# Computer-supported Exploration of a Categorical Axiomatization of Modeloids

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraïssé games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fraïssé games.

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## 1 Introduction

Modeloids have been introduced by M. Benda [1]. They can be seen as an abstraction from a structure to a partial automorphism semigroup created in the attempt to study properties of structures from a different, more general angle which is independent of the language that is defining the structure. We do not follow Benda’s original formulation in terms of an equivalence relation but treat modeloids as a certain set of partial bijections. Our recent interest in them was triggered by D. Scott’s suggestion to look at the modeloidal concept form a categorical perspective. The new approach aims at establishing a framework in which the relationship between different structures of same vocabulary can be studied by means of their partial isomorphisms. The overall project is work in progress, but as a first result we obtained a purely algebraic formulation of Ehrenfeucht-Fraïssé games.

Throughout the project, computer-based theorem proving is employed in order to demonstrate and explore the virtues of automated and interactive theorem proving in context. The software used is Isabelle/HOL [12] in the 2019 Edition. We are generally interested in conducting as many proofs of lemmas and theorems as possible by using only the sledgehammer command [3], and to study how far full proof automation scales in this area. Reporting on these practically motivated studies, however, will not be the focus of this paper. We only briefly mention here how we encoded, in Isabelle/HOL, an inverse semigroup and an inverse category, and we present a summary of our practical experience.

Inverse semigroups (see e.g. [9] for more information) play a major role in this paper. They serve as a bridge between modeloids and category theory. The justification for this is given by the fact that an inverse semigroup can be faithfully embedded into a set of partial bijections by the Wagner-Preston representation theorem. This opens up the possibility of generalizing modeloids, which are sets of partial bijections, to the language of inverse semigroup theory.

Once there, we have a natural transition from an inverse semigroup to an inverse category (for further reference see [11]). We introduce the theory of inverse categories by an equational axiomatization that enables computer-supported reasoning. This serves as the basis for our formulation of a categorical modeloid.

In each stage of generalization the derivative, a central operation in the theory of modeloids, can be adapted and reformulated. This operation is essentially about the possibility of extending the elements of a modeloid. As it turns out, the derivative on a categorical modeloid on the category of -structures, where is a finite relational vocabulary, is equivalent to playing an Ehrenfeucht-Fraïssé game.

This paper is organized in the following way. In section 2 we define a modeloid and encounter the derivative operation. We then turn to inverse semigroups in section 3 and develop the axiomatization of a modeloid in inverse semigroup language. Section 4 shows how to represent a category in Isabelle/HOL and defines the categorical modeloid. After the derivative operation is established in this context, we give an introduction to Ehrenfeucht-Fraïssé games in Section 5 and present the close connection between the categorical derivative and Ehrenfeucht-Fraïssé games. Proofs for the stated theorems, propositions and lemmas are presented in the Appendix (cf. also [14]) and the Isabelle/HOL source files are available online.

## 2 Modeloids

As a start let be a finite non-empty set. We then define

 F(Σ):={f:Σ→Σ∣f is a partial bijection} (1)

as the set of all partial bijections on . It is the set that we define a modeloid on.

###### Definition 1 (Modeloid [1])

Let . is called a modeloid on if, and only if, it satisfies the following axioms:

1. Closure of composition:

2. Closure of taking inverses:

3. Inclusion property: and implies

4. Identity:

As such, a modeloid is a set of partial bijections which is closed under composition and taking inverses, which has the identity on as a member, and which satisfies the inclusion property. The inclusion property can be seen a downward closer in regards of function restriction.

In order to further illustrate the definition, we present a motivating example from model theory.

###### Example 1

Let be a relational finite structure. The set of all partial isomorphisms on forms a modeloid.

Note that the name modeloid originates from the above example since is also called a model. For further motivation, background information and details on modeloids, we refer to Benda’s paper [1]; a nice example in there is the construction of a Scott Sentence presented through modeloidal glasses [1, p. 82]. We, on the other hand, turn to the core concept of the derivative which is defined in the following way.

