e-Learning Platform Ranking Method using a Symbolic Approach based

on Preference Relations

Soraya Chachoua, Nouredine Tamani, Jamal Malki and Pascal Estraillier

L3i Laboratory, University of La Rochelle, Avenue Michel Cr

´

epeau, La Rochelle, France

Keywords:

e-Learning Platforms, Qualitative Weight and Sum (QWS), Aggregation and Ranking.

Abstract:

e-Learning platforms are of a great help in teaching and learning ﬁelds given their ability to improve training

activity quality. Subsequently, several e-Learning systems have been developed in many domains. The diver-

sity of such platforms in a single ﬁeld makes it arduous to select the optimal platform in terms of tools and

services that meet users’ requirements. Therefore, we propose in this paper a ranking approach of e-Learning

platforms relying on symbolic values, borrowed from the Qualitative Weight and Sum method (QWS) (Stufﬂe-

beam, 1994), preference relations and aggregating operators providing a total order among the considered

e-Learning platforms.

1 INTRODUCTION AND

MOTIVATION

The past decade has seen tremendous changes in ed-

ucational and industrial training methods along with

the increasing of the number of users having diverse

needs and objectives. Indeed, there are a huge num-

ber of free and commercial e-Learning systems which

have been developed in different areas such as ed-

ucation (Venkataraman and Sivakumar, 2015), lan-

guage learning (Ba

˜

nados, 2013), business training

(Colace et al., 2006; Ubell, 2000), medicine (Schnei-

der et al., 2015; Hannan, 2013) and public administra-

tions (Stoffregena et al., 2015), etc. which provide on-

line and remote training making user learning tasks

more ﬂexible and easier. The multitude of e-learning

platforms developed for a single domain (such as in

language learning, for instance, we can distinguish

tens of e-learning applications and on-line platforms

like babel, busuu, duolingo, ef, tell me more, Pim-

sleur, etc.) makes it difﬁcult to pick the more suitable

one according to one’s needs and objectives.

The choice of a suitable system in compliance

with user’s needs and goals based on some cri-

teria is important. Some criteria are mandatory

to choose platform but they are insufﬁcient, such

as the compatibility of the e-Learning system on

hand to certain norms and standards like SCORM

1

,

1

SCORM: Sharable Content Object Reference Model,

http://scorm.com

QTI

2

, IMS

3

, etc. These standards ensure a struc-

tured learning object creation and e-Learning qual-

ity through properties, such as adaptability, sustain-

ability, interoperability and reusability. We refer the

reader to (Garc

´

ıa and Jorge, 2006) for an e-Learning

platform evaluation based on the SCORM speciﬁca-

tion.

Besides, many other evaluation approaches have

been proposed such as (Britain and Liber, 2004), in

which the framework considers two models. The for-

mer addresses the different ways to produce learning

processes in an e-Learning system, which has been

reused in (Laurillard, 2013), and the latter character-

izes the different evaluation criteria of learning mod-

els as introduced in (Liber et al., 2000).

Qualitative methods have also been considered for

e-Learning systems evaluation; the most commonly

used one is Qualitative Weight and Sum, denoted

by QWS (Stufﬂebeam, 1994). It relies on a list of

weighted criteria (Graf and List, 2005; Hamtini and

Fakhouri, 2012) for the evaluation of e-Learning sys-

tems. In practice, it is based on qualitative weight

symbols expressing six levels of importance, namely:

E for essential, ∗ for extremely valuable, # for very

valuable, + for valuable, | for marginally valuable,

and 0 for not valuable. Hence, e-Learning system’s

2

QTI: Question and Test Interoperability, http://www.ims

global.org

3

IMS: Instructional Management Systems, http://www.ims

global.org

114

Chachoua, S., Tamani, N., Malki, J. and Estraillier, P.

e-Learning Platform Ranking Method using a Symbolic Approach based on Preference Relations.

In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 1, pages 114-122

ISBN: 978-989-758-179-3

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

performance is measured by symbolic weights at-

tached to some criteria as described in (Graf and List,

2005), such that low-weighted criteria cannot over-

power high-weighted ones. For instance, if a crite-

rion weighted #, the platform can only be judged #

or lesser (+, | or 0) but not ∗ or higher. To obtain

a global evaluation for a platform, QWS approach

aggregates the symbols attached to criteria through

a simple counting, which is ﬁnally used to rank the

considered e-learning systems. Because of the naive

aggregation function used by the approach, the result

may be counterintuitive and not clear to explain and

justify. For example, let us suppose three e-Learning

systems, denoted by e

1

, e

2

, e

3

respectively, for which

the aggregation function delivers the results as sum-

marized in Table 1. It is easy to conclude that e

1

is

Table 1: Example of e-Learning system aggregation results.

