^{\prime},\mu^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

a commutative diagram in Flow${}_{R\textit{-}Mod}$. Therefore,
$\bar{\alpha}_{(N,\eta_{|_{N}})}^{(M,\eta)}$ is a preradical on
Flow${}_{R\textit{-}Mod}$. ∎

######  Proposition 3.3.

Let $(M,\eta)$ be an object in Flow${}_{R\textit{-}Mod}$ and
$(N,\eta_{|_{N}})$ a subobject of $(M,\eta)$. Then, for any $(K,\mu)$ in
Flow${}_{R\textit{-}Mod}$

$\bar{\omega}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K,\mu)=\big{(}\underset{f}{\cap}\
f^{-1}(N),\mu_{|}\big{)}$

defines a preradical on Flow${}_{R\textit{-}Mod}$.
Here the intersection is taken over all flow morphisms
$f\colon(K,\mu)\longrightarrow(M,\eta)$.

######  Proof.

Let $(M,\eta)$ be an object in Flow${}_{R\textit{-}Mod}$ and
$(N,\eta_{|_{N}})$ a subobject of $(M,\eta)$. We will first notice that, for
each flow morphism $f\colon(K,\mu)\longrightarrow(M,\eta)$ the pair
$\big{(}f^{-1}(N),\mu_{|}\big{)}$ defines a subobject of $(K,\mu)$. For any
flow morphism $f\colon(K,\mu)\longrightarrow(M,\eta)$, we get a commutative
diagram in $R$-Mod of the form

$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{f}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{M.}$

If we take into consideration the submodule $f^{-1}(N)\subseteq K$, then we
have that

$\displaystyle\begin{split}f\big{(}\mu(f^{-1}(N))\big{)}=(f\circ\mu)(f^{-1}(N))=(\eta\circ
f)(f^{-1}(N))\\\ =\eta\big{(}f(f^{-1}(N))\big{)}\subseteq\eta(N)\subseteq
N,\end{split}$ (1)

where the last subset relation follows from the fact that $(N,\eta_{|_{N}})$
is an object in Flow${}_{R\textit{-}Mod}$. Thus, equation (1) implies that
$\mu(f^{-1}(N))\subseteq f^{-1}(N)$, and so

$\textstyle{f^{-1}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{|}}$$\scriptstyle{\iota}$$\textstyle{f^{-1}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{K}$

is a commutative diagram in $R$-Mod. This shows that $(f^{-1}(N),\mu_{|})$
defines a subobject of $(K,\mu)$. Now, when considering all flow morphisms
$f\colon(K,\mu)\longrightarrow(M,\eta)$, for the submodule
$\underset{f}{\cap}\ f^{-1}(N)$ of $K$ we also have that
$\mu\big{(}\underset{f}{\cap}\ f^{-1}(N)\big{)}\subseteq\underset{f}{\cap}\
f^{-1}(N)$. Hence, we can say that $\big{(}\underset{f}{\cap}\
f^{-1}(N),\mu_{|}\big{)}$ defines a subobject of $(K,\mu)$.

Let $(K,\mu)\overset{h}{\longrightarrow}(K^{\prime},\mu^{\prime})$ be a flow
morphism. On the one hand, observe that for any flow morphism
$f^{\prime}:(K^{\prime},\mu^{\prime})\longrightarrow(M,\eta)$ the composition
$(f^{\prime}\circ h):(K,\mu)\longrightarrow(M,\eta)$ is also a flow morphisms.
With this in mind, we have that

$\Big{\\{}\underset{f}{\cap}\
f^{-1}(N)\mid(K,\mu)\overset{f}{\longrightarrow}(M,\eta)\mbox{ in
}\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}$

$\subseteq\Big{\\{}\underset{f^{\prime}}{\cap}\ (f^{\prime}\circ
h)^{-1}(N)\mid(K^{\prime},\mu^{\prime})\overset{f^{\prime}}{\longrightarrow}(M,\eta)\mbox{
in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}$

$=\Big{\\{}\underset{f^{\prime}}{\cap}\
h^{-1}(f^{\prime-1}(N))\mid(K^{\prime},\mu^{\prime})\overset{f^{\prime}}{\longrightarrow}(M,\eta)\mbox{
in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}$

$=h^{-1}\Big{(}\Big{\\{}{\underset{f^{\prime}}{\cap}\
f^{\prime-1}(N)|(K^{\prime},\mu^{\prime})\overset{f^{\prime}}{\longrightarrow}(M,\eta)\mbox{
in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}\Big{)}}$,

from which we deduce that

$h\Big{(}\Big{\\{}\underset{f}{\cap}\
f^{-1}(N)\mid(K,\mu)\overset{f}{\longrightarrow}(M,\eta)\mbox{ in
}\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}\Big{)}$
$\subseteq\Big{\\{}{\underset{f^{\prime}\ \ \ \ \ \ \ \ \ \ }{\cap\
f^{\prime-1}(N)}|(K^{\prime},\mu^{\prime})\overset{f^{\prime}}{\longrightarrow}(M,\eta)\mbox{
in }\textbf{Flow}_{R\textit{-}Mod}\Big{\\}}}$.

On the other hand, since $h$ is a flow morphism, we have the commutative
diagram

$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{h}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}}$$\textstyle{K^{\prime}}$

in $R$-Mod. This diagram, when restricted to $\underset{f}{\cap}\ f^{-1}(N)$,
induces the commutative diagram

$\textstyle{\underset{f}{\cap}\
f^{-1}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{|}}$$\scriptstyle{h_{|}}$$\textstyle{\underset{f}{\cap}\
f^{-1}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{|}}$$\textstyle{\underset{f^{\prime}}{\cap}\
f^{\prime-1}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}_{|}}$$\textstyle{\underset{f^{\prime}}{\cap}\
f^{\prime-1}(N).}$

The latter shows that $h_{|}:\big{(}\underset{f}{\cap}\
f^{-1}(N),\mu_{|}\big{)}\longrightarrow\big{(}\underset{f^{\prime}}{\cap}\
f^{\prime-1}(N),\mu^{\prime}_{|}\big{)}$ is a flow morphism which makes

$\textstyle{(K,\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(K^{\prime},\mu^{\prime})}$$\textstyle{\bar{\omega}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K,\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{h_{|}}$$\textstyle{\bar{\omega}_{(N,\eta_{|_{N}})}^{(M,\eta)}(K^{\prime},\mu^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

a commutative diagram in Flow${}_{R\textit{-}Mod}$. Therefore,
$\bar{\omega}_{(N,\eta_{|_{N}})}^{(M,\eta)}$ is a preradical on
Flow${}_{R\textit{-}Mod}$. ∎

We will denote by $\textbf{Flow}_{R}\textit{-}pr$ the collection of all
preradicals on Flow${}_{R\textit{-}Mod}$. As in the case of $R\textit{-}pr$,
we can define a partial order on $\textbf{Flow}_{R}$-pr as follows: given
$\bar{\sigma},\bar{\tau}\in\textbf{Flow}_{R}\textit{-}pr$, we say that
$\bar{\sigma}\preceq\bar{\tau}$ if and only if $\bar{\sigma}(M,\eta)$ is a
subobject of $\bar{\tau}(M,\eta)$, for each
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$. Observe that by Remark 1, the
latter condition amounts to saying that
$U\big{(}\overline{\sigma}(M,\eta)\big{)}$ is a subobject of
$U\big{(}\overline{\tau}(M,\eta)\big{)}$ in $R$-Mod. Thus, in this case we
might assume that $U\big{(}\overline{\sigma}(M,\eta)\big{)}$ is a submodule of
$U\big{(}\overline{\tau}(M,\eta)\big{)}$.
As we next see, the assignment defined on Proposition 3.1 is injective and
order preserving.

######  Proposition 3.4.

Let $\phi:R\textit{-}pr\longrightarrow\textbf{Flow}_{R}\textit{-}pr$ be the
assignment defined by $\sigma\longmapsto\overline{\sigma}$ as in Proposition
3.1. Then $\phi$ is order preserving and an object-injective assignment.