###### Definition 2 (Derivative)

Let be a modeloid on . Then the derivative is defined by

 {(x1,y1),..,(xn,yn)}∈D(M)\vcentcolon⇔∀a∈Σ∃b∈Σ:{(x1,y1),..,(xn,yn),(a,b)}∈M∧∀a∈Σ∃b∈Σ:{(x1,y1),..,(xn,yn),(b,a)}∈M

A derivative is thus a set which only contains partial bijections that can be extended by an arbitrary element from and which then still belong to . This extension can take place either in the domain or in the range of the function. The next two results give some insight into why modeloids and the derivative operation are in harmony.

###### Lemma 1

Let be a modeloid on and the derivative. Then we have that .

###### Proposition 1

If is a modeloid then so is .

The importance of these results is essentially due to the fact that they enable us to apply the derivative several times.

## 3 Inverse Semigroups and Modeloids

In this section we show how the Wagner-Preston representation theorem justifies our generalization of a modeloid to inverse semigroup language. We also discuss how well proof automation performs in the context of inverse semigroups. Some familiarity with the Isabelle/HOL proof assistant [12, 3] is assumed.

### 3.1 Inverse Semigroups in Isabelle/HOL

###### Definition 3 (Inverse semigroup [5])

Let be a set equipped with the binary operation and the unary operation . is called an inverse semigroup if, and only if, it satisfies the axioms

1. for all ,

2. for all ,

3. for all and

4. for all

We encode this definition as follows in Isabelle/HOL.

[width=]InverseSemigroup

The domain for individuals is chosen to be , which is a type variable. This means we have encoded a polymorphic version of inverse semigroups.

Using this implementation almost all results needed for proving the Wagner-Preston representation theorem, which we will discuss shortly, can be found by automated theorem provers. Occasionally, however, some additional lemmas to the ones usually presented in a textbook (e.g. [9]) are needed. By automated theorem proving we here mean the use of sledgehammer [3] for finding the proofs of the given statements without any further interaction. Regarding equivalent definitions of an inverse semigroup, we were able to automate the proofs of the following theorem (except for , which is due to a Skolemization issue).

###### Theorem 3.1 ([9])

Let be a semigroup. Then the following are equivalent:

1. is an inverse semigroup.

2. Every element of has a unique inverse.

3. Every element of is regular, meaning , and idempotents in commute.

Our experiments confirm that automated theorem proving (and also model finding) can well support the exploration of an axiomatic theory as presented. However, the intellectual effort needed to model and formulate the presented mathematics in the first place is of course still crucial, and a great deal of work has gone into this intuitive aspect of the development process. A more technical challenge also is to find suitable intermediate steps that can be proven by sledgehammer fully automatically.

### 3.2 Modeloid as Inverse Semigroup

We now show that every modeloid is an inverse semigroup. We make use of Theorem 3.1 by using the third characterization. For this task regard as a semigroup. This is clear since composition of partial functions is associative. Since the partial identities of are exactly the idempotent elements in , commutativity is ensured by referring to the next proposition. Furthermore, also by using the next proposition, the closure of taking inverses required by a modeloid implies regularity for all elements in . Hence, is an inverse semigroup.

###### Proposition 2 ([9])

Let and be sets, and let be a partial bijection.

1. For a partial bijection , the equations and hold if, and only if,

2. for all partial identities and where

Not only is every modeloid an inverse semigroup, but by the Wagner-Preston representation theorem also every inverse semigroup can be faithfully embedded into , which is itself a modeloid. This motivates the idea of formulating the axioms for a modeloid in inverse semigroup language. Our aim is to restate the derivative operation in this context. In order to achieve this, we shall translate the axioms from Definition 1, examining them one by one.

1. Closure of Composition: Because of the embedding, the composition of partial functions will simply be the -operation in an inverse semigroup.

2. Closure of taking inverses: By Theorem 3.1 an inverse semigroup is such that the inverse exists for every element and is unique, hence resembling the inverses of partial functions and in particular the closure property.

3. The inclusion property: Here it is not apparent from first sight how this can be expressed within an inverse semigroup. We shall see that the natural partial order is capable of exactly that.

4. The identity on : The identity sure will be an idempotent element in an inverse semigroup. It will lead us to the notion of an inverse monoid.

It is Axiom 3 that we focus our attention on next. We state the natural partial order and will then see that the Wagner-Preston representation theorem establishes a connection to function restriction in . We introduce notation for such a restriction. For two partial functions we write to say that and .