E ∗ # + | 0

e

1

- 3 4 - 2 -

e

2

- 2 4 - 2 -

e

3

- 2 8 1 2 -

better than e

2

, since e

1

is better than e

2

on symbol ∗

and both tied the score for the other symbols. But, it is

not that easy to say whether e

1

is better than e

3

or not,

because even though e

1

performs well on symbol ∗, e

3

is much more better than e1 on symbols # and +. In

the latter case, further analysis has to be conducted to

conclude. As some e-learning systems are not compa-

rable, then the approach delivers a pre-order over the

evaluated platforms.

To deal with this issue, one can consider the Ana-

lytic Hierarchy Process (AHP) method (Hamtini and

Fakhouri, 2012). AHP is used to deal with complex

decision-making processes. It translates the symbols

deﬁned in QWS into values as detailed in Table 2, bor-

rowed from (Stufﬂebeam, 1994). Thus, AHP captures

both subjective and objective values, checks their con-

sistency and reduces bias decision making in testing

and evaluating e-Learning systems (Maruthur et al.,

2015). The criteria are gathered up by category and

sub-category. The results of the feature’s category or

subcategory evaluation computed by the weight cal-

culation functions are percentages of the form of a

real number as described in (Hamtini and Fakhouri,

2012). For example, let us say that the percentage

returned for the feature Chat is 14%. Then, accord-

ing to Table 2, the judgment of this result is between

“marginally valuable” and “valuable”, but which is it?

The percentages returned can be difﬁcult to interpret

for comparing e-learning platforms when several at-

tributes have to be dealt with.

As these methods return numerical values or av-

Table 2: QWS symbols translated into AHP weights.

QWS Weight in

AHP

Essential E 5

Extremely valuable ∗ 4

Very valuable # 3

Valuable + 2

Marginally valuable | 1

Not valuable 0 0

erages (Graf and List, 2005), which can be less ex-

pressive and non-intuitive enough from a user stand-

point for system quality assessment and ranking, then

we propose in this paper a hybrid approach for sys-

tem assessment and ranking combining QWS values,

symbolic preference relations and formal comparison

operators, which have been proved to be total orders

allowing the distinction of optimal e-Learning plat-

forms from the user standpoint.

The remainder of the paper is structured as fol-

lows. Section 2 details our symbolic-based approach

for e-Learning systems evaluation. Section 3 presents

an illustrative example of our approach to evaluate

and to rank a set of open-source e-Learning systems.

Finally, section 4 concludes the paper and introduces

some future work.

2 HYBRID E-LEARNING

SYSTEM EVALUATION

APPROACH

In this section, we detail our approach for e-Learning

platform evaluation and ranking relying on symbols

borrowed from QWS method and qualitative prefer-

ence relation and comparison operators. In Subsec-

tion 2.1, we introduce our evaluation approach and in

Subsection 2.2, we show the use of our approach for

e-learning platform ranking.

2.1 Symbolic Approach for e-Learning

Platforms Evaluation

We deﬁne the evaluation symbols as follows.

Deﬁnition 1 (Evaluation Symbols). The evaluation

symbols as deﬁned in QWS approach are:

E = essential, * = extremely valuable, # = very valu-

able, + = valuable, | = marginally valuable and 0 =

not valuable. We denote by S = {E, ∗, #, +, |, 0} an

ordered set of evaluation symbols.

We deﬁne a preference relations more preferred

than or equal to, denoted by , and less preferred

e-Learning Platform Ranking Method using a Symbolic Approach based on Preference Relations

115

than or equal to, denoted by , over the evaluation

symbol set S as follows.