######  Proof.

Let $\sigma,\tau\in R\textit{-}pr$ with $\sigma\leq\tau$, and let
$\phi(\sigma)=\overline{\sigma}$ and $\phi(\tau)=\overline{\tau}$ be their
respective preradicals on $\textbf{Flow}_{R}\textit{-}pr$. As $\sigma\leq\tau$
in $R\textit{-}pr$, then $\sigma(M)\leq\tau(M)$ for every $R$-module $M$.
Moreover, if for any endomorphism $\eta\colon M\longrightarrow M$ we denote by
$\overline{\sigma}(M,\eta)=(\sigma(M),\eta_{|})$ and
$\overline{\tau}(M,\eta)=(\tau(M),\eta_{|})$, then the commutative diagram

$\textstyle{\sigma(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{|}}$$\scriptstyle{\iota}$$\textstyle{\sigma(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\tau(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{|}}$$\textstyle{\tau(M)}$
(2)

in $R\textit{-}Mod$ implies that $(\tau(M),\eta_{|})$ is a subobject of
$(\sigma(M),\eta_{|})$ in $\textbf{Flow}_{R}\textit{-}pr$. Thus,
$\overline{\sigma}\preceq\overline{\tau}$.

For the second statement, suppose that $\sigma,\tau\in R\textit{-}pr$ are such
that

$\overline{\sigma}=\phi(\sigma)=\phi(\tau)=\overline{\tau}$.

Then for any object $(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$ we have that

$(\sigma(M),\eta_{|})=\overline{\sigma}(M,\eta)=\overline{\tau}(M,\eta)=(\tau(M),\eta_{|})$,

which in turn implies that $\sigma(M)=\tau(M)$. Since this happens for any
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$, we can infer that
$\sigma(M)=\tau(M)$ for all $M\in R\textit{-}Mod$. Therefore, $\sigma=\tau$ in
$R\textit{-}pr$, and thus, $\phi$ is an object-injective assignment. ∎

Next we show that every preradical on $\textbf{Flow}_{R}\textit{-}pr$ induces
a preradical in $R\textit{-}pr$. This induced preradical is obtained via the
forgetful functor $U$ together with the embedding functor $E$. Before we begin
with the proof, note that for any $M\in R$-Mod one can consider the object
$E(M)=(M,Id_{M})$ in Flow${}_{R\textit{-}Mod}$, and then evaluate any
preradical $\overline{\sigma}\in\textbf{Flow}_{R}\textit{-}pr$ on
$(M,Id_{M})$. This way we will get a subobject
$\overline{\sigma}(M,Id_{M})=(M^{\prime},Id_{M^{\prime}})$ of $(M,Id_{M})$.
Also observe that, by the way subobjects are defined in
Flow${}_{R\textit{-}Mod}$, we can assume that
$M^{\prime}\overset{\iota}{\hookrightarrow}M$, with which the following
diagram

$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id_{M^{\prime}}}$$\scriptstyle{\iota}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id_{M}}$$\textstyle{M}$
(3)

commutes in $R$-Mod.

######  Proposition 3.5.

Each preradical $\bar{\sigma}$ on Flow${}_{R\textit{-}Mod}$ induces a
preradical $\sigma$ on $R$-Mod. This preradical is given by
$\sigma:=(U\circ\overline{\sigma}\circ E)$ and satisfies that, for each $M\in
R$-Mod,

$\sigma(M)=M^{\prime}$ where $M^{\prime}$ is such that
$\bar{\sigma}(M,Id_{M})=(M^{\prime},Id_{M^{\prime}})$.

######  Proof.

Let $\bar{\sigma}$ be a preradical on Flow${}_{R\textit{-}Mod}$ and let us
take $M,L\in R$-Mod with $g\colon M\longrightarrow L$ an $R$-morphism. For the
objects $M$ and $L$ in $R$-Mod along with the morphism $g\colon
M\longrightarrow L$, their image under the embedding functor
$E:R\textit{-}Mod\longrightarrow\textbf{Flow}_{R\textit{-}Mod}$ is given by
the objects $(M,Id_{M})$ and $(L,Id_{L})$ together with the flow morphism
$g\colon(M,Id_{M})\longrightarrow(L,Id_{L})$. Thus, when applying the
preradical $\bar{\sigma}\in\textbf{Flow}_{R}\textit{-}pr$, we obtain the
commutative diagram

$\textstyle{(M,Id_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(L,Id_{L})}$$\textstyle{(M^{\prime},Id_{M^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{|}}$$\scriptstyle{\iota}$$\textstyle{(L^{\prime},Id_{L^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$
(*)

in Flow${}_{R\textit{-}Mod}$. Here
$\bar{\sigma}(M,Id_{M})=(M^{\prime},Id_{M^{\prime}})$ and
$\bar{\sigma}(L,Id_{L})=(L^{\prime},Id_{L^{\prime}})$. Also, observe that
$(M^{\prime},Id_{M^{\prime}})\overset{g_{|}}{\longrightarrow}(L^{\prime},Id_{L^{\prime}})$
is a flow morphism, which implies that $g_{|}:M^{\prime}\longrightarrow
L^{\prime}$ is a morphisms in $R$-Mod that makes

$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id_{M^{\prime}}}$$\scriptstyle{g_{|}}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{|}}$$\textstyle{L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id_{L^{\prime}}}$$\textstyle{L^{\prime}}$

a commutative diagram. Finally, as all functors preserve commutative diagram,
if we apply functor $U$ to diagram (* ‣ Proof) we obtain the commutative
diagram in $R$-Mod:

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{L}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{|}}$$\scriptstyle{\iota}$$\textstyle{L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\iota}$

Therefore, $\sigma\colon=(U\circ\overline{\sigma}\circ E)$ defined on objects
as $\sigma(M)=M^{\prime}$ and on morphisms as $\sigma(g)=g_{|}$, defines a
preradical on $R$-Mod. ∎

######  Corollary 3.6.

The assignment
$\textbf{Flow}_{R}\textit{-}pr\longrightarrow\textbf{R}\textit{-}pr$ given by
$\overline{\sigma}\longmapsto\sigma$ is order preserving.

######  Proof.

Let $\overline{\sigma}$ and $\overline{\tau}$ be preradicals on
Flow${}_{R\textit{-}Mod}$ such that $\overline{\sigma}\preceq\overline{\tau}$.
Thus, for every $M\in R\textit{-}Mod$ one has that
$\overline{\sigma}(M,Id_{M})$ is a subobject of $\overline{\tau}(M,Id_{M})$.
As noted before, this is equivalent to saying that
$U\big{(}\overline{\sigma}(M,Id_{M})\big{)}$ is a subobject of
$U\big{(}\overline{\tau}(M,Id_{M})\big{)}$ in $R$-Mod, which in turn is
equivalent to $U\big{(}\overline{\sigma}(M,Id_{M})\big{)}\leq
U\big{(}\overline{\tau}(M,Id_{M})\big{)}$. Therefore, as $E(M)=(M,Id_{M})$ we
have that

$\sigma(M)=(U\circ\overline{\sigma}\circ
E)(M)=U\big{(}\overline{\sigma}(M,Id_{M})\big{)}\leq
U\big{(}\overline{\tau}(M,Id_{M})\big{)}=(U\circ\overline{\tau}\circ
E)(M)=\tau(M)$.

Thus, $\sigma(M)\leq\tau(M)$ for all $M\in R\textit{-}Mod$, and therefore
$\sigma\leq\tau$ in $R\textit{-}pr$. ∎

##  4 Entropy for preradicals

We start this section recalling the notion of an invariant function on $R$-Mod
along with some notation and basic facts which lead to the definition of
entropy for endomorphisms and objects in $R$-Mod. Afterwards, we will define
entropy for preradicals on $R$-Mod after for preradicals on
Flow${}_{R}\textit{-}pr$ as a consequence. For a complete introduction to
invariants and entropy functions on $R$-Mod, see [4], [5],[6],[8], [9] and
[15].

######  Definition 4.1.

[8][Definition 1.1] An invariant in $R$-Mod is a function
$i:R\textit{-}Mod\longrightarrow\mathbb{R}_{\geq 0}\cup\infty$ such that
$i(0)=0$ and $i(M)=i(M^{\prime})$ whenever $M$ and $M^{\prime}$ are isomorphic
objects in $R$-Mod.

From now on we will consider invariants with the two following conditions:

  * (a)

$i(M_{1}+M_{2})\leq i(M_{1})+i(M_{2})$ for all submodules $M_{1},M_{2}$ of
$M$.

  * (b)

$i(M/N)\leq i(M)$ for any submodule $N$ of $M$.

An invariant $i$ satisfying $(a)$ and $(b)$ is called a subadditive invariant.

Let $(M,\eta)$ be an object in $\textbf{Flow}_{R\textit{-}Mod}$ and let $L$ be
a subset of $M$. For any positive integer $n$, we define the $n$-th
$\eta$-trajectory of $L$ as

$T_{n}(L,\eta)=L+\eta(L)+\cdots+\eta^{n-1}(L)$.

