# A modified model for topic detection from a corpus and a new metric
evaluating the understandability of topics

Tomoya Kitano
Graduate School of Information
Science and Technology
Osaka University
<EMAIL_ADDRESS>
& Yuto Miyatake
Cybermedia Center
Osaka University
<EMAIL_ADDRESS>
& Daisuke Furihata
Cybermedia Center
Osaka University
<EMAIL_ADDRESS>
###### Abstract

This paper presents a modified neural model for topic detection from a corpus
and proposes a new metric to evaluate the detected topics. The new model
builds upon the embedded topic model incorporating some modifications such as
document clustering. Numerical experiments suggest that the new model performs
favourably regardless of the document’s length. The new metric, which can be
computed more efficiently than widely-used metrics such as topic coherence,
provides variable information regarding the understandability of the detected
topics.

_Keywords_ natural language processing $\cdot$ neural network $\cdot$ neural
topic model $\cdot$ document clustering

##  1 Introduction

Topic analysis is a statistical technique used to uncover latent themes within
a corpus. Let $\mathcal{D}$ represent an observed corpus, i.e., a collection
of documents, $\mathcal{V}$ a set of observed words, and $\mathcal{T}$ a
collection of known topics. Each document
${\bm{w}}_{d}=(w_{di})_{i}\in\mathcal{D}$ consists of a sequence of words in
$\mathcal{V}$, with $w_{di}$ considered as a realization of a
$\mathcal{V}$-valued random variable $W_{di}$. Throughout the paper, $d$ and
$i$ consistently denote the indices for documents and words, respectively. A
topic for each word $w_{di}$ is typically modelled by a $\mathcal{T}$-valued
random variable $Z_{di}$. Topic analysis aims to conduct unsupervised
inference of a hidden relationship between words and topics modelled as
probability distributions, specifically $p(W_{di}=v,Z_{di}=t)$.

Latent Dirichlet allocation (LDA) [1] is the most prevalent topic model, which
treats both the word distribution $p(W_{di}=v\mid Z_{di}=t)$ and topic
distribution $p(Z_{di}=t)$ as unknown parameters. These parameters are assumed
to be drawn from Dirichlet priors and are estimated by using Bayesian
inference. Although LDA is a powerful model, posterior computation is usually
expensive. Furthermore, since LDA relies on word co-occurrence, its
performance can be limited in the analysis of short texts due to the lack of
frequency information.

In recent years, neural topic models, a new class of models that utilizes deep
learning techniques, have been extensively studied. These models build upon
embeddings and variational inference [2]. An embedding maps a discrete symbol,
such as a word, to a dense vector in a finite-dimensional vector space.
Specifically, a word embedding refers to an embedding map
$\mathcal{E}_{\mathcal{V}}\colon\mathcal{V}\to\mathbb{R}^{H}$, where $H$ is a
pre-specified natural number that must be substantially smaller than
$\\#\mathcal{V}$ (the symbol $\\#$ always represents the number of elements of
a set). Word embedding techniques are incorporated into topic models because
they capture the semantic similarity between words more accurately than
frequency-based approaches. Variational inference, an optimization method, is
employed in topic models with the expectation of improving the accuracy of the
posterior distribution approximation and the model’s fit to data. These
techniques are expected to well perform the learning from short texts and can
take advantage of parallel computing resources, such as GPUs.

However, the training process requires considerable trial and error for
setting optimization options, such as regularizers, as suitable choices depend
heavily on the dataset. Moreover, unlike classification problems, true labels
are not provided in topic analysis, making model evaluation being a long-
standing issue [3]. The most commonly used evaluation metric is topic
coherence, among others like perplexity and topic diversity. Topic coherence
measures the semantic consistency within each identified topic. Nonetheless,
the length of each document could influence the metric’s value, potentially
limiting the reliability of evaluation for relatively short documents.

In light of the two aforementioned issues concerning modelling and evaluation,
we shall propose a modified version of neural topic models and also introduce
a new metric with the potential to assess the understandability of detected
topics.