###### Definition 4 (Natural partial order)

Let be an inverse semigroup. Let . We define

 s≤t\vcentcolon⇔s=t∗e

for some idempotent .

###### Theorem 3.2 (Wagner-Preston representation theorem [9])

Define to be an inverse semigroup. Then there is an injective homomorphism such that for

 a≤b⟺Ω(a)⊆Ω(b).

From this theorem it is clear what we mean by a faithful embedding of an inverse semigroup into the set of partial bijections . Faithfulness corresponds to the fact that the natural partial order in light of the representation theorem is equivalent to the partial order which function restriction defines. This nicely opens up the possibility to capture the essence of the inclusion property from Definition 1 by the natural partial order.

With the new notation we write this property as

 (f∈M∧g⊆f)⟹g∈M. (2)

At this point the above axiom is in a form that can quite naturally be seen in inverse semigroup language. But if we consider , the problem arises that the domain of is not given explicitly anymore in an inverse semigroup. Therefore, it is necessary to state what is an element of. Indeed, the above statement can be written as

 ∀f∈M∀g∈F(Σ):g⊆f⟹g∈M. (3)

In this form the dependency of a modeloid on is seen explicitly. In translating 3 this has to be part of the statement. These considerations lead to the fact that a modeloid in semigroup theory is the subset of an inverse semigroup. So let where is an inverse semigroup. Then 3 can be stated as

 ∀f∈M∀g∈S:g≤f⟹g∈M. (4)

It is immediate that a modeloid, seen as an inverse semigroup, fulfills 4 by the following proposition.

###### Proposition 3

Let be a modeloid on . Then for

 g≤f⟺g⊆f

In a modeloid the inclusion property implies that the empty partial bijection, which we denote by , is also included in . As a result we want to establish a similar behavior in the generalized modeloid. The deeper reason for this is found in the definition of the derivative operation. Seeing as an inverse semigroup is an idempotent element for which the following property holds: . Hence, we will call the idempotent with this property the zero element. When defining a modeloid in semigroup language we require the zero element to be part of it.

Turning to Axiom 4, which is , we examine which element of an inverse semigroup is most suitable for this task. To evaluate we again look at the modeloid regarded as an inverse semigroup. In this semigroup will be an idempotent satisfying . Such an element is known as a neutral element in the context of group theory. We require for the inverse semigroup, which we eventually call a modeloid, that is part of it. What we get is known as an inverse monoid in the literature.

###### Remark 1

Given an inverse monoid, denoted by , and the element with consider the representation theorem again. This theorem does not give uniqueness of the embedding and in fact there can be several. As a result we can not suppose that will be mapped to the identity . However, for all idempotent we have that . Hence, is an element that always resembles the upper bound of all idempotents in .

We have prepared everything needed for defining a modeloid again. We shall call it a semimodeloid. Note, as mentioned before, that a modeloid is a subset of for some finite non-empty set and, as discussed, we keep this subset property to state the inclusion axiom.

###### Definition 5 (Semimodeloid)

Let be a finite inverse monoid. Then is called a semimodeloid if, and only if,

###### Remark 2

A semimodeloid is again an inverse monoid with the zero element.

###### Proposition 4

Every semimodeloid can be faithfully embedded into a modeloid. Furthermore, by the considerations above, every modeloid is a semimodeloid.

Now we develop the derivative operation in the setting of a semimodeloid. Consider again Definition 2 in which we have introduced the derivative operation. It is evident that the elements of are of crucial importance. Furthermore, we are required to be able to extend the domain of a function by one element at a time. This poses a challenge because in an inverse monoid this information is not directly accessible. But as we shall see, it is possible to obtain.

First we characterize the elements of . Therefore, consider and realize that all the singleton-identities for are in natural bijection to the elements of . The special property of such a singleton-identity is that

 ∀f∈F(Σ):f⊆id{a}⇒(f=id{a}∨f=0) (5)

since . Seeing as an inverse monoid with zero element leads to the following definition.

###### Definition 6 (atomic)

Let be an inverse monoid with zero element . Then a non-zero element is called atomic if, and only if,

 ∀f∈S1:f≤x⇒(f=x∨f=0)

Our plan is to use the notion of atomic to define the derivative. The next lemma justifies this usage.