When $L$ happens to be a submodule of $M$, then $T_{n}(L,\eta)$ is in fact a
submodule of $M$. Furthermore, the sum

$\underset{n\geq 1}{\sum}\ T_{n}(L,\eta)=\underset{n\geq 0}{\sum}\
\eta^{n}(L)$

defines the submodule of $M$ named the $\eta$-trajectory of $L$, which is
denoted by $T(L,\eta)$.

Let us now suppose that $i$ is a subadditive invariant on $R$-Mod. Then, for
every submodule $L$ of $M$, with $i(L)<\infty$, one has that
$i\Big{(}T_{n}(L,\eta)\Big{)}<\infty$ for every $n\geq 1$. Indeed, by
proceeding with induction over $n$, we will start considering the base case
$n=2$ since $n=1$ is trivial. As $\eta(L)\cong\Big{(}L/ker(\eta)\Big{)}$ and
$i$ is a subadditive invariant, we have that

$i\big{(}\eta(L)\big{)}=i\Big{(}L/ker(\eta)\Big{)}\leq i(L)<\infty$.

Hence,

$i\big{(}T_{2}(L,\eta)\big{)}=i\big{(}L+\eta(L)\big{)}\leq
i(L)+\eta(L)<\infty$.

Assume now that the inductive hypothesis is valid for $n>2$. Thus, since
$\eta^{n}(L)\cong\Big{(}L/ker(\eta^{n})\Big{)}$ we have that

$i\big{(}\eta^{n}(L)\big{)}=i\Big{(}L/ker(\eta^{n})\Big{)}\leq i(L)<\infty$.

Therefore, as

$T_{n+1}(L,\eta)=L+\eta(L)+\cdots+\eta^{n-1}(L)+\eta^{n}(L)=T_{n}(L,\eta)+\eta^{n}(L)$,

by the inductive hypothesis we have that

$i\big{(}T_{n+1}(L,\eta)\big{)}=i\Big{(}T_{n}(L,\eta)+\eta^{n}(L)\Big{)}$
$\leq i\Big{(}T_{n}(L,\eta)\Big{)}+i\big{(}\eta^{n}(L)\big{)}<\infty$.

We now observe that, for every subadditive invariant $i$ and any submodule $L$
of $C$, one has

$\displaystyle
T_{n+m}(C,\eta)=L+\eta(L)+\cdots+\eta^{n-1}(L)+\eta^{n}(L)+\cdots+\eta^{n+m-1}(L)$
$\displaystyle=T_{n}(L,\eta)+\eta^{n}(L)+\cdots+\eta^{n+m-1}(L)$
$\displaystyle=T_{n}(L,\eta)+\eta^{n}\big{(}L+\eta(L)+\cdots+\eta^{m-1}(L)\big{)}=T_{n}(L,\eta)+\eta^{n}\big{(}T_{m}(L,\eta)\big{)}.$

Since $i$ is a subadditive invariant, it follows that

$\displaystyle
i\Big{(}T_{n+m}(L,\eta)\Big{)}=i\Big{(}T_{n}(L,\eta)+\eta^{n}\big{(}T_{m}(L,\eta)\big{)}\Big{)}$
$\displaystyle\leq
i\Big{(}T_{n}(L,\eta)\Big{)}+i\Big{(}\eta^{n}\big{(}T_{m}(L,\eta)\big{)}\Big{)}$
$\displaystyle\leq i\Big{(}T_{n}(L,\eta)\Big{)}+i\Big{(}T_{m}(L,\eta)\Big{)},$

where the last relation follows from $(b)$ and the fact that
$\eta^{n}\big{(}T_{m}(L,\eta)\big{)}\cong\Big{(}T_{m}(L,\eta)/ker(\eta^{n})\Big{)}$.

######  Remark 2.

If $L$ is a submodule of $M$ such that $i(L)<\infty$, then every endomorphism
$\eta$ of $M$ induces a sequence of positive reals
$\\{a_{i}\\}_{n\in\mathbb{N}}$ such that $a_{n+m}\leq a_{n}+a_{m}$.

We lastly note that every sequence of positive reals
$\\{a_{i}\\}_{n\in\mathbb{N}}$ such that $a_{n+m}\leq a_{n}+a_{m}$ converges
in $\mathbb{R}$. Indeed, as $a_{2}\leq a_{1}+a_{1}=2\cdot a_{1}$ by
mathematical induction we have that $a_{k}\leq k\cdot a_{1}$ for all $k\geq
1$. This in turn implies that the sequence $\\{\frac{a_{k}}{k}|k\geq 1\\}$ is
bounded above by $a_{1}$ and bounded below by $0$. Therefore,
$inf\\{\frac{a_{k}}{k}|k\geq 1\\}$ exist. Furthermore, one exactly has
$\underset{n\to\infty}{lim}\ \frac{a_{n}}{n}$ coincides with
$inf\\{\frac{a_{k}}{k}|k\geq 1\\}$ as the next Proposition states:

######  Proposition 4.2.

[18, Exercise 6.5] Let $\\{a_{i}\\}_{i\in\mathbb{N}}$ be a sequence of
positive real numbers such that $a_{n+m}\leq a_{n}+a_{m}$ for all
$n,m\in\mathbb{N}$. Then the sequence $\\{\frac{a_{k}}{k}|k\geq 1\\}$
converges to $inf\\{\frac{a_{k}}{k}|k\geq 1\\}$.

The above proposition is the prelude that allows giving a formal definition of
$i$-entropy for each invariant $i$ on $R$-Mod. In doing this, one must
consider the collection of submodules $M^{\prime}$ of $M$ such that
$i(M^{\prime})<\infty$, which is denoted by

$\mathcal{F}_{i}(M)\colon=\\{M^{\prime}\leq M\mid i(M^{\prime})<\infty\\}$.

######  Definition 4.3.

[9, Definition 5.25] Let $i$ be a subadditive invariant, $M$ an $R$-module and
$M\overset{\eta}{\longrightarrow}M$ an endomorphism. The algebraic $i$-entropy
of $\eta$ with respect to $M^{\prime}\in\mathcal{F}_{i}(M)$ is

$H_{i}(M^{\prime},\eta)\colon=\underset{n\to\infty}{Lim}\frac{i(T_{n}(M^{\prime},\eta))}{n}.$

The algebraic $i$-entropy of $\eta$ is

$ent_{i}(\eta)\colon=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}(M)\Big{\\}}$.

######  Definition 4.4.

[15, Definition 3] Let $i$ be a subadditive invariant of $R\textit{-}Mod$ and
let $End(M)$ denote the set of all endomorphism of the module $M\in
R\textit{-}Mod$. The $i$-entropy of $M$ is

$ent_{i}(M)\colon=sup\Big{\\{}ent_{i}(\eta)\mid\eta\in
End(M)\Big{\\}}=\underset{\eta\in
End(M)}{sup}\Big{\\{}ent_{i}(\eta)\Big{\\}}$.

We now present the definition of $i$-entropy for preradicals on $R$-Mod:

######  Definition 4.5.

Let $i$ be a subadditive invariant on $R\textit{-}Mod$, and let $\sigma\in
R\textit{-}pr$. If $M\in R\textit{-}Mod$, then the algebraic $i$-entropy of
$\sigma$ with respect to $M$ is given by

$ent_{i}(\sigma)|_{M}\colon=sup\Big{\\{}H_{i}((M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}(\sigma(M))\mbox{ and }\eta\in End(M)\Big{\\}}$
$=\underset{\eta\in End(M)}{sup}\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}(\sigma(M))\Big{\\}}$

where
$H_{i}(M^{\prime},\eta)=\lim_{n\to\infty}\frac{i(T_{n}(M^{\prime},\eta))}{n}$.