##  2 Preliminaries

Embedded topic model (ETM) treats both words and topics as unknown embeddings.
ETM models a word distribution as

$p_{\mathcal{E}}(W_{di}=v\mid
Z_{di}=t)=\frac{\exp\left[\mathcal{E}_{\mathcal{V}}(v)^{\top}\mathcal{E}_{\mathcal{T}}(t)\right]}{\displaystyle\sum_{v^{\prime}\in\mathcal{V}}\exp\left[\mathcal{E}_{\mathcal{V}}(v^{\prime})^{\top}\mathcal{E}_{\mathcal{T}}(t)\right]},$
(1)

where $\mathcal{E}_{\mathcal{T}}:\mathcal{T}\to\mathbb{R}^{H}$ is a topic
embedding. The number of topics $\\#\mathcal{T}$ is pre-specified, and $H$ is
typically set such that $\\#\mathcal{T}<H<\\#\mathcal{V}$. A prior
distribution for a topic in a single document is usually set, with the normal
distribution, to

$p(Z_{di}=t\mid{\bm{x}}_{d})=\mathrm{softmax}({\bm{x}}_{d})\mid_{t},\quad{\bm{x}}_{d}\sim\mathrm{N}({\bm{0}},\mathit{I}_{\\#\mathcal{T}}).$
(2)

Then, the posterior distribution of the unknown parameter ${\bm{x}}_{d}$ reads

$\pi^{*}_{\mathcal{E}}({\bm{x}}_{1},\dots,{\bm{x}}_{\\#\mathcal{D}}\mid\mathcal{D})=\frac{1}{\mathcal{M}_{\mathcal{E}}}\left[\prod_{d=1}^{\\#\mathcal{D}}p_{\mathcal{E}}({\bm{w}}_{d}\mid{\bm{x}}_{d})\pi_{\mathrm{N}}({\bm{x}}_{d})\right],$

where $\pi_{\mathrm{N}}$ is the probability density function of a standard
normal distribution,

$\mathcal{M}_{\mathcal{E}}=\int\left[\prod_{d=1}^{\\#\mathcal{D}}p_{\mathcal{E}}({\bm{w}}_{d}\mid{\bm{x}}_{d})\pi_{\mathrm{N}}({\bm{x}}_{d})\right]\mathrm{d}{\bm{x}}_{1}\cdots\mathrm{d}{\bm{x}}_{\\#\mathcal{D}}$

is a marginal likelihood, and

$p_{\mathcal{E}}({\bm{w}}_{d}\mid{\bm{x}}_{d})=\prod_{i=1}^{\\#{\bm{w}}_{d}}\sum_{t\in\mathcal{T}}p_{\mathcal{E}}(w_{di}\mid
t)p(t\mid{\bm{x}}_{d}).$

In the learning process of ETM, we need to infer the embedding
$\mathcal{E}_{\mathcal{V}}$ and $\mathcal{E}_{\mathcal{T}}$, and the
parameters ${\bm{x}}_{1},\dots,{\bm{x}}_{\\#\mathcal{D}}$ based on the above
settings.

Variational inference simplifies the learning process of ETM. The key concept
involves approximating the posterior distribution $\pi_{\mathcal{E}}^{\ast}$,
which is difficult to assess in its current form, by a normal distribution.
The optimization problem is then formulated by using the Kullback–Leibler
divergence.

The posterior distribution $\pi_{\mathcal{E}}^{\ast}$ is approximated by

$r_{{\bm{\eta}}}({\bm{x}}_{1},\dots,{\bm{x}}_{\\#\mathcal{D}}\mid\mathcal{D}):=\prod_{d=1}^{\\#\mathcal{D}}r_{{\bm{\eta}}}^{(d)}({\bm{x}}_{d}\mid{\bm{w}}_{d}),$

where each $r_{{\bm{\eta}}}^{(d)}$ is the probability distribution function of
a normal distribution
$\mathrm{N}({\bm{m}}^{(d)}_{{\bm{\eta}}},\mathrm{diag}({\bm{s}}^{(d)}_{{\bm{\eta}}})^{2})$.
Here, the mean and variance are realized by a neural network
$\mathrm{NN}_{{\bm{\eta}}}^{(1)}\colon\mathcal{D}\to\mathbb{R}^{2\\#\mathcal{T}}$
with an unknown parameter ${\bm{\eta}}$, i.e.,
$\big{[}({\bm{m}}^{(d)}_{{\bm{\eta}}})^{\top},(\log{\bm{s}}^{(d)}_{{\bm{\eta}}})^{\top}\big{]}^{\top}=\mathrm{NN}_{{\bm{\eta}}}^{(1)}({\bm{w}}_{d})$,
where $\log{\bm{s}}_{{\bm{\eta}}}^{(d)}=(\log
s_{{\bm{\eta}}}^{(d)}(t))_{t\in\mathcal{T}}$.