###### Lemma 2

The idempotent atomic elements in are exactly the singleton-identities.

This suffices to define the derivative for semimodeloids. We then ensure that the definition matches Definition 2 if the semimodeloid is on .

###### Definition 7 (Derivative on semimodeloid)

Let be a semimodeloid on the inverse monoid with zero element . Then we define the derivative of to be

 D(M):={f∈M|∀ idempotent and atomic a∈S1∃x∈M:(f≤x∧a≤x−1x)∧∀ idempotent and atomic b∈S1∃y∈M:(f≤y∧b≤yy−1)}
###### Proposition 5

The derivative on a modeloid produces the same result as the semimodeloidal derivative on .

With this result we conclude this section and move on to the categorical setting.

## 4 Categorical Axiomatization of a Modeloid

We use an axiomatic approach to category theory using free logic [13, 7, 8] enabling an implementation in Isabelle/HOL which was proposed by Benzmüller and Scott [2]. This encoding is extended to represent an inverse category. We then formulate a modeloid and its derivative in this setting.

### 4.1 Category Theory in Isabelle/HOL

When looking at the definition of a category , one can realize that the objects are in natural bijection with the identity morphisms because those are unique. This enables a characterization of a category just by its morphisms and their compositions which is used to establish a formal axiomatization. However, in this axiomatic approach one has to deal with the challenge of partiality because the composition between two morphisms is defined if, and only if,

 dom(g)=cod(f). (6)

As a result the composition is a partial operation.

An elegant way to deal with this issue is by changing the underlying logic to free logic. We introduce an explicit notion of existence for the objects in the domain that we quantify over. In our case the domain consists of the morphisms of a category. The idea now is to define the composition total, that is, any two morphisms can always be composed, but only those compositions “exist” that satisfy 6. Because we can distinguish between existing and non-existing morphisms, we are able to formulate statements that take only existing morphisms into account. Due to the achievement of finding a shallow embedding of free logic in Isabelle/HOL by Benzmüller and Scott, first order axiomatic category theory could also be implemented. We refer to [2] for more information.

Using this work, a category in Isabelle/HOL is defined as follows.

[width=]DefinitionCategory

Due to our construction we are tied to small categories. Therefore, we use notation from set theory. As a result, a category for us consist only of a set of morphisms which satisfy the above axiom schema. For notation, we may write to mean that is a morphism from the category . In addition, it says that and , so is the domain of and the codomain. The identity morphisms and , which are representing objects in the usual sense, are characterized by the property that , respectively for . Hence every satisfying or is representing an object and we refer to such a morphism as an object.

We want a categorical generalization of an inverse semigroup, so let’s turn to the question on how to introduce generalized inverses to a category. In the above setting we found that by adding the axioms of an inverse semigroup, which are responsible for shaping these inverses (Definition 3, Axioms 2-4), we arrive at a notion that is equivalent to the usual definition of an inverse category. Note that this definition is adopted by us to the usage with free logic by using the Kleene equality denoted by . We emphasize again that this equality between terms states that, if either term is existing, so is the other one and they are equal.

###### Definition 8 (Inverse category [6])

A small category is called an inverse category if for any morphism there exists a unique morphisms such that and .

For the representation in Isabelle/HOL we skolemized the definition.

[width=]InverseCategory1

Next, we see the quantifier free definition.

[width=]InverseCategory2

The equivalence between the two formulations has been shown by interactive theorem proving. Again, a significant number of the required subproofs could be automated by sledgehammer.

For us the setting of an inverse category is interesting because of the following proposition.

###### Proposition 6

Let be an inverse category with exactly one object. Then is an inverse semigroup.

This allows us to generalize a semimodeloid to an inverse category by formulating the new axioms in a way that this categorical construction will collapse to a semimodeloid under the condition of having just one object.

### 4.2 Categorical Axiomatization of a Modeloid

The notion of the natural partial order is rediscovered in an inverse category. To state it, we first introduce a definition for idempotence.

###### Definition 9 (Idempotence)

Let be a small category. Then a morphism is called idempotent if, and only if,

 e⋅e≅e.