######  Remark 3.

If $i$ is an invariant on $R$-Mod, then for every preradical $\sigma\in
R\textit{-pr}$ and for every $M\in R$-Mod one has that

$ent_{i}(\sigma)|_{M}\leq_{\mathbb{R}}ent_{i}(\sigma(M)).$

######  Proposition 4.6.

Let $i$ be an invariant on $R$-Mod. If $\sigma,\tau\in R\textit{-}pr$ are such
that $\sigma\leq\tau$, then for any $M\in R\textit{-}Mod$, one has

$ent_{i}(\sigma)|_{M}\leq_{\mathbb{R}}ent_{i}(\tau)|_{M}.$

######  Proof.

Let $M\in R\textit{-}Mod$. As $\sigma(M)\leq\tau(M)$, then for any
endomorphism $M\overset{\eta}{\longrightarrow}M$ one has that
$\mathcal{F}_{i}(\sigma(M))\subseteq\mathcal{F}_{i}(\tau(M))$. Hence,

$\big{\\{}H_{i}(M^{\prime},\eta)\ |\
M^{\prime}\in\mathcal{F}_{i}(\sigma(M))\big{\\}}\leq_{\mathbb{R}}\big{\\{}H_{i}(M^{\prime},\eta)\
|\ M^{\prime}\in\mathcal{F}_{i}(\tau(M))\big{\\}},$

and thus

$\underset{\eta\in End(M)}{sup}\big{\\{}H_{i}(M^{\prime},\eta)\ |\
M^{\prime}\in\mathcal{F}_{i}(\sigma(M))\big{\\}}\leq_{\mathbb{R}}\underset{\eta\in
End(M)}{sup}\big{\\{}H_{i}(M^{\prime},\eta)\ |\
M^{\prime}\in\mathcal{F}_{i}(\tau(M))\big{\\}}.$

Therefore, $ent_{i}(\sigma)|_{M}\leq_{\mathbb{R}}ent_{i}(\tau)|_{M}$.

A well known property in the big lattice $R$-pr that relates the four basic
operations between preradicals is: given $\sigma,\tau\in R$-pr one always has
that

$(\sigma\cdot\tau)\leq(\sigma\wedge\tau)\leq(\sigma\vee\tau)\leq(\sigma:\tau)$.

As a straightforward result, we have the next

######  Proposition 4.7.

Let $\sigma,\tau\in R\textit{-}pr$ and $M\in R\textit{-}Mod$. Then

$ent_{i}(\sigma\cdot\tau)|_{M}\leq_{\mathbb{R}}ent_{i}(\sigma\wedge\tau)|_{M}\leq_{\mathbb{R}}ent_{i}(\sigma\vee\tau)|_{M}\leq_{\mathbb{R}}ent_{i}(\sigma:\tau)|_{M}$.

Next we display two examples of invariants for a fix preradical on $R$-Mod.

######  Example.

Let $R=\mathbb{Z}$ -so the category $R$-Mod coincides with the category of
abelian groups $\mathbf{Ab}$\- and let us consider the preradical $Tor$ on
$\mathbb{Z}\text{-Mod}$ which assigns to each abelian group $M$ its torsion
subgroup $Tor(M)$, this is, its subgroup consisting of all elements of finite
order.

On the other hand, we can consider the algebraic entropy on $\mathbb{Z}$-Mod
defined by the invariant
$i\colon\mathbb{Z}\textit{-}Mod\longrightarrow\mathbb{Z}\textit{-}Mod$, where
$i(M)=|log(M)|$ whenever $M$ is finite, otherwise $i(M)=\infty$. We recall
that if $\mathcal{F}(M)$ denotes all finite subgroups of the abelian group
$M$, then for any endomorphism $M\overset{\eta}{\longrightarrow}M$ and any
$L\in\mathcal{F}(M)$, we define for each $n\in\mathbb{N}$

$H_{n}(L,\eta)=log|T_{n}(L,\eta)|$.

In other words, $H_{n}(L,\eta)$ is the logarithm of the $n\textit{-}th$
trajectory. Thus, the algebraic entropy of $\eta$ with respect to the finite
subgroup $L$ is

$H(L,\eta)=\lim_{n\to\infty}\frac{log|(T_{n}(L,\eta))|}{n}$,

and hence, the algebraic entropy of $\eta$ is

$ent_{i}(\eta)=sup\Big{\\{}H(L,\eta)\mid L\in\mathcal{F}(M)\Big{\\}}$.

Taking the latter into consideration, we define the algebraic entropy of $M$
as

$ent_{i}(M)=sup\Big{\\{}ent_{i}(\eta)|\eta\in End(M)\Big{\\}}$.

Let us define $M=\oplus_{i\in\mathbb{N}}\ \mathbb{Z}_{p}^{i}$, where
$\mathbb{Z}_{p}^{i}$ denotes the cyclic group of order $p$, for each
$i\in\mathbb{N}$. Note that, as $M$ is clearly a torsion group, we have that
$Tor(M)=M$. Furthermore, if $\beta:M\longrightarrow M$, defined by

$\beta(x_{1},x_{2},x_{3},\cdots)\longmapsto(0,x_{1},x_{2},x_{3},\cdots)$

is the Bernoulli shift endomorphism, then one has that $ent(\beta)=|log(p)|$
and $ent(Tor)\mid_{M}>0$ as a consequence. Indeed, if we take the subgroup
$\mathbb{Z}_{p}^{1}$ of $M$, for each $n\in\mathbb{N}$ we have that

$H_{n}(\mathbb{Z}_{p}^{1},\beta)=log|\oplus_{i\leq n}\
\mathbb{Z}_{p}^{i}|=log(|p^{n}|)=n\times log(p)$.

Hence,

$\frac{H(\mathbb{Z}_{p}^{1},\beta)_{n}}{n}=log(p)$

for all $n\in\mathbb{N}$. Thus, $H(\mathbb{Z}_{p}^{1},\beta)=log(p)$, which in
turn implies that $ent_{i}(\beta)\geq log(p)$. Moreover, since the
$\beta$-trajectory of the finite subgroup $\mathbb{Z}_{p}^{1}$ covers $M$,
then it also covers any other $\beta$-trajectory for any finite subgroup $F$
of $M$. Consequently, we have that $H(F,\beta)\leq
H(\mathbb{Z}_{p}^{1},\beta)$, and therefore

$ent_{i}(\beta)=H_{i}(\mathbb{Z}_{p}^{1},\beta)=log(p)$.

Observe that the latter implies that $ent_{i}(Tor)\mid_{M}>0$.

On the other hand, if we consider the rank invariant on $\mathbb{Z}$-Mod we
will get that $ent_{\textbf{rk}}(Tor)|_{M}=0$ for all
$M\in\mathbb{Z}\textit{-}$Mod. Recall that the rank invariant is defined as
follows: for each $M\in\mathbb{Z}\textit{-}Mod$,
$\textbf{rk}(M)=dim_{\mathbb{Q}}(M\otimes\mathbb{Q})$ when this dimension is
finite, otherwise we set $\textbf{rk}(M)=\infty$. Thus, we have that
$\textbf{rk}(M)=0$ for any torsion group $M$. This in turn implies that

$ent_{\textbf{rk}}(Tor)|_{M}=0$

for all $M\in\mathbb{Z}\textit{-}$Mod.

We will now define the $i$-entropy for preradicals on
Flow${}_{R\textit{-}Mod}$ object-wise, this is, in terms of flows, and then we
will prove some correlations between the entropy of a preradical on $R$-Mod
with the entropy of its corresponding preradical on the category of flows
Flow${}_{R\textit{-}Mod}$.

######  Definition 4.8.

Let $i$ be a subadditive invariant on $R\textit{-}Mod$ and let
$\overline{\sigma}$ be a preradical on $\textbf{Flow}_{R}\textit{-}pr$. By
denoting $\overline{\sigma}(M,\eta)=(\sigma(M),\eta_{|})$ for each flow object
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$, we have that the algebraic
$i$-entropy of $\overline{\sigma}$ with respect to $\eta$ and
$M^{\prime}\in\mathcal{F}_{i}(\sigma(M))$ is

$H_{i}(M^{\prime},\eta)\colon=\lim_{n\to\infty}\frac{i(T_{n}(M^{\prime},\eta))}{n}.$

The algebraic $i$-entropy of $\overline{\sigma}$ regarding to the object
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$ is

$ent_{i}(\overline{\sigma})_{(M,\eta)}\colon=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}(\sigma(M))\Big{\\}}$.

In other words, the algebraic $i$-entropy of $\overline{\sigma}$, with respect
to the flow object $(M,\eta)$, is the same as the algebraic $i$-entropy of the
endomorphism obtained by restricting $\eta$ to $\sigma(M)$, this is,
$\eta_{|}:\sigma(M)\longrightarrow\sigma(M)$. By how
$\psi:R\textit{-}pr\longrightarrow\textbf{Flow}_{R}\textit{-}pr$ is defined,
we have the straightforward result:

######  Proposition 4.9.

Let $\sigma,\tau\in R\textit{-}pr$ and
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$. If $\overline{\sigma}$ and
$\bar{\tau}$ denote the preradicals induced by $\sigma$ and $\tau$,
respectively, via the assignment
$\psi:R\textit{-}pr\longrightarrow\textbf{Flow}_{R}\textit{-}pr$, then for
each invariant $i$ on $R$-Mod

  * •

The $i$-entropy for $(\overline{\sigma\wedge\tau})$ with respect to $(M,\eta)$
is

$ent_{i}(\overline{\sigma\wedge\tau})_{(M,\eta)}=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\Big{\\}},$

that is, where $M^{\prime}\leq\sigma(M)\cap\tau(M)$ and
$i(M^{\prime})<\infty$.