We then formulate a maximization problem for the following objective function

$\mathcal{L}({\bm{\eta}},\mathcal{E}_{\mathcal{V}},\mathcal{E}_{\mathcal{T}})=-D_{\mathrm{KL}}[r_{{\bm{\eta}}}\|\pi^{*}_{\mathcal{E}}]+\log\mathcal{M}_{\mathcal{E}},$

where $D_{\mathrm{KL}}[r_{{\bm{\eta}}}\|\pi^{*}_{\mathcal{E}}]$ is a
Kullback–Leibler divergence. This problem is typically solved by gradient
descent. In solving the problem, the approximation accuracy for the posterior
distribution and the model’s fit to the data should be improved
simultaneously. It should be noted that the word embedding can be kept fixed
during training if it has been pre-trained by other models, such as skip-gram.
In most topic models, it is reported that pre-training a word embedding
improves the performance in short text analysis [4].

Topic coherence [5], the most commonly used evaluation metric in topic
analysis, is defined as

$\mathrm{TC}=\frac{1}{\\#\mathcal{T}}\sum_{t\in\mathcal{T}}\left[\frac{1}{N(N-1)}\sum_{\begin{subarray}{c}u,v\in
U(t,N)\\\ u\neq v\end{subarray}}\frac{\log\frac{P(u,v)}{P(u)P(v)}}{-\log
P(u,v)}\right],$ (3)

where $U(t,N)$ is a set consisting of $N$ words corresponding to the $N$
largest $p_{\mathcal{E}}(v\mid t)$ values, $P(u,v)$ is the probability of
words $u$ and $v$ co-occurring in a document, and $P(u)$ is the marginal
probability of word $u$. In general, computing $P(u,v)$ and $P(u)$ for all
possible combinations of $u$ and $v$ is highly expensive. Following [6], in
the numerical experiments presented below, we will employ the following
approximations, which are relatively inexpensive among other approximations,
taking in mind that approximation accuracy can be limited:

$\displaystyle
P(u,v)\approx\frac{1}{\\#\mathcal{D}}\sum_{d=1}^{\\#\mathcal{D}}\mathbb{I}[u\in\bm{w}_{d}]\mathbb{I}[v\in\bm{w}_{d}],$
$\displaystyle
P(u)\approx\frac{1}{\\#\mathcal{D}}\sum_{d=1}^{\\#\mathcal{D}}\mathbb{I}[u\in\bm{w}_{d}]$

are distributions of a word in the corpus, where $\mathbb{I}$ is the indicator
function. We note that their definitions depend on $\\#\mathcal{D}$.

##  3 A modified model

Since neural topic models, including ETM, have a high degree of freedom and
admit numerous candidate solutions as local minimum, these models sometimes
fail to capture the semantic structure of the corpus. To address this issue,
training neural topic models necessitates regularization techniques, such as
weight decay, dropout, or batch normalization, to guide the model towards
better results. In this section, we introduce three modifications that lead to
additional constraints to ETM to facilitate capturing the differences in
meaning among documents. Hereafter, we assume that the corpus $\mathcal{D}$ is
partitioned into clusters $C_{1},\dots,C_{\\#\mathcal{T}}$ by using k-means
and their centres are denoted by
$\bm{c}(C_{1}),\dots,\bm{c}(C_{\\#\mathcal{T}})$.

First, we replace the topic embedding $\mathcal{E}_{\mathcal{T}}$ with a
neural network that inputs cluster centres. In the case of standard ETM, the
output of the topic embedding as $H$-dimensional vectors does not effectively
represent the relationship between topics. Besides, the relationship of the
meanings in a document to each other can be measured by computing the inner
product similarity (cosine similarity) between cluster centres. To utilize
this prior knowledge, we define a new topic embedding as follows. Since the
number of topics and clusters are the same, $\\#\mathcal{T}$, a one-to-one
correspondence can be established between topics and clusters. If a topic
$t_{j}$ corresponds to a cluster $C_{j}$, we define the topic embedding as

$\tilde{\mathcal{E}}_{\mathcal{T}}(t_{j})=\mathrm{NN}^{(2)}_{\bm{\xi}}(\bm{c}(C_{j})),$

where $\mathrm{NN}^{(2)}_{\bm{\xi}}$ is a neural network parametrized by an
unknown parameter $\bm{\xi}$.

Second, we propose a new prior distribution for the parameter $\bm{x}_{d}$
based on the clustering outcomes. In ETM, the prior distribution for topics is
defined by (2). However, this does not incorporate any information about the
topic occurrence within the document. To represent a prior knowledge that the
same topic is likely to appear in documents belonging to the same cluster, we
define a new prior distribution as

$\mathrm{N}(\bm{m}_{0}^{(d)},\mathit{I}_{\\#\mathcal{T}}),\quad\bm{m}_{0}^{(d)}=\lambda^{(d)}\bm{e}_{n(d)},$

where $\bm{\lambda}=\\{\lambda^{(d)}\\}_{d=1}^{\\#\mathcal{D}}$ is a positive
unknown parameter, $\bm{e}_{i}$ is a unit vector with only the $i$-th
component being 1, and $n(d)$ is the cluster number to which the document
$\bm{w}_{d}$ belongs.