Deﬁnition 2 (Preference relations and ). Let

S = {E, ∗, #, +, |, 0} be an ordered set of evaluation

symbols such that:

• Position 1 is for symbol E, denoted by pos

S

(E)

• Position 2 is for symbol ∗, denoted by pos

S

(∗)

• Position 3 is for symbol #, denoted by pos

S

(#)

• Position 4 is for symbol +, denoted by pos

S

(+)

• Position 5 is for symbol |, denoted by pos

S

(|)

• Position 6 is for symbol 0, denoted by pos

S

(0)

We deﬁne the preference relation more preferred

than or equal to over S as follows.

∀(a, b) ∈ S

2

: a b iff pos

S

(a) ≤ pos

S

(b) (1)

The preference relation less preferred than or

equal to, denoted , is deﬁned as follows.

∀(a, b) ∈ S

2

: a b iff pos

S

(a) ≥ pos

S

(b) (2)

We can easily prove that the preference relation

is a total order.

Property 1. (Total order properties). The preference

relations and are a total order.

Proof. The proof of property 1 is detailed in Ap-

pendix A.

Based on the above deﬁned preference relations,

we deﬁne two comparison operators named pre f Min

and pre f Max, so that it will be possible to compare

systems on each criterion describing them. These op-

erators will serve as means to aggregate the evalua-

tions obtained for system criteria.

Deﬁnition 3. (pre f Max and pre f Min comparison

operators). pre f Max and pre f Min operators are de-

ﬁned by formulas (3) and (4) respectively.

The function pre f Max is deﬁned by the following

formula (3).

S × S → S

(a, b) 7→ max(a, b) =

(

a i f (a b)

b otherwise.

(3)

The function pre f Min is deﬁned by the following

formula (4).

S × S → S

(a, b) 7→ min(a, b) =

(

a i f (a b)

b otherwise.

(4)

When we apply the comparison operators

pre f Max and pre f Min over our symbolic set S , we

obtain Table 3.

Property 2. (pre f Max properties). pre f Max is as-

sociative, commutative, idempotent, it has E as ab-

sorbent element and 0 as neutral element.

Proof. Proofs of pre f Max properties are detailed in

Appendix B.

Property 3. (pre f Min properties). pre f Min is as-

sociative, commutative, idempotent, it has 0 as ab-

sorbent element and E as neutral element.

Proof. Proofs of pre f Min properties are detailed in

Appendix C.

2.2 Using our Comparison Operators to

Rank e-Learning Systems

The evaluation of e-Learning platforms is based on

categories, each of which deﬁnes some criteria as de-

ﬁned in (Atthirawong and MacCarthy, 2002), for ex-

ample the category Communication tools, and their

criterion such as Chat. Categories and their criteria

are summarized in Table 4. The ﬁve categories con-

sidered in platform evaluation are the following:

• Communication tools

• Software and installation

• Administrative tools and security

• Hardware presentation tools

• Management features

To evaluate each category, we use the comparison op-

erators pre f Max and pre f Min. But to evaluate a

considered e-Learning system, we need the evalua-

tion of the ﬁve categories. For that purpose, we de-

ﬁne two aggregation operators, called pre f MinMax

and pre f MaxMin, which are based on our compari-

son operators.

Deﬁnition 4. (pre f MinMax). Let A be a matrix of n

lines and m columns of evaluation symbols of S . We

deﬁne the minimum guaranteed satisfaction value as

follows.

We denote a matrix from A as:

A = (a

i j

)

1≤i≤m

1≤ j≤n

and a

i j

∈ S

We deﬁne pre f MinMax of A as:

S

m×n

→ S

A 7→ pre f MinMax(A) =

pre f Min

1≤i≤m

(pre f Max

1≤ j≤n

(a

i j

))

(5)

CSEDU 2016 - 8th International Conference on Computer Supported Education

116

Table 3: The operators pre f Max and pre f Min table.

pre f −

Max

E ∗ # + | 0

E E E E E E E

∗ E ∗ ∗ ∗ ∗ ∗

# E ∗ # # # #

+ E ∗ # + + +

| E ∗ # + | |

0 E ∗ # + | 0

pre f −

Min

E ∗ # + | 0

E E ∗ # + | 0

∗ ∗ ∗ # + | 0

# # # # + | 0

+ + + + + | 0

| | | | | | 0

0 0 0 0 0 0 0

Table 4: Overview of the evaluation hierarchy categories and their criteria.