  * •

The $i$-entropy for $(\overline{\sigma\vee\tau})$ with respect to $(M,\eta)$
is

$ent_{i}(\overline{\sigma\vee\tau})_{(M,\eta)}=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\Big{\\}},$

that is, where $M^{\prime}\leq\sigma(M)+\tau(M)$ and $i(M^{\prime})<\infty$.

  * •

The $i$-entropy for $(\overline{\sigma\cdot\tau})$ with respect to $(M,\eta)$
is

$ent_{i}(\overline{\sigma\cdot\tau})_{(M,\eta)}=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\cdot\tau)(M)\big{)}\Big{\\}},$

that is, $M^{\prime}\leq\sigma(\tau(M))$ and where $i(M^{\prime})<\infty$.

  * •

The $i$-entropy for $(\overline{\sigma:\tau})$ with respect to $(M,\eta)$ is

$ent_{i}(\overline{\sigma:\tau})_{(M,\eta)}=sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma:\tau)(M)\big{)}\Big{\\}},$

that is, where $M^{\prime}\leq\tau(M/\sigma(M))$ and $i(M^{\prime})<\infty$.

######  Proposition 4.10.

Let $\sigma,\tau\in R\textit{-}pr$ and let
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$. Then, we have that

$ent_{i}(\overline{\sigma\cdot\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma\wedge\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma\vee\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma:\tau})_{(M,\eta)}$.

where $\overline{\sigma}$ represents the induced preradical by $\sigma$, via
the assignment $\phi$.

######  Proof.

Let $\sigma,\tau\in R\textit{-}pr$ and let
$(M,\eta)\in\textbf{Flow}_{R\textit{-}Mod}$.

  * (i)

Since $(\sigma\cdot\tau)(M)\leq(\sigma\wedge\tau)(M)$ for all $M\in
R\textit{-}Mod$, it follows that

$\mathcal{F}_{i}\big{(}(\sigma\cdot\tau)(M)\big{)}\subseteq\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}$.

The latter in turn implies that

$\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\cdot\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\Big{\\}}$.

Therefore,

$sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\cdot\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\Big{\\}}$.

Hence,
$ent_{i}(\overline{\sigma\cdot\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma\wedge\tau})_{(M,\eta)}$.

  * (ii)

Since $(\sigma\wedge\tau)(M)\leq(\sigma\vee\tau)(M)$ for all $M\in
R\textit{-}Mod$, then we have that

$\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\subseteq\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}$.

Thus,

$\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\Big{\\}}$,

which implies that

$sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\wedge\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\Big{\\}}$.

Therefore,
$ent_{i}(\overline{\sigma\wedge\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma\vee\tau})_{(M,\eta)}$

  * (iii)

Since $(\sigma\vee\tau)(M)\leq(\sigma:\tau)(M)$, for all $M\in
R\textit{-}Mod$, then

$\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\subseteq\mathcal{F}_{i}\big{(}(\sigma:\tau)(M)\big{)}$.

Thus,

$\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma:\tau)(M)\big{)}\Big{\\}}$,

which in turn implies

$sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma\vee\tau)(M)\big{)}\Big{\\}}\leq_{\mathbb{R}}sup\Big{\\{}H_{i}(M^{\prime},\eta)\mid
M^{\prime}\in\mathcal{F}_{i}\big{(}(\sigma:\tau)(M)\big{)}\Big{\\}}$.

Hence,
$ent_{i}(\overline{\sigma\vee\tau})_{(M,\eta)}\leq_{\mathbb{R}}ent_{i}(\overline{\sigma:\tau})_{(M,\eta)}$.

∎

We close this subsection with a result that relates the algebraic entropy
$log$ defined on categories of the form $S$-Mod and $R$-Mod, whenever there is
a ring homomorphism between the base rings. With this in mind, we will show
how the entropy values for preradicals on $S$-Mod relate to the entropy values
of preradicals on $R$-Mod, whenever the ring homomorphism $t:R\longrightarrow
S$ is surjective.
Let us then suppose that $t\colon R\longrightarrow S$ denotes a ring
homomorphism. Then each $S$-module $M$ admits an $R$-module structure via the
homomorphism $t$. Indeed, given $M\in S$-Mod we have that $t$ induces an
action on $M$ \- as an $R$-module-, given by the correspondence rule $r\cdot
m:=t(r)\cdot m$, for any $r\in R$ and $m\in M$. Moreover, $t$ induces a
functor $F_{t}:S\textit{-}Mod\longrightarrow R\textit{-}Mod$ that assigns to
each $M\in S\textit{-}Mod$, the $R$-module $F_{t}(M)=M$, and assigns to each
morphism $f:M\longrightarrow M^{\prime}$ in $S\textit{-}Mod$, the $R$-morphism
$F_{t}(f)=f$.

######  Proposition 4.11.

Let $log_{s}$ and $log_{r}$ be the invariant $log()$ defined on
$S\textit{-}Mod$ and $R\textit{-}Mod$ respectively. If $t\colon
R\longrightarrow S$ is a ring homomorphism, then for any $M\in S\textit{-}Mod$
and its respective $R$-module $F_{t}(M)$, one has that

$ent_{log_{s}}(M)\leq ent_{log_{r}}(F_{t}(M))$.

######  Proof.

Let $t\colon R\longrightarrow S$ be a ring homomorphism. For the sake of
clarity, we will denote an $S$-module by $M_{S}$ and to its image under
functor $F_{t}$ by $M_{R}$. We first note that, if
$L_{S}\in\mathcal{F}_{log_{s}}(M_{S})$ then
$L_{R}\in\mathcal{F}_{log_{r}}(M_{R})$. Indeed, the functor $F_{t}$ send the
inclusion map $\iota_{S}:L_{S}\longrightarrow M_{S}$ into the inclusion map
$i_{R}:L_{R}\longrightarrow M_{R}$ in $R$-Mod. Hence, as
$L_{S}\in\mathcal{F}_{log_{s}}(M_{S})$ then
$log(L_{R})=Log|L|=log(L_{S})<\infty$, and thus
$L_{R}\in\mathcal{F}_{log_{r}}(M_{R})$ for each
$L_{S}\in\mathcal{F}_{log_{s}}(M_{S})$. Consequently, we have that for each
endomorphism $\eta:M_{S}\longrightarrow M_{S}$ and each
$L_{S}\in\mathcal{F}_{log_{s}}(M_{S})$ one has that

$\displaystyle T_{n}(L_{S},\eta)=L_{S}+\eta(L)_{S}+\cdots+\eta^{n-1}(L)_{S}$
$\displaystyle=L_{R}+\eta(L)_{R}+\cdots+\eta^{n-1}(L)_{R}=T_{n}(L_{R},\eta_{R}),$

where $\eta_{R}$ denotes the endomorphism $\eta_{R}:M_{R}\longrightarrow
M_{R}$ resulting from evaluating functor $\mathcal{F}_{t}$ on $\eta$.
Considering the above we have that

$H_{log_{s}}(L_{S},\eta)=\lim_{n\to\infty}\frac{log|(T_{n}(L_{S},\eta))|}{n}$
$=\lim_{n\to\infty}\frac{log|(T_{n}(L_{R},\eta_{R}))|}{n}=H_{log_{r}}(L_{R},\eta_{R})$,

and thus,

$ent_{log_{s}}(\eta)=sup\\{H_{log_{s}}(L_{S},\eta)\mid
L_{S}\in\mathcal{F}_{log_{s}}(M_{S})\\}$

$=sup\\{H_{log_{r}}(L_{R},\eta_{R})\mid
L_{R}\in\mathcal{F}_{log_{r}}(M_{R})\\}$

$\leq_{\mathbb{R}}sup\\{H_{log_{r}}(N,\eta_{R})\mid
N\in\mathcal{F}_{log_{r}}(M_{R})\\}=ent_{log_{r}}(\eta_{R}).$