Third, we modify the word probability within a topic based on the word
probability in the entire document. In ETM, since the word distribution in a
topic is defined by (1), the word-topic similarity
$\mathcal{E}_{\mathcal{V}}(v)^{\top}\mathcal{E}_{\mathcal{T}}(t)$ may become
large even for words with limited information. To circumvent this issue, we
modify the word distribution within a topic as follows:

$\displaystyle\tilde{p}_{\mathcal{E},\bm{\xi}}(W_{di}=v\mid Z_{di}=t)$
$\displaystyle\quad=\frac{\exp\left[\mathcal{E}_{\mathcal{V}}(v)^{\top}\tilde{\mathcal{E}}_{\mathcal{T}}(t)\right]G_{0}(v)}{\displaystyle\sum_{v^{\prime}\in\mathcal{V}}\exp\left[\mathcal{E}_{\mathcal{V}}(v^{\prime})^{\top}\tilde{\mathcal{E}}_{\mathcal{T}}(t)\right]G_{0}(v^{\prime})},$

where
$G_{0}(v)=\sum_{d=1}^{\\#\mathcal{D}}\sum_{i=1}^{\\#\bm{w}_{d}}\mathbb{I}[w_{di}=v]\big{/}\sum_{d=1}^{\\#\mathcal{D}}\\#\bm{w}_{d}$
represents the relative frequency of the word $v\in\mathcal{V}$ in a whole
document $\mathcal{D}$. This modification is expected to make it easier to
differentiate between low-information words and words that characterize the
topic.

Let $\pi^{*}_{\mathcal{E},\bm{\xi},\bm{\lambda}}$ be the posterior
distribution of $\bm{x}_{d}$ and
$\mathcal{M}_{\mathcal{E},\bm{\xi},\bm{\lambda}}$ be the marginal likelihood
in the modified ETM. Then, the variational inference is a maximization problem
with the following objective function:

$\displaystyle\mathcal{L}(\bm{\eta},\mathcal{E}_{\mathcal{V}},\bm{\xi},\bm{\lambda})$
$\displaystyle=-D_{\mathrm{KL}}[r_{\bm{\eta}}\|\pi^{*}_{\mathcal{E},\bm{\xi},\bm{\lambda}}]+\log\mathcal{M}_{\mathcal{E},\bm{\xi},\bm{\lambda}}$
$\displaystyle=\sum_{d=1}^{\\#\mathcal{D}}\left(\int\left[\log\tilde{p}_{\mathcal{E},\bm{\xi}}(\bm{w}_{d}\mid\bm{x}_{d})\right]r_{\bm{\eta}}(\bm{x}_{d}\mid\bm{w}_{d})\mathrm{d}\bm{x}_{d}-\frac{\\#\mathcal{T}}{2}\right.$
$\displaystyle\phantom{=}\quad-\left.\frac{1}{2}\left(\|\bm{m}_{\bm{\eta}}^{(d)}-\bm{m}_{0}^{(d)}\|^{2}+\|\bm{s}_{\bm{\eta}}^{(d)}\|^{2}\right)+\sum_{t\in\mathcal{T}}\log
s_{\bm{\eta}}^{(d)}(t)\right).$

##  4 A new metric

In topic modelling, in addition to the coherence, the understandability of the
detected topics is also a crucial factor. We here introduce an evaluation
metric that could measure the topic understandability based on the so-called
word familiarity[7], which is a measure of the relative understandability of
words for humans. The higher the word familiarity scores, the more easily
understandable the word is. Tanaka et al. [7] reported a strong correlation
between the word log frequency and word familiarity, with more frequent words
being more familiar to people. In this study, we define a topic
understandability metric using the log relative frequency $\log G_{0}$.
Intuitively, a topic is deemed easier to understand if the sum or average of
$\log G_{0}$ in the topic is higher.

For a topic model, we define a weighted sum of word familiarity as

$\mathrm{WSWF}=\frac{1}{\mathcal{T}}\sum_{t\in\mathcal{T}}\left[\sum_{v\in
U(t,N)}\hat{p}(v\mid t)\log G_{0}(v)\right],$ (4)

where $\hat{p}(v\mid t)$ is the estimated word distribution in the topic $t$.
We consider the model with a higher value of WSWF to have more understandable
topics. We note that WSWF can be computed instantly once $\hat{p}$ is
established.