Category Communication

tools

Software &

Installation

Administrative

tools and Security

Hardware

Presentation

tools

Management

features

Criterion

Chat

Forum

Mail

Video conference

Calendar

Downloading

Installation

Assistance

Documentation

Courses administration

Tracking progress

Online registration

Learning path creation

Report

Learning path organisation

Test evaluation

Security

Announcements

Learning Objects

Exercices

Content import

Multi course management

Multi user management

Evaluation management

User Group

Deﬁnition 5. (pre f MaxMin). We deﬁne the

maximum possible satisfaction value of S

m×n

as

pre f MaxMin:

S

m×n

→ S

A 7→ pre f MaxMin(A) =

pre f Max

1≤i≤m

(pre f Min

1≤ j≤n

(a

i j

))

(6)

The pre f MinMax operator computes the least

optimistic value amongst the criteria, whereas

pre f MaxMin operator computes the greatest pes-

simistic value amongst the criteria.

3 ILLUSTRATIVE EXAMPLE

We apply our e-Learning systems evaluation approach

to a set of nine open-source e-Learning enumerated

below, and which have been tested and compared their

(Lebrun et al., 2008),(Reiter et al., 2006) (Dogbe Se-

manou et al., 2007) (Laforcade and Oubahssi, 2014).

1. Claroline: version 1.9.2, http://www.claroline.net

2. Dokeos: version 2.1.1, http://www.dokeos.com/fr

3. eFront: version 3.6.11,

http://www.efrontlearning.net

4. ILIAS: version 4.1.3, http://www.ilias.de

5. Open ELMS: version 7, http://www.openelms.org

6. Ganesha: version 4.5, http://www.ganesha.fr

7. Olat: version 7.2.1, http://www.olat.org

8. AnaXagora: version 3.5,

http://www.anaxagora.tudor.lu

9. Sakai: version 10.4, https://sakaiproject.org

Table 5: pre f Min and pre f Max results for Communication

Tools category.

Category Communication

tools

pre f Min

pre f Max

Criterion

chat

Forum

Mail

Conference Video

Calendar

Claroline # # + # + + #

Dokeos ∗ + + ∗ ∗ + ∗

eFront | # # + + | #

ILIAS # + + 0 + 0 #

Open ELMS 0 0 ∗ 0 0 0 ∗

Ganesha # # + 0 0 0 #

Olat ∗ ∗ ∗ 0 ∗ 0 ∗

AnaXagora # # # 0 + 0 #

Sakai ∗ # ∗ ∗ # # ∗

e-Learning Platform Ranking Method using a Symbolic Approach based on Preference Relations

117

Table 6: Results of pre f MaxMin computation over the set of e-learning platform.

Category

Communication tools

Software & Installation

Administrative tools and Security

Hardware presentation tools

Management features

pre f MaxMin

Platform pre f Min pre f Min pre f Min pre f Min pre f Min

Claroline + + + # + #

Dokeos + | | # + #

eFront | # | | | #

ILIAS 0 | | 0 | |

Open ELMS 0 | | 0 0 |

Ganesha 0 | + 0 # #

Olat 0 | | 0 | |

AnaXagora 0 | | 0 | |

Sakai # | | 0 | #

Table 7: Results of pre f MinMax computation over the set of e-learning platform.

Category

Communication tools

Software & Installation

Administrative tools and Security

Hardware presentation tools

Management features

pre f MinMax

Platforms pre f Max pre f Max pre f Max pre f Max pre f Max

Claroline # ∗ ∗ ∗ # #

Dokeos ∗ # E ∗ E #

eFront # ∗ ∗ + + +

ILIAS # + # # + +

Open ELMS ∗ + ∗ ∗ # +

Ganesha # + E # # +

Olat ∗ # ∗ ∗ ∗ #

AnaXagora # + ∗ # ∗ +

Sakai ∗ ∗ E ∗ E ∗

Each criterion takes a symbolic value from the set S

based on users opinions community. To obtain the

evaluation of each criterion, we have carried out sur-

veys in our university involving under-graduated stu-

dents (small group of 10 students), who have tested

each e-learning platform during a training session (2

CSEDU 2016 - 8th International Conference on Computer Supported Education

118

Table 8: E-Learning platform’s features evaluation.