Therefore,

$ent_{log_{s}}(M_{S})=sup\\{ent_{log_{s}}(\eta)\mid\eta\in End_{S}(M_{S})\\}$

$\leq_{\mathbb{R}}sup\\{ent_{log_{r}}(\eta_{R})\mid\eta_{R}\in
End_{R}(M_{R})\\}$

$\leq_{\mathbb{R}}sup\\{ent_{log_{r}}(\mu)\mid\mu\in
End_{R}(M_{R})\\}=ent_{log_{r}}(M_{R}).$

∎

In case that $t\colon R\longrightarrow S$ is a surjective ring homomorphism,
one always has that the induced functor $F_{t}\colon
S\textit{-}Mod\longrightarrow R\textit{-}Mod$ is full ([10, Theorem 2.5]).
Furthermore, as $F_{t}$ respects inclusion maps and is injective on objects,
by [10, Theorem 2.4] we have that $F_{t}$ induces an injective assignment
$\phi_{t}:S\textit{-}pr\longrightarrow R\textit{-}pr$ that is order
preserving. This assignment is given by

$\phi_{t}(\sigma)=\underset{M\in S\textit{-}Mod}{\bigvee}\
\alpha_{\sigma(M)}^{M}$

where $\alpha_{\sigma(M)}^{M}$ denotes the alpha preradical on $R$-pr defined
by the objects $\sigma(M)=F_{t}(\sigma(M))$ and $M=F_{t}(M)$. With these facts
in mind, we have the next

######  Proposition 4.12.

Let $log_{s}$ and $log_{r}$ be the invariant $log()$ defined on
$S\textit{-}Mod$ and $R\textit{-}Mod$ respectively. If $t\colon
R\longrightarrow S$ is a ring epimorphism, then

$ent_{log_{s}}(\sigma)|_{M_{S}}\leq_{\mathbb{R}}ent_{log_{r}}(\phi_{t}(\sigma))|_{M_{R}}$

for each preradical $\sigma\in S\textit{-}pr$.
Here, $M_{S}\in S\textit{-}Mod$ and $M_{R}$ is the image of $M_{S}$ under
functor $F_{t}$.

######  Proof.

Let $M_{S}\in S\textit{-}Mod$ and let $M_{R}=F_{t}(M_{S})$. In order to avoid
confusion, we will denote $\sigma(M_{S})=\sigma(M)_{S}$ and thus, we denote
$\sigma(M)_{R}$ to its corresponding $R$-module under functor $F_{t}$. Now,
for $\sigma(M)_{R}\leq M_{R}$ one has that

$\alpha_{\sigma(M_{R})}^{M_{R}}\leq\underset{M\in S\textit{-}Mod}{\bigvee}\
\alpha_{\sigma(M)}^{M}$.

Further, we have that $\alpha_{\sigma(M)_{R}}^{M_{R}}(M_{R})=\sigma(M)_{R}$.
Therefore, since $M_{S}^{\prime}\in\mathcal{F}_{log_{s}}(\sigma(M_{S}))$
implies that
$M_{R}^{\prime}\in\mathcal{F}_{log_{r}}(\alpha_{\sigma(M_{R})}^{M_{R}}(M_{R}))\subseteq\mathcal{F}_{log_{r}}\Big{(}\big{(}\underset{M\in
S\textit{-}Mod}{\bigvee}\ \alpha_{\sigma(M)}^{M}\big{)}(M_{R})\Big{)}$, we
then have that

$ent_{log_{s}}(\sigma)|_{M_{S}}\leq_{\mathbb{R}}ent_{log_{r}}(\alpha_{\sigma(M_{R})}^{M_{R}})|_{M_{R}}\leq_{\mathbb{R}}ent_{log_{r}}(\phi_{t}(\sigma))|_{M_{R}}$

∎

###  4.1 Functorial entropy between $\mathcal{L}\textit{-}pr$ and
$\mathcal{L_{M}}\textit{-}pr$

We will now use the notion of functorial entropy described in [9] to define
entropy for lattice preradicals (See [3]). In doing this, we will show some
results that relates entropy for lattice preradicals with entropy for module
preradicals.

It is well known that each $R$-module $M$ induces a complete modular lattice
$L(M)$, whose elements correspond to all submodules $N$ of $M$. The meet and
the join operations in $L(M)$ ( denoted by $\wedge$ and $\vee$ respectively)
are given by the intersection and the sum of submodules. Further, each $L(M)$
is bounded above by $M$ and below by $\\{0\\}$. As the authors show in [2],
the collection of all bounded modular lattices are the objects of a category,
denoted by $\mathcal{L_{M}}$, whose morphisms are called linear morphisms:
these summon the property of having a kernel $ker(\eta)$ for every morphisms
$\eta:M\longrightarrow M^{\prime}$ in $R$-mod, so that the First Isomorphisms
Theorem holds; this is:

$M/ker(\eta)\cong Im(\eta)$.

Moreover, for each $R$-morphisms $M\overset{\eta}{\longrightarrow}M^{\prime}$
one can define a linear morphism
$L(M)\overset{\mathcal{X}_{\eta}}{\longrightarrow}L(M^{\prime})$ as follows:
if $N\in L(M)$ -that is, $N$ is a submodule of $M$\- then
$\mathcal{X}_{\eta}(N)=\eta(N)$, where $\eta(N)$ is the submodule of
$M^{\prime}$ generated by the image of $N$ under $\eta$. The above can be
summarized by saying that, for each unital ring $R$, one has a functor
$\mathcal{X}:R\textit{-}Mod\longrightarrow\mathcal{L_{M}}$ with correspondence
$M\longmapsto L(M)$ on objects and $\eta\longmapsto\mathcal{X}_{\eta}$ on
morphisms. We note here that the image of the functor $\mathcal{X}$ defines a
non-full subcategory of $\mathcal{L_{M}}$, which we will denote by
$\mathcal{SL_{M}}$. In this case, the set of morphisms from $L(M)$ to
$L(M^{\prime})$ within $\mathcal{SL_{M}}$ is given by the set

$Hom_{\mathcal{SL_{M}}}\big{(}L(M),L(M^{\prime})\big{)}=\Big{\\{}\mathcal{X}_{\eta}\mid
M\overset{\eta}{\longrightarrow}M^{\prime}\mbox{ is a morphism in
}R\textit{-}Mod\Big{\\}}$.

When considering preradicals, in [3] the authors show the existence of an
assignment, denoted here by $\phi:\mathcal{L_{M}}\textit{-}pr\longrightarrow
R\textit{-}pr$, between the big lattice of lattice preradicals and the big
lattice of $R$-module preradicals. This assignment is given by the
correspondence $\tilde{\sigma}\longmapsto\sigma$, where $\sigma$ evaluated in
an $R$-module $M$ coincides with the respective submodule obtained by
evaluating $\tilde{\sigma}$ in $L(M)$. In other words, for any $M\in R$-Mod,
we have $\sigma(M)=X_{L(M)}^{\tilde{\sigma}}$ where
$X_{L(M)}^{\tilde{\sigma}}$ is such that
$\tilde{\sigma}(L(M))=X_{L(M)}^{\tilde{\sigma}}/0$444A lattice preradical on
$\mathcal{L_{M}}$ assigns to each lattice $L$ a sublattice of the form $X/0$.
Here, we have written $X_{L(M)}^{\tilde{\sigma}}$ with a capital letter to
emphasize that $X_{L(M)}^{\tilde{\sigma}}$ defines a submodule of $M$. .
Lastly, as shown in [11], $\phi$ is an order preserving assignment which also
respects the join, the product and the coproduct operations.

Let us now suppose that $i$ is a subadditive invariant on $R$-Mod, and let
$M\in R$-Mod. As noted before, $M$ induces a complete modular lattice
$\big{(}L(M),\wedge,\vee,\subseteq\big{)}$. This complete modular lattice
contains the set

$\mathcal{F}_{i}(L(M))=\\{N\in L(M)\mid i(N)<\infty\\}$.

Further, as $i$ is an subadditive invariant, one has that

$i(N_{1}+N_{2})\leq i(N_{1})+i(N_{2})<\infty$.