##  5 Numerical experiment

We study the following corpora.

  1. 1.

20News–Groups:
http://qwone.com/~jason/20Newsgroups
/20news-18828.tar.gz (accessed on 9 January 2023)

  2. 2.

Search–Snippets:
http://jwebpro.sourceforge.net/data-web-
snippets.tar.gz (accessed on 9 January 2023)

Table 1 lists the details of each dataset. 20–NewsGroups is a standard dataset
in NLP community and consists of multiple news articles in each of 20
categories. On the other hand, Search–Snippets is a collection of related
keywords for each Wikipedia article and the length of each document is
relatively short. As preprocessing, numbers, special characters, pronouns,
prepositions, and low-frequent words were removed. In addition, words were
lowercased and lemmatized.

Table 1: Information for each dataset.  | $\\#\mathcal{D}$ | $\\#\mathcal{V}$ | Avg. length
---|---|---|---
20–NewsGroups | 18827 | 7558 | 81.9
Search–Snippets | 10059 | 9577 | 15.8

We compared five models with $\\#\mathcal{T}=50$: LDA, ETM, ETM with a pre-
trained word embedding, our modified model and our modified model with a pre-
trained word embedding. We pre-trained the word embedding using the skip-gram
model [8] for each corpus with the embedding dimension $H=200$. We evaluated
all the models by topic coherence and WSWF with $N=10$. Additionally, we
provided subjective accuracy for the 20-NewsGroups dataset. We compared the
content of $100$ randomly sampled documents with the top $10$ most probable
words from the predictive distribution for each document. The evaluation was
conducted in a subjective manner based on the consistency of the words, the
presence of keywords in the document, and the effectiveness of the summary.
Since each document in the Search-Snippets dataset consists of a list of
keywords, making the comparative evaluation difficult, subjective evaluation
was not performed.

Table 2: Model evaluation in 20–NewsGroups.Higher is better. The row pretrained indicates whether the pretrained word embedding is Available or NOT. Subj. indicates subjective accuracy.  model | LDA | ETM | modified
---|---|---|---
pretrained |  | N | A | N | A
TC | 0.274 | 0.085 | 0.232 | 0.245 | 0.254
WSWF | -0.833 | -2.867 | -1.793 | -1.053 | -0.962
Subj. [%] | 55 | 4 | 44 | 54 | 55
Figure 1: Model comparison by topic coherence and WSWF in 20–NewsGroups.
Higher is better.

Table 2 lists the value of each evaluation metric for 20-NewsGroups, and Fig.
1 visualize the comparison between TC and WSWF for each model. We observe that
all models except for ETM without pre-trained word embedding achieve high
scores. In addition, whether or not the pre-trained word embedding is
incorporated, our modified model outperforms ETM in both metrics, which
indicates that the modification improves the topic coherence and
understandability simultaneously for this dataset. Furthermore, our modified
model achieves almost the same level as LDA in subjective evaluation.

Figure 2: Plot of TC and WSWF pairs for each topic detected by LDA (blue
circles) and ETM with word embedding pretraining (orange triangles),
respectively.

Note that both TC (3) and the proposed metric WSWF (4) are defined as the
average of the ratings for each topic. However, comparing the values of TC and
WSWF before averaging may give us some insights into the quality of the
detected topics. To illustrate this, we depict the values of LDA, ETM with
pre-training and the modified model with pre-training in Fig. 2 (the modified
model without pre-training exhibited similar results that with pre-training).
This figure suggests that topics detected by LDA tend to consist of more
common words, while those by ETM are more likely to comprise less familiar
terms, such as technical jargon. In contrast, the modified model detects
topics with a higher proportion of familiar words than ETM.

Table 3: Model evaluation in Search–Snippets.Higher is better. The row pretrained indicates whether the pretrained word embedding is Available or NOT.  model | LDA | ETM | modified
---|---|---|---
pretrained |  | N | A | N | A
TC | -0.191 | -0.659 | -0.336 | -0.115 | 0.014
WSWF | -1.590 | -3.927 | -2.528 | -1.830 | -1.723

Table 3 displays the values of each metric for Search–Snippets. LDA, a
frequency-based model, shows law coherence because of the lack of co-
occurrence information. In contrast, the modified model exhibits the highest
coherence. This observation suggests that the modified model seems effective
even when the length of each document is relatively short.