Whenever we do not assume that both sides of the equation exist then we use the Kleene equality.

###### Definition 10 (Natural partial order [11])

Let be an inverse category and let be morphisms in . Then we define

 s≤t\vcentcolon⇔∃ idempotent e∈EndC(X):s≅t⋅e

where is called an Endoset.

When defining a categorical modeloid on an inverse category , we will see that for each object in , is a semimodeloid. As in the case of a semimodeloid we require the inverse category to be finite meaning that the underlying set is finite. We also require the category to have a zero element in each of its Endosets. For this we simply write that has all zero elements.

###### Definition 11 (categorical modeloid)

Let be a finite inverse category with all zero elements. Then a categorical modeloid on is such that satisfies the following axioms.

It is evident that this definition is close by its appearance to a semimodeloid. However, we are now dealing with a network of semimodeloids and have thus reached a much more expressive definition.

###### Proposition 7

Let be a finite inverse category with all zero elements and be a categorical modeloid on . Then for each object in we get that is a semimodeloid (on itself).

###### Remark 3

Every semimodeloid can easily be seen as a categorical modeloid by the fact that an inverse monoid with zero element is an one-object inverse category.

We have achieved to formulate a generalization of a modeloid in category theory. What is left now is to define the derivative in this context. We will need the notion of a Homset and of an atomic element, which we already introduced for semigroups.

###### Definition 12 (Homset)

Let be a small category. Then the Homset between two elements , satisfying and , is defined as

 HomC(X,Y):={m∈C|m:X→Y}

Hence an Endoset is a special case of a Homset. We only assume zero elements to be present in Endosets and as a result an atomic element needs to be part of an Endoset.

###### Definition 13 (Atomic)

Let be an inverse category with all zero elements. Then an element for some object is called atomic if, and only if, the existence of implies that is not the zero element and

 ∀e∈EndC(X):e≤a implies that e≅a∨e≅0EndC(X).

This concludes the preliminaries for defining the derivative on a Homset.

###### Definition 14 (Derivative on Homset)

Let be a finite inverse category with all zero elements and let be a categorical modeloid on . We define the derivative on for as

 D(HomM(X,Y)):={f∈HomM(X,Y)|∀ idempotent atomic a∈EndM(X)∃h∈HomM(X,Y):(f≤h∧a≤h−1h)∧∀ idempotent atomic b∈EndM(Y)∃g∈HomM(X,Y):(f≤g∧b≤gg−1)}
###### Remark 4

Done on a finite inverse category with just one object and a zero element by Proposition 5 reduces to the definition of the derivative on a semimodeloid.

Now the key property of this operation is that it produces a categorical modeloid again if we apply it to all Homsets simultaneously.

###### Theorem 4.1

Let be a finite inverse category with all zero elements and let be a categorical modeloid on . Then

 ⋃X,Y∈MD(HomM(X,Y))

is a categorical modeloid on .

As a result we define this to be the derivative operation on categorical modeloids.

###### Definition 15 (Derivative on a categorical modeloid)

Let be an inverse category with all zero elements and let be a categorical modeloid on . Then we set the derivative as

 D(M):=⋃X,Y∈MD(HomM(X,Y)).

At this point we also explore what it means to take the derivative -times because it will be needed in the next section. This is, however, straight forward. Let be a categorical modeloid. Then we define

 D0(M):=M and Dn+1(M):=D(Dn(M)) (7)

for . As a result will state to take the derivative -times. We shall investigate what our established framework is capable of in finite model theory.

## 5 Algebraic Ehrenfeucht-Fraïssé games

When moving from classical model theory to the finite case, some machinery for proving inexpressibility results in first-order logic, such as the compactness theorem, fails. However, Ehrenfeucht-Fraïssé (EF) games are still applicable and, therefore, play a central notion in finite model theory due to the possibility to show that a property is first-order axiomatizable. For more information see [10].

In this section we explicitly show the connection that derivatives on categorical modeloids and EF games share.