Category Communication

tools

Software &

Installation

Administrative

tools and Security

Hardware

presentation

tools

Management

features

Criterion

chat

Forum

Mail

Video conference

Calendar

Downloading

Installation

Assistance

Documentation

Courses administration

Tracking progress

Online registration

Learning path creation

Report

Learning path organisation

Test evaluation

Security

Announcements

Learning Objects

Exercises

Content import

Multi course management

Multi user management

Evaluation management

User group

Claroline # # + # + ∗ + # ∗ # # + + ∗ + # # ∗ ∗ # ∗ + + + #

Dokeos ∗ + + ∗ ∗ # | + + + E + | E | # # ∗ ∗ # ∗ + + E +

eFront | # # + + ∗ # # # | ∗ + # ∗ # # # | + + + + | + |

ILIAS # + + 0 + + | | | | + # | | | | # 0 # + 0 + | + |

Open ELMS 0 0 ∗ 0 0 + | + + | # + + # | ∗ | 0 ∗ 0 # 0 0 # #

Ganesha # # + 0 0 + | + + # E | + E # # + 0 # 0 # # # # #

Olat ∗ ∗ ∗ 0 ∗ + | | # # ∗ | + # | # # 0 ∗ ∗ ∗ | + ∗ ∗

AnaXagora # # # 0 + + | + | # ∗ | ∗ + # # + 0 + # | | # # ∗

Sakai ∗ # ∗ ∗ # ∗ | ∗ | E ∗ | E ∗ ∗ ∗ | ∗ # 0 # | # E ∗

e-Learning Platform Ranking Method using a Symbolic Approach based on Preference Relations

119

hours). We are aware that the process is subjective

and a different panel of students or users can express

different opinions about the e-Learning platforms. We

recall that this data collection aims at illustrating the

use of our approach. The values obtained for each cri-

terion in its category are summarized in Table 8. The

application of our approach on the set of considered

systems is performed as follows.

1. for each category in Table 4 we calculate values

of pre f Min and pre f Max for all functionalities

based on Deﬁnition 3. In Table 5, we display the

results obtained by applying our approach on the

category “Communication Tools” for our consid-

ered set of e-learning platforms.

2. for all categories in Table 8 we calculate values of

pre f MaxMin and pre f MinMax. Results of both

calculus are displayed in Table 6 and 7 respec-

tively.

According to Table 6, we obtain the following ranking

over the set of e-learning system considered.

1. Claroline, Dokeos, eFront, Ganesha and Sakai.

2. Ilias, Open ELMS, Olat and AnaXagora.

According to Table 7, we obtain the following

ranking over the set of e-learning system considered.

1. Sakai

2. Claroline, Dokeos and Olat

3. eFront, ILIAS, Open ELMS, Ganesha and

AnaXagora

Finally, users can make a choice based on either

pre f MaxMin or pre f MinMax operators or can com-

bine the result returned by both. For instance, in

our illustrative example, Claroline, Dokeos, eFront,

Ganesha and Sakai are all optimal platforms accord-

ing to pre f MaxMin operator, whereas Sakai is the op-

timal one according to pre f MinMax operator. But,

we can notice that Sakai performs better since it is

optimal according to both operators.

4 CONCLUSION AND FUTURE

WORK

In this paper, we have presented an e-Learning sys-

tems evaluation approach based on a symbolic set of

value, a total order preference relation and compar-

ison operators. To describe e-Learning system, we

have used categories, each of which deﬁnes some cri-

terion of well-known properties of these systems. We

apply our approach on a set of open source e-Learning

systems for which you have gathered through small

surveys their evaluation on the considered criteria.

The proposed approach assesses the quality of an e-

Learning system amongst a set of e-Learning plat-

forms by considering a maximum possible satisfac-

tion and/or a minimum guaranteed satisfaction. Once

this value is obtained, it becomes easy to rank the set

of e-learning systems considered from the most to the

least satisfactory, and to deliver to the user the one or

several optimal systems.

Our approach brings a solution to the problem of

choosing a system according to well-deﬁned criteria.

It is still to perform a larger survey to obtain values as

accurate as possible for the criteria. It is also worthy

to consider user proﬁles when performing surveys in

such a way that we obtain different values for different

proﬁles. A proﬁle can be deﬁned over a population of

users based on their interests and training objectives.

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Appendix

A Proof of Property 1 Total Order

We only prove hereinafter the property for the

preference relation . The proof of the property

for the preference relation is similar to the one of .