##  6 Discussions

In this paper, we have introduced the modified model for topic detection and
proposed a new metric, WSWF, that measures the understandability of the
detected topics. Numerical experiments suggest that the new model,
particularly with pre-training, outperforms ETM irrespective of the length of
each document. Further, WSWF is efficiently computed, independently of the
length of each document. If the user prioritizes understandability, the new
metric could be highly beneficial. We also emphasize that examining both TC
and WSWF could offer variable insights into the quality of detected topics.

## References

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  * [2] D. P. Kingma and M. Welling, Auto–encoding variational Bayes, in: Proc. of the International Conference on Learning Representations, 2014.
  * [3] T. N. Doan and T. A. Hoang, Benchmarking neural topic models: An empirical study, in: Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pp. 4363–4368, 2021.
  * [4] R. Murakami and B. Chakraborty, Investigating the efficient use of word embedding with neural–topic models for interpretable topics from short texts, Sensors, 22 (2022), 852.
  * [5] J. H. Lau, D. Newman and T. Baldwin, Machine reading tea leaves: Automatically evaluating topic coherence and topic model quality, in: Proc. of the 14th Conference of the European Chapter of the Association for Computational Linguistics, pp. 530–539, 2014.
  * [6] A. B. Dieng, F. J. R. Ruiz and D. M. Blei, Topic modeling in embedding spaces, Transactions of the Association for Computational Linguistics, 8 (2020), 439–453.
  * [7] K. Tanaka–Ishii and H. Terada, Word familiarity and frequency, Studia Linguistica, 65 (2011), 96–116.
  * [8] T. Mikolov, K. Chen, G. Corrado and J. Dean, Efficient estimation of word representations in vector space, in: Proc. of the International Conference on Learning Representations, 2013.


# Rotations of Gödel algebras with modal operators

Tommaso Flaminio1 and Lluis Godo1 and Paula Menchón2 and Ricardo O. Rodriguez3
1Artificial Intelligence Research Institute (IIIA - CSIC), Barcelona, Spain.
2Nicolaus Copernicus University, Toruń, Poland, CONICET, Univ. Nacional del
Centro de la Provincia de Buenos Aires, Tandil, Argentina.
3 UBA-FCEyN, Departamento de Computación,
CONICET-UBA, Inst. de Invest. en Cs. de la Computación, Buenos Aires,
Argentina.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The present paper is devoted to study the effect of connected and disconnected
rotations of Gödel algebras with operators grounded on directly indecomposable
structures. The structures resulting from this construction we will present
are nilpotent minimum (with or without negation fixpoint, depending on whether
the rotation is connected or disconnected) with special modal operators
defined on a directly indecomposable algebra. In this paper we will present a
(quasi-)equational definition of these latter structures. Our main results
show that directly indecomposable nilpotent minimum algebras (with or without
negation fixpoint) with modal operators are fully characterized as connected
and disconnected rotations of directly indecomposable Gödel algebras endowed
with modal operators.

##  1\. Introduction

Fuzzy modal logic is an active and rapidly growing area of research that aims
at generalizing classical modal logic to a many-valued or fuzzy framework.
Although the birth of Mathematical Fuzzy logic as a discipline is usually
taken with the publication of Hájek’s book [13] in 1998, the first attempts to
generalize modal logic to the setting of many-valued logic can be traced back
to the 90’s of the last century. Indeed, in 1991 and 1992 Fitting published
two fundamental papers [9, 10] in which, he investigates two families of many-
valued modal logics: on the side of their relational semantics, the first one
is characterized by Kripke models in which, at each possible world, formulas
are evaluated by a finite Heyting algebra; the second one allows also the
accessibility relation to be many-valued.

Fitting’s work on many-valued modal logics paved the way to the birth of fuzzy
modal logic as a field and inspired several other researchers who, following
his ideas, further generalized classical modal logic to the ground of
infinite-valued fuzzy logics. In this respect it is worth to remember the
general approaches collected in the papers by Priest [23], by Hájek [14], by
Bou, Esteva, Godo and Rodriguez [3] and by Diaconescu, Metcalfe and Schnüriger
[7]. Further works which follow Fitting’s ideas of generalizing Kripke models
to the fuzzy environment, but focus on modal expansions of specific
propositional logics, are also worth to recall. In particular the ones by
Caicedo and Rodriguez [5, 6] who investigate modal expansions of Gödel logic,
the paper [15] by Hansoul and Teheux who instead consider modal expansions of
propositional Łukasiewicz logic, and the one by Vidal, Esteva and Godo [24]
where the propositional base is product fuzzy logic.