### 5.1 Rules of EF game

To play an EF game, two -structures and , where is a finite relational vocabulary, are needed. Note, that an EF game is not restricted to the finite case but for our purpose we shall only deal with this case. In order to give an intuitive understanding we imagine two players, which we call the spoiler and the duplicator, playing the game. The rules are quite simple. In rounds the spoiler tries to show that the two structures are not equal while the duplicator tries to disprove the spoiler every time. A round consists of the following:

• The spoiler picks either or and then makes a move by choosing an element from that structure, so or .

• After the spoiler is done the duplicator picks an element of the other structure and the round ends.

Next we define what the winning condition for each round will be. For convenience let be the set of all partial isomorphisms from to .

###### Definition 16 (Winning position [10])

Suppose the EF game was played for rounds. Then there are moves picked from and picked from . For this to be a winning position we require that the map

 {(a1,b1),..,(an,bn),(cA1,cB1),..,(cAr,cBr)}∈Part(A,B)

where the are all constant symbols of interpreted by the structure and likewise for the structure .

In order to win, the duplicator needs to defeat the spoiler in every possible course of the game. We say the duplicator has an n-round winning strategy in the Ehrenfeucht-Fraïssé game on and [10] if the duplicator ends the game in a winning position regardless of what the spoiler does. This is made precise by the back-and-forth method due to Fraïssé.

###### Definition 17 (back-and-forth relation [4])

We define a binary relation on all -structures by iff there is a sequence for such that

• Every is a non-empty set of partial isomorphisms from to

• (Forth property) we have

• (Back property) we have

Hence means that the duplicator has a -round winning strategy.

### 5.2 The derivative and Fraïssé’s method

We relate the categorical derivative to Fraïssé’s method which we have just seen. In order to do this, we define a categorical modeloid on the category of -structures, where is a finite relational vocabulary. For that let and be two -structures. Denote by the set

 ⋃(X,Y)∈{A,B}2Part(X,Y)

and let be an arbitrary element. Then define .

We construct two functions and such that for a partial isomorphism we set and and for the element we define and .

Next we define a binary operation by

 f⋅g={f∘g,if$dom(f)=cod(g)$and$f,g≠⋆%$⋆,else

where denotes the composition of partial functions.

###### Proposition 8

is an inverse category where denotes the inverse of each partial isomorphism and . The existing elements are exactly all elements in and the compositions in case for .

What we have just seen shows a general procedure for creating a category in our sense of having free logic underlying the definition.

###### Corollary 1

is also a categorical modeloid on itself.

###### Remark 5

Hence we have that every inverse category having a zero element for each of its Endosets is also a categorical modeloid and thus admits a derivative.

At this point we are able to use the derivative on . The final theorem draws the concluding connection between modeloids and Fraïssé’s method. We show that in the established setting an -round winning strategy between and is given by the sets which the derivative produces if applied times. Mind the abuse of notation in the way we are using here.

###### Theorem 5.1

Let be the categorical modeloid . Then

 ∃h:X→Y∈Dm(M) with h≠⋆⟺X≡mY,m∈N

## 6 Conclusion

In this paper we have shown how to arrive at the notion of a categorical modeloid using axiomatic category theory. We started out with a set of partial bijections abstracting from a structure, then interpreted this set as an inverse semigroup by the embedding due to the Wagner-Preston representation theorem, and, finally, we were able to axiomatize a modeloid in an inverse category. The key feature needed is the natural partial order which also enabled us to present the derivative operation in each step of abstraction. The categorical derivative on the category of structures of a finite vocabulary can then be used to play an Ehrenfeucht-Fraïssé game between two structures. As a result a more abstract representation of these games is possible.

Using the inverse category presented in Isabelle/HOL, we are currently working on implementing a categorical modeloid together with its derivative operation. This naturally results in formulating basic definitions from category theory in the framework established so far [2]. Furthermore, an investigation of the, in a sense, generalized Ehrenfeucht-Fraïssé games in terms of applicability has to be conducted. We believe that the notion of a categorical modeloid will continue to play a role when connecting model theoretical and categorical concepts.

## Appendix

This appendix provides the proofs of all the claims we made during the paper. We will go through them in the order in which they appear.

### 6.1 Proofs of section 2

###### Lemma 1

Let be a modeloid on and the derivative. Then we have that .

###### Proof

We want to show . Fix . Next fix . We know we can find such that . Since is a modeloid, the inclusion property implies .