Hence, $N_{1}\vee N_{2}=N_{1}+N_{2}\in\mathcal{F}_{i}(L(M))$, and thus the
tuple $(\mathcal{F}_{i}(L(M)),\vee,\subseteq)$ defines a join semi-lattice
555A semi-lattice is a set $S$ with a binary operation $*$ which is
commutative, associative, and satisfies $s*s=s$ for all $s\in S$. In
particular, any lattice $L$ is a semi-lattice under $\vee$ and also under
$\wedge$.. Moreover, according to [9, Section 5], we have that the tuple
$(\mathcal{F}_{i}(L(M)),\vee,\subseteq,i)$ defines a subadditive normed semi-
lattice666 A normed semi-lattice is a semi-lattice $S$ equipped with a norm
$i:S\longrightarrow\mathbb{R}_{\geq 0}$.

######  Proposition 4.13.

Let $i$ be an invariant on $R$-Mod, and let $\eta\colon M\longrightarrow
M^{\prime}$ be an $R$-morphism. Then $\eta$ induces a morphism
$\mathcal{F}_{i}(\eta)$ between the join semi-lattices $\mathcal{F}_{i}(L(M))$
and $\mathcal{F}_{i}(L(M^{\prime}))$. This morphism is given by
$\big{(}\mathcal{F}_{i}(\eta)\big{)}(N)=\eta(N)$ for each
$N\in\mathcal{F}_{i}(L(M))$.

######  Proof.

Let $i$ be an invariant on $R$-Mod, and let $\eta\colon M\longrightarrow
M^{\prime}$ be an $R$-morphism. We will first see that
$\mathcal{F}_{i}(\eta)\colon\mathcal{F}_{i}(L(M))\longrightarrow\mathcal{F}_{i}(L(M^{\prime}))$,
with correspondence rule $\big{(}\mathcal{F}_{i}(\eta)\big{)}(N)=\eta(N)$ for
each $N\in\mathcal{F}_{i}(L(M))$, is well defined. Indeed, as $i$ is an
invariant, for each $N\in\mathcal{F}_{i}(L(M))$ we have that

$i(\eta(N))=i(N/ker(\eta))\leq i(N)<\infty.$

Thus, $i(\eta(N))<\infty$, which implies that
$\big{(}\mathcal{F}_{i}(\eta)\big{)}(N)=\eta(N)\in\mathcal{F}_{i}(L(M^{\prime}))$.
Now, if $N_{1},N_{2}\in\mathcal{F}_{i}(L(M))$, then

$\displaystyle\big{(}\mathcal{F}_{i}(\eta)\big{)}(N_{1}\vee
N_{2})=\eta(N_{1}\vee N_{2})=\eta(N_{1}+N_{2})=\eta(N_{1})+\eta(N_{2})$
$\displaystyle=\big{(}\mathcal{F}_{i}(\eta)\big{)}(N_{1})+\big{(}\mathcal{F}_{i}(\eta)\big{)}(N_{2})=\big{(}\mathcal{F}_{i}(\eta)\big{)}(N_{1})\vee\big{(}\mathcal{F}_{i}(\eta)\big{)}(N_{2})\in\mathcal{F}_{i}(L(M^{\prime})).$

Therefore,
$\mathcal{F}_{i}(\eta):\mathcal{F}_{i}(L(M))\longrightarrow\mathcal{F}_{i}(L(M^{\prime}))$
is a morphism. ∎

In this way, we have a functor $\mathcal{S}\colon
R\textit{-Mod}\longrightarrow\mathcal{L}^{{\dagger}}$, where
$\mathcal{L}^{{\dagger}}$ is the category whose objects are normed join semi-
lattices and whose morphisms are the contractive777These morphisms are named
contractive since $i(\eta(N))<i(N)$, for all $N$. join semi-lattice morphisms.
Here $\mathcal{S}$ assigns to each $R$-module $M$ the join semi-lattice
$(\mathcal{F}_{i}(L(M)),\vee,\subseteq,i)$, and assigns to each $R$-morphism
$\eta$ the join semi-lattice morphism $\mathcal{F}(\eta)$. Similarly, we can
define a functor
$\mathcal{F}\colon\mathcal{SL_{M}}\longrightarrow\mathcal{L}^{{\dagger}}$
where $\mathcal{F}$ assigns to each object
$\big{(}L(M),\wedge,\vee,\subseteq\big{)}$ the object
$(\mathcal{F}_{i}(L(M)),\vee,\subseteq,i)$ and assigns to each morphism
$\mathcal{X}_{\eta}$ the join semi-lattice morphism $\mathcal{F}(\eta)$. As we
now see, the entropy for objects and endomorphisms within the category
$\mathcal{SL_{M}}$ is equivalent to that defined on objects and endomorphisms
within $R$-Mod:

######  Definition 4.14.

Let $i$ be a subadditive invariant on $R$-Mod, $M$ an $R$-module and $L(M)$
the complete lattice of all submodules of $M$. If $\eta:M\longrightarrow M$ is
an endomorphism and $\mathcal{X}_{\eta}$ is the induced endomorphism of
$L(M)\in\mathcal{SL_{M}}$, then the algebraic $i$-entropy of
$\mathcal{X}_{\eta}$ with respect to $N\in\mathcal{F}_{i}(L(M))$ is

$H_{i}(N,\mathcal{X}_{\eta})=\lim_{n\to\infty}\frac{i(T_{n}(N,\mathcal{X}_{\eta}))}{n}$.

Hence, the algebraic $i$-entropy of $\mathcal{X}_{\eta}$ is

$ent_{i}(\mathcal{X}_{\eta})=sup\Big{\\{}H_{i}(N,\mathcal{X}_{\eta})\mid
N\in\mathcal{F}_{i}(L(M))\Big{\\}}$.

A first consequence of Definition 4.14 is that the entropy for a complete
modular lattice of the form $L(M)\in\mathcal{SL_{M}}$ (with $M\in R$-Mod) is:

$ent_{i}(L(M))=sup\big{\\{}ent_{i}(\mathcal{X}_{\eta})\mid\mathcal{X}_{\eta}\in
End_{\mathcal{SL_{M}}}(L(M))\big{\\}}$

$=sup\big{\\{}ent_{i}(\mathcal{X}_{\eta})\mid\eta\in End(M)\big{\\}}$.

Therefore, as $N\in\mathcal{F}_{i}(L(M))$ if, and only if,
$N\in\mathcal{F}_{i}(M)$ and the fact that $\mathcal{X}_{\eta}(N)=\eta(N)$, it
follows the next

######  Proposition 4.15.

Let $M\in R\textit{-}Mod$ and $L(M)\in\mathcal{L_{M}}$. Then

$ent_{i}(M)=ent_{i}(L(M))$.

Furthermore, according to [9, Theorem 5.26], we have that for each
endomorphism $M\overset{\eta}{\longrightarrow}M$ in $R$-Mod,

$\textbf{h}_{\mathcal{S}}(\eta)=ent_{i}(\eta)=ent_{i}(\mathcal{X}_{\eta})=\textbf{h}_{\mathcal{F}}$,

where $\textbf{h}_{\mathcal{S}}$ is the functorial entropy defined by the
functor $\mathcal{S}:R\textit{-}Mod\longrightarrow\mathcal{L}^{{\dagger}}$ and
$\textbf{h}_{\mathcal{F}}$ is the functorial entropy defined by the functor
$\mathcal{F}:\mathcal{SL_{M}}\longrightarrow\mathcal{L}^{{\dagger}}$.

Considering the above, we will define entropy for lattice preradicals on
objects of the category $\mathcal{SL_{M}}$, and we will show that this
definition is compatible with the definition of entropy for preradicals on
$R$-Mod.

######  Definition 4.16.

Let $i$ be a subadditive invariant on $R$-Mod, $M$ an $R$-module, $L(M)$ the
complete lattice of all submodules of $M$, and $\tilde{\sigma}$ a lattice
preradical on $\mathcal{L_{M}}$. Then the algebraic $i$-entropy of
$\tilde{\sigma}$ with respect to $L(M)\in\mathcal{SL_{M}}$ is

$ent_{i}(\tilde{\sigma})|_{L(M)}\colon=\underset{\mathcal{X}_{\eta}\in
End_{\mathcal{SL_{M}}}(L(M))}{sup}\Big{\\{}H_{i}(N,\mathcal{X}_{\eta})\mid
N\in\mathcal{F}_{i}(\tilde{\sigma}(L(M)))\Big{\\}}$

where
$H_{i}(N,\mathcal{X}_{\eta})=\lim_{n\to\infty}\frac{i(T_{n}(N,\mathcal{X}_{\eta}))}{n}$.