Proof 1. ( is total order) The preference relation

is a total order iff:

1. is reﬂexive

2. is antisymmetric

3. is transitive

1. Relation is reﬂexive iff ∀a ∈ S : a a. There-

fore, a a iff pos

S

(a) ≤ pos

S

(a) which is veriﬁed

for the comparison operator ≤ since ≤ is reﬂex-

ive. Then is reﬂexive.

2. Relation is antisymmetric iff ∀a, b ∈ S : a = b.

Then:

a b ∧ b a iff pos

S

(a) ≤ pos

S

(b) ∧ pos

S

(b) ≤

pos

S

(a) which is veriﬁed for ≤ since ≤ is anti-

symmetric.

Then is antisymmetric.

3. is transitive iff ∀a, b, c ∈ S : a b ∧ b c ⇒

a c. As a b ∧ b c then pos

S

(a) ≤ pos

S

(b)

∧ pos

S

(b) ≤ pos

S

(c). Therefore, pos

S

(a) ≤

pos

S

(c) since ≤ is transitive. That means that

a c and is transitive.

B Proof of pre f Max Properties

Proof 2. (pre f Max properties).

1. pre f Max is associative on S:

∀a, b, c ∈ S , then: pre f Max(pre f Max(a, b), c) =

pre f Max(a, pre f Max(b, c)). We denote by I the

left term Pre f Max(Pre f Max(a, b), c) and by II

the right term Pre f Max(a, Pre f Max(b, c)).

Table 9 shows results of evaluation of the left and

the right terms, which are identical. Therefore

pre f Max is associative.

Table 9: The formula results.

I II

a b ∧ a c a ⇒a pre f Max(b, c) ⇒ II = a

a b ∧ c a c ⇒ c b (transivity)

pre f Max(b, c) = c ⇒ II = c

since c a

b a ∧ b c b pre f Max(b, c) = b ⇒ II = b

since b a

b a ∧ c b c pre f Max(b, c) = c ∧ c a

(transitivity) ⇒ II = c

2. pre f Max is commutative iff ∀a, b ∈ S :

pre f Max(a, b) = pre f Max(b, a).

From table 3, the pre f Max matrix is symmetric

so pre f Max is commutative.

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3. pre f Max is idempotent iff ∀a ∈ S :

pre f Max(a, a) = a.

From the main diagonal of table 3, we conclude

that pre f Max is idempotent.

4. pre f Max has 0 as neutral element iff ∀a ∈ S :

pre f Max(a, 0) = a.

Table 3 shows that pre f Max has 0 as neutral ele-

ment.

5. pre f Max has E as absorbent element iff ∀a ∈ S :

pre f Max(a, E) = E.

Table 3 also shows that pre f Max has E as ab-

sorbent element.

C Proof of pre f Min Properties

Proof 3. (pre f Min properties)

1. pre f Min is associative on S :

∀a, b, c ∈ S : pre f Min(pre f Min(a, b), c) =

pre f Min(a, pre f Min(b, c)). We denote by I the

left term Pre f Min(Pre f Min(a, b), c) and by II

the right term Pre f Min(a, Pre f Min(b, c)).

Table 10 shows results of evaluation of the

left and the right terms, which are identical.

Therefore pre f Min is associative.

Table 10: The formula results.

I II

a b ∧ b c c ⇒ a pre f Min(b, c) ⇒ II = c

since a c

a b ∧ c b b ⇒a pre f Min(b, c) ⇒ II = b

since a b

b a ∧ a c c ⇒ b c (transivity)

pre f Min(b, c) = c ⇒ II = c

since a c

b a ∧ c a a a is the smallest symbol

between a, b and c, so II = a

2. pre f Min is commutative iff ∀a, b ∈ S :

pre f Min(a, b) = pre f Min(b, a).

From table 3, the pre f Max matrix is symmetric

so pre f Min is commutative.

3. pre f Min is idempotent iff ∀a ∈ S :

pre f Min(a, a) = a.

From the main diagonal of table 3, we conclude

that pre f Min is idempotent.

4. pre f Min has E as neutral element iff ∀a ∈ S :

pre f Min(a, 0) = a.

Table 3 shows that pre f Min has E as neutral ele-

ment.

5. pre f Min has 0 as absorbent element iff ∀a ∈ S :

pre f Min(a, E) = E.

Table 3 also shows that pre f Min has 0 as ab-

sorbent element.

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