###### Proposition 1

If is a modeloid then so is .

###### Proof

We will prove the modeloidal axioms for in the order in which they appear in the definition of a modeloid.

Hence, we start with the closure of composition. Let . We need to show that for a fixed we can find such that . The second conjunct of the derivative follows by analogy. Therefore, fix . We can find such that . Then we can find such that . Because is closed under the composition, it follows that . As a result, is closed under composition.

Up next is the closure of taking inverses. Let and fix . Because of the second conjunct of the derivative, we get that for some the statement holds. Since is a modeloid, this implies that . By analogy we also get . Thus .

The inclusion property is evident. Fix and a satisfying .

 ∀a∈Σ∃b∈Σ:f∪{(a,b)}∈M

immediately implies

 ∀a∈Σ∃b∈Σ:g∪{(a,b)}∈M.

Analogously we get and hence .

The fact that can be seen by noting and . This concludes the proof.

### 6.2 Proofs of section 3

###### Proposition 2

Let be a modeloid on . Then for

 g≤f⟺g⊆f
###### Proof

Let be an alphabet and a functional modeloid on . We have already established that is an inverse semigroup. Fix . Supposing that holds we know where is an idempotent. As such is a partial identity in . As a result

 dom(g)=e−1(dom(f)∩cod(e))=dom(f)∩cod(e)

and hence . Furthermore, for . This yields .
Conversely suppose that . Since we know that . In addition, the partial identity is idempotent. Now and because since and for . As a result holds.

###### Proposition 3

Every semimodeloid can be faithfully embedded into a modeloid. Furthermore, by the considerations above, every modeloid is a semimodeloid.

###### Proof

This proposition basically summarizes the work done. By taking the modeloid , it is clear that every semimodeloid on can be faithfully embedded into it by the Wagner-Preston representation theorem because the inclusion property only depends on the faithfulness of the embedding with respect to the natural partial order. Proposition 2 establishes that every modeloid is a semimodeloid.

###### Lemma 2

The idempotent atomic elements in are exactly the singleton-identities.

###### Proof

Idempotent elements in are the partial identities. So it suffices to show that if and only if the idempotent is atomic. Assume is atomic and idempotent. Suppose now that . Then we can find with . But then we have that and . This is a contradiction to atomic. The case implies that but an atomic element is unequal to the zero element. As such that case is also taken care of.
Conversely assume for some non-zero partial identity . Then implies that . If it is the first option we have . And if it is the second we get .

###### Proposition 4

The derivative on a modeloid produces the same result as the semimodeloidal derivative on .

###### Proof

Let be a functional modeloid on for an alphabet . We want to show that the two definitions of the derivative are equivalent in this case. It suffices to show that for fixed

 ∀a∈Σ∃b∈Σ:f∪{(a,b)}∈M⟺∀ idempotent and atomic b∈F(Σ)∃x∈M:(f≤x∧b≤x−1x)

because the second part of the definition of the derivative follows by analogy.

To start remember the natural bijection

 Ψ:Σ≃{p∈F(Σ)|p atomic and idempotent }.

Suppose the upper formula holds. Fix . Then setting yields that by proposition 2. Furthermore, . But then . Hence we get that . But now we can quantify over instead of which yields the desired result.
Conversely suppose the bottom formula holds. Then fix an idempotent and atomic element and let be the element with . It holds that because of the inclusion property. But this already yields that

 f∪{(Ψ−1(b),x(Ψ−1(b)))}∈M.

Quantifying over instead of concludes the proof.

### 6.3 Proofs of section 4

###### Proposition 5

Let be a finite inverse category with all zero elements and be a categorical modeloid on . Then for each object in we get that is a semimodeloid (on itself).

###### Proof

Fix an object in . Once we have proven that is an inverse monoid it will follow that is a semimodeloid. That is because then is closed under composition and taking inverses. Furthermore, the definition of the partial order defined on morphisms will simply reduce to the natural partial order. As a result the inclusion axiom also holds. And at last we have that with the property that for all hence giving us the neutral element required by a semimodeloid.