In the recent paper [11], taking inspiration on intuitionistic modal logics,
we made the first steps towards an algebraic approach to modal expansions of
Gödel propositional logic by introducing finite Gödel algebras with operators
(GAOs for short) and a class of relational models which, in contrast to the
original Fitting’s approach, are not Kripke-like structures. Indeed, the
relational duals of GAOs turn out to be frames based on forests rather than
sets, that is, posets in which the downset of each element is totally ordered.

In the present paper, we are interested in studying nilpotent minimum algebras
with modal operators. Nilpotent minimum algebras (NM-algebras for short) are
the algebraic semantics of the so-called nilpotent minimum fuzzy logic,
denoted by NM, which turns out to be sound and complete w.r.t. the algebra on
$[0,1]$ defined by the so-called nilpotent minimum t-norm and its residuum
[8], that is a left-continuous, but not continuous, t-norm. The results on NM-
algebras with modal operators are obtained by extending the well-known
construction of connected and disconnected rotations of Gödel algebras [18, 4]
to the modal framework.

This paper is organized as follows. Section 2 is devoted to recall basic
algebraic notions and, in particular, to introduce Gödel and NM-algebras. Also
we recall how NM-algebras with or without negation fixpoint can be constructed
as connected or disconnected rotations of directly indecomposable Gödel
algebras. In Section 3, after briefly recalling Gödel algebras with modal
operators, we study modal expansions of NM+ and NM--algebras obtained by
extending connected and disconnected rotations to the modal setting. In
particular, Subsection 3.1 is on modal NM+-algebras, while NM--algebras are
the subject of Subsection 3.2. Our main results, namely Theorems 3.7 and 3.14,
show that for every Gödel algebra with operators whose Gödel reduct is
directly indecomposable and satisfying certain properties, one can build both
an NM+ and an NM--algebra with operators whose NM-reduct still is directly
indecomposable. Indeed, each such modal NM-algebra arises in those ways. We
conclude with Section 4 where we present some conclusions and some prospects
for our future work.

##  2\. Gödel and nilpotent minimum algebras

Most of the algebraic structures we consider in this paper lay in the variety
of MTL-algebras [8], namely, integral, commutative, bounded and prelinear
residuated lattices which are structures of the form ${\bf
A}=(A,\ast,\to,\wedge,\vee,\bot,\top)$ of type $(2,2,2,2,0,0)$ where:
$(A,\ast,\top)$ is a commutative monoid; $(A,\wedge,\vee,\bot,\top)$ is a
bounded lattice; and the following conditions hold for all $x,y,z\in A$:

${\bf(Res)}$:

$x\ast y\leq z\mbox{ iff }x\leq y\to z$, where $\leq$ denotes the lattice
order of ${\bf A}$;

${\bf(Pre)}$:

$(x\to y)\vee(y\to x)=\top.$

In every MTL-algebra, a negation operator $\neg$ can be defined as: $\neg
x=x\to\bot$.

Let ${\bf A}$ be an MTL-algebra. A subset $f$ of ${\bf A}$ is said to be a
filter provided that: (1) $\top\in f$, (2) if $x,y\in f$, then $x\ast y\in f$,
(3) if $x\in f$ and $y\geq x$ then $y\in f$. A filter $f\neq A$ (that is, a
proper filter) is said to be prime if $x\vee y\in f$ implies that either $x\in
f$ or $y\in f$. A filter $f$ is principal (or principally generated) if there
exists an element $x\in A$ such that $f={\uparrow}x=\\{y\in A\mid y\geq x\\}$.

An MTL-algebra ${\bf A}$ is said to be directly indecomposable (d.i.,
henceforth) if it cannot be factorized as a non-trivial direct product
$\prod_{i\in I}{\bf A}_{i}$ of MTL-algebras ${\bf A}_{i}$.

######  Definition 2.1.

A Gödel algebra is MTL-algebra which further satisfies the following equation
which expresses the idempotency property:

${\bf(Idem)}$:

$x\ast x=x$.

The idempotency of Gödel algebras allows us to present them in the simplified
signature in which $\ast=\wedge$, which coincides with the signature of
Heyting algebras [16, 17]. Indeed, an equivalent definition of Gödel algebras
is to present them as those Heyting algebras, i.e., integral, commutative,
bounded and idempotent residuated lattices, that further satisfy the
prelinearity equation ${\bf(Pre)}$ above.

A Gödel algebra is d.i. iff $\bot$ is meet irreducible. Equivalently a Gödel
algebra is d.i. iff it has a join-irreducible co-atom and hence a unique
maximal filter that is in fact principally generated by its unique co-atom. In
what follows we will need the following easy result.