######  Proposition 4.17.

Let $i$ be an invariant on $R$-Mod, and let
$\phi\colon\mathcal{L_{M}}\textit{-}pr\longrightarrow R\textit{-}pr$. If
$\tilde{\sigma}\in\mathcal{L_{M}}\textit{-}pr$ and
$\phi(\tilde{\sigma})=\sigma$ denotes its induced preradical on $R$-pr, then
given $M\in R\textit{-}Mod$ and $L(M)\in\mathcal{SL_{M}}$ one has that

$ent_{i}(\sigma)|_{M}=ent_{i}(\tilde{\sigma})|_{L(M)}$.

######  Proof.

First observe that $N\in\mathcal{F}_{i}(\sigma(M))$ if, and only if,
$N\in\mathcal{F}_{i}(\tilde{\sigma}(L(M)))$. Indeed, by definition of
$N\in\mathcal{F}_{i}(\sigma(M))$ one has that $N$ is a submodule of
$\sigma(M)=X_{L(M)}^{\tilde{\sigma}}$ with $i(N)<\infty$. Recall that
$X_{L(M)}^{\tilde{\sigma}}$ is such that

$\tilde{\sigma}(L(M))=\Big{(}X_{L(M)}^{\tilde{\sigma}}/0\Big{)}$.

This in turn implies that $N$ is an element of the lattice
$\Big{(}X_{L(M)}^{\tilde{\sigma}}/0\Big{)}$, and hence
$N\in\mathcal{F}_{i}(\tilde{\sigma}(L(M)))$.
Conversely, if $N\in\mathcal{F}_{i}(\tilde{\sigma}(L(M)))$ then $i(N)<\infty$
and $N$ is an element of the lattice
$\Big{(}X_{L(M)}^{\tilde{\sigma}}/0\Big{)}$. Since by definition
$X_{L(M)}^{\tilde{\sigma}}=\sigma(M)$, then $N\leq\sigma(M)$ and thus,
$N\in\mathcal{F}_{i}(\sigma(M))$.

Lastly, since

$H_{i}(N,\eta)=\lim_{n\to\infty}\frac{i(T_{n}(N,\eta))}{n}=\lim_{n\to\infty}\frac{i(T_{n}(N,\mathcal{X}_{\eta}))}{n}=H_{i}(N,\mathcal{X}_{\eta}),$

for each endomorphism $M\overset{\eta}{\longrightarrow}M$ in $R$-Mod and its
induced endomorphism $L(M)\overset{\mathcal{X}_{\eta}}{\longrightarrow}L(M)$
in $\mathcal{SL_{M}}$, we have that

$ent_{i}(\sigma)|_{M}=\underset{\eta\in
End(M)}{sup}\Big{\\{}H_{i}((N,\eta)\mid
N\in\mathcal{F}_{i}(\sigma(M))\Big{\\}}$

$=\underset{\mathcal{X}_{\eta}\in
End_{\mathcal{SL_{M}}}(L(M))}{sup}\Big{\\{}H_{i}(N,\mathcal{X}_{\eta})\mid
N\in\mathcal{F}_{i}(\tilde{\sigma}(L(M)))\Big{\\}}$

$=ent_{i}(\tilde{\sigma})|_{L(M)}$.

∎

Finally, since the assignment
$\phi\colon\mathcal{L_{M}}\textit{-}pr\longrightarrow R\textit{-}pr$ preserves
the four operations between preradicals (see [11, Section 8]), it is
straightforward the next

######  Corollary 4.18.

Let $i$ be an invariant on $R$-Mod, and let
$\tilde{\sigma}\in\mathcal{L_{M}}\textit{-}pr$ with
$\phi(\tilde{\sigma})=\sigma\in R\textit{-}pr$ its induced preradical under
the assignment $\phi$. If $M\in R\textit{-}Mod$ and $L(M)\in\mathcal{SL_{M}}$
then

  * (i)

$ent_{i}(\tilde{\sigma}\wedge\tilde{\tau})|_{L(M)}=ent_{i}(\sigma\wedge\tau)|_{M}$

  * (ii)

$ent_{i}(\tilde{\sigma}\vee\tilde{\tau})|_{L(M)}=ent_{i}(\sigma\vee\tau)|_{M}$

  * (iii)

$ent_{i}(\tilde{\sigma}\cdot\tilde{\tau})|_{L(M)}=ent_{i}(\sigma\cdot\tau)|_{M}$

  * (iv)

$ent_{i}(\tilde{\sigma}:\tilde{\tau})|_{L(M)}=ent_{i}(\sigma:\tau)|_{M}$

######  Corollary 4.19.

Let $i$ be an invariant on $R$-Mod, and let
$\tilde{\sigma}\in\mathcal{L_{M}}\textit{-}pr$ with
$\phi(\tilde{\sigma})=\sigma\in R\textit{-}pr$ its induced preradical under
the assignment $\phi$. If $M\in R\textit{-}Mod$ and $L(M)\in\mathcal{SL_{M}}$
then

$ent_{i}(\tilde{\sigma}\cdot\tilde{\tau})|_{L(M)}\leq_{\mathbb{R}}ent_{i}(\tilde{\sigma}\wedge\tilde{\tau})|_{L(M)}\leq_{\mathbb{R}}ent_{i}(\tilde{\sigma}\vee\tilde{\tau})|_{L(M)}\leq_{\mathbb{R}}ent_{i}(\tilde{\sigma}:\tilde{\tau})|_{L(M)}$.

######  Proof.

It follows from Corollary 4.7 and Corollary 4.18 ∎

## Acknowledgments

This work was supported by unrestricted funds to the Center for Engineered
Natural Intelligence at the University of California San Diego.

## References

  * [1] Adler R., Konheim, A., McAndrew M.; Topological entropy; Trans. Am. Math. Soc.; 114; (1965); 309–319.
  * [2] Albu T.; Iosif M.; The category of linear modular lattices,  Bull. Math. Soc. Sci. Math. Roum. 56 (104); (2013); 33-46.
  * [3] Albu T., Iosif M.; Lattice preradicals versus module preradicals, Ann. Univ. Buchar. Math. Ser. 6(1); (2015); 19–34.
  * [4] Giordano B., Salce L.; A soft introduction to algebraic entropy; Arab. J. Math; 1(1); (2012); 69–87.
  * [5] Dikranjan D., Giordano Bruno A.; Entropy on abelian groups; Adv. Math; 298, (2016); 612-653.
  * [6] Dikranjan D., Goldsmith B., Salce L., Zanardo P.; ; ebraic; Entropy for Abelian Groups; Transac. Amer Math Soc.; 361; (2009); 3401-3434.
  * [7] Bican L., Kepka T., Němec P., Rings, Modules and Preradicals , Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York; (1982).
  * [8] Dikranjan D., Giordano B.; Entropy in a Category, Appl Categor Struct, 21; (2013); 67-101.
  * [9] Dikranjan D., Bruno A. G.; Entropy on normed semigroups;  Disser. Mathem.; 542; (2019); 1-90.
  * [10] Fernández-Alonso R., Gavito S., Pérez-Terrazas J; On the connection between the representation type of an algebra and its lattice of preradicals; Comm. in Alg., 46(1); (2018); 176-190.
  * [11] Pardo-Guerra S., Rincón H., Zorrilla-Noriega M.; Some isomorphic big lattices and some properties of lattice preradicals,  J. Algebra Appl.; 19; (2020); 2050140.
  * [12] Peters, J; Entropy on discrete Abelian groups; Adv. Math. 33; (1979); 1–13.
  * [13] Raggi F., Ríos J., Rincón H., Fernández-Alonso R., Signoret C.; The lattice structure of preradicals;  Comm. Alg; 30(3); (2002); 1533-1544.
  * [14] Raggi, F., Ríos J., Rincón H., Fernández-Alonso R., Signoret C.; The lattice structure of preradicals II: Partitions; J. Alg. Appl; 1; (2002); 201-214.
  * [15] Salce L., Zanardo P.; A general notion of algebraic entropy and the rank-entropy; Forum Mathem. 21(4), (2009); 579-599.
  * [16] Stenström, Bo.; Rings of quotients, An introduction to methods of ring theory, Die Grundlehren der Mathematischen Wissenschaften, Band 217. Springer-Verlag, New York-Heidelberg; (1975).
  * [17] Silva G., Pardo-Guerra S., Information flows as a path of least resistance: A category theory approach using preradicals, arXiv:2012.02886 [math.CT].
  * [18] Ward T.; Entropy of compact group automorphisms, Notes for the Graduate course, Ohio State University, (1994).
  * [19] Weiss, M.; Algebraic and other entropies of group endomorphisms; Math. Syst. Theo.; 8(3); (1974); 243–248.