Let’s now prove that is an inverse monoid. First we will show the closure of the composition. For that fix two elements . We know that . We distinguish two cases. First, we assume that exists, so . Therefore, holds and as a result exists. Hence, we have . This implies

 dom(a⋅b)≃cod(dom(a⋅b)) (8)

and the existence of from which we get . By using 8 we deduce but this holds . Similarly one obtains . As a result, .

Now assume does not exist. This yields that does not exist since otherwise, and would exist which contradicts . As a result, also and do not exist and therefore, holds since none of the terms is existing. Hence .

The closure of inverses is immediate because taken an element , by assumption but and as a result . Now one can regard the inverse function on as a restriction of on . As a result the inverses are unique. Associativity follows by the fact that the composition in really is the composition in restricted to . Above we have already taken care of the neutral element.

###### Theorem 6.1

Let be a finite inverse category with all zero elements and let be a categorical modeloid on . Then

 ⋃X,Y∈MD(HomM(X,Y))

is a categorical modeloid on .

###### Proof

Assume the assumptions formulated above. Then we define

 H:=⋃X,Y∈MD(HomM(X,Y)).

We will prove (1) that all objects from are in , (2) the closure of taking inverses from H, (3) the closure of the composition on and (4) the inclusion property hold which is

 ∀s∈C∀t∈H:s≤t⇒s∈H.
1. Fix an object . Since is a categorical modeloid, and, furthermore . We need to prove that . So fix an idempotent and atomic . We show that and . Note that, since , is idempotent by axiom of a category. As a result and hence . On the other hand by definition of we have that . Because is idempotent it commutes with and hence by definition . The second part of the condition posed by the derivative holds simply because .

2. Next take an element . Then for some . As a result we can write as . We know that and want to show that .

Fix an idempotent and atomic element . Then, since , we find . Also and . Because we know that . In total that yields

 ∀ idempotent atomic a∈EndM(Y) ∃h′∈HomM(Y,X):(s−1≤h′∧a≤h′−1h′)

The second condition required by the derivative follows by an analogous construction. Hence .

3. Up now is the closure of the composition. Let . We want to show that . We will do this by case analysis.

Case 1: or does not exist. W.l.o.g. is non-existent. Then and because if existed, so would and .

Case 2: and exist but . This implies that there are two different objects in . This means that , the unique non-existing element. Because by definition and is closed for composition this yields .

But then . Now we show that . Note that and is idempotent. As a result is by default atomic. But because we get that as desired since the second part of the derivative reduces to what we have just shown. As a result we have .

Case 3: and exist and . As a result the composition exists and we can write as and as for . As a result and . As a result we want to show that . First we will prove

 ∀ idempotent atomic a∈EndM(X)∃h∈HomM(X,Z):((t⋅s)≤h∧a≤h−1h). (9)

For that fix such an idempotent and atomic . We use the assumptions about now. That yields

 ∃f∈HomM(X,Y):s≤f∧a≤f−1f. (10)

The idea is now to construct something which can be thought of as applying to which will be idempotent and atomic in . This construction is .

First note that . We get that and . As a result .

Next we wish to show that is idempotent and atomic. For idempotence see that by using that from 10 and the fact that is idempotent.

In order to show that is atomic in we assume for and show that where is the zero element of . So let . Then

 c≤faf−1⇒f−1c≤af−1⇒f−1cf≤a.

But because is atomic by assumption it follows that . The later implies that . We prove this by contraposition.

Suppose . and hence since otherwise . Since is idempotent by the fact that is idempotent, and by axiom of an inverse category . But and as a result .

We may now assume that . As a result is idempotent and atomic. We wish to show now that is an inverse of because this will yield that by the fact that is its own inverse and as such unique.

It is immediate that

 faf−1⋅c⋅faf−1≃faaaf−1≃faf−1.

Furthermore,

 c⋅faf−1⋅c ≃ faf−1c⋅faf−1⋅faf−1c ≃ faf−1c⋅faf−1c ≃ faf−1c ≃ c.

We conclude that is indeed atomic, idempotent and an element of . We now use the assumption about which yields

 ∃k∈HomM(Y,Z):t≤k∧faf−1≤k−1k. (11)

We are now in the position to say that we can find such that 9 is satisfied. For this set . Because and by 10 and 11 respectively we have that . Furthermore, by 11 it holds that

 faf−1≃k−1kfaf−1