######  Lemma 2.2.

In every d.i. Gödel algebra, if $x>\bot$, $\neg x=\bot$.

######  Definition 2.3.

A nilpotent minimum algebra ${\bf A}$ is a MTL-algebra which further satisfies
the following equations:

${\bf(Inv)}$:

$\neg\neg x=x$,

${\bf(NM)}$:

$\neg(x\ast y)\vee(x\wedge y\to x\ast y)=\bot$.

In linearly ordered nilpotent minimum algebras, condition (NM) implies that
either $x*y=\bot$ or $x*y=\min(x,y)$. Since the negation operator of NM-
algebras is involutive, if it has a fixpoint, it is unique. The property of
having a negation fixpoint or not, can be captured equationally. An NM-algebra
${\bf A}$ is said to be without negation fix-point (and we write that ${\bf
A}$ is an NM--algebra) if it satisfies the further equation:

${\bf(NM^{-})}$:

$\neg(\neg(x\ast x))\ast\neg(x\ast x)=(\neg(\neg x\ast\neg x))\ast(\neg(\neg
x\ast\neg x))$.

As for those NM-algebras with a negation fixpoint, we need to expand their
language by a new constant ${\bf f}$ and an algebra ${\bf
A}=(A,\ast,\to,\wedge,\vee,{\bf f},\bot,\top)$ is said to be with negation
fix-point (and we write that ${\bf A}$ is an NM+-algebra) if its $\\{{\bf
f}\\}$-free reduct is a NM-algebra and it further satisfies:

${\bf(NM^{+})}$:

$\neg{\bf f}={\bf f}$.

Similarly to Gödel algebras, a NM+ or NM--algebra is directly indecomposable
iff it has a unique maximal filter. In this case, however, the unique maximal
filter is not principally generated by a unique co-atom of ${\bf A}$. Indeed,
as we will see below, there exist d.i. NM+ and NM--algebras without a unique
co-atom.

Now we recall a general construction which allows to define NM- and
NM+-algebras as disconnected and connected rotations of a directly
indecomposable Gödel algebra. The original ideas are from [18, 4] while the
proofs of next results, namely Propositions 2.4 and 2.5 and Theorems 2.6 and
2.7, can be found in [4, §4].

Let ${\bf A}$ be a Gödel algebra and define

:

$NM^{+}({\bf A})=\\{(a^{-},a^{+})\in A\times A\mid a^{-}\wedge a^{+}=\bot\\}$

:

$NM^{-}({\bf A})=\\{(a^{-},a^{+})\in A\times A\mid(a^{-}\wedge
a^{+})\vee\neg(a^{-}\vee a^{+})=\bot\\}.$

Further, for every $(a^{-},a^{+}),(b^{-},b^{+})\in NM^{+}(A)\cup NM^{-}(A)$,
define:

:

$(a^{-},a^{+})\ast(b^{-},b^{+})=((a^{+}\vee b^{+})\to(a^{-}\vee
b^{-}),a^{+}\wedge b^{+})$;

:

$(a^{-},a^{+})\wedge(b^{-},b^{+})=(a^{-}\vee b^{-},a^{+}\wedge b^{+})$;

:

$\neg(a^{-},a^{+})=(a^{+},a^{-})$.

Now, denote by ${\bf NM}^{+}({\bf A})$ and ${\bf NM}^{-}({\bf A})$ the
structures:

:

${\bf NM}^{+}({\bf A})=(NM^{+}({\bf
A}),\ast,\wedge,\neg,(\bot,\bot),(\top,\bot),(\bot,\top)),$

:

${\bf NM}^{-}({\bf A})=(NM^{-}({\bf
A}),\ast,\wedge,\neg,(\top,\bot),(\bot,\top)).$

It is worth pointing out that, in case ${\bf A}$ is a d.i. Gödel algebra, the
above constructions that build the algebras ${\bf NM}^{+}({\bf A})$ and ${\bf
NM}^{-}({\bf A})$ coincide, respectively, with the well-known connected and
disconnected rotations of ${\bf A}$ (see [18] and [4]).

######  Proposition 2.4.

For every Gödel algebra ${\bf A}$, ${\bf NM}^{+}({\bf A})$ is a NM+-algebra
with negation fixpoint $(\bot,\bot)$ and ${\bf NM}^{-}({\bf A})$ is a NM--
algebra without negation fixpoint. Moreover, ${\bf A}$ is d.i. iff so are
${\bf NM}^{+}({\bf A})$ and ${\bf NM}^{-}({\bf A})$.

According to the above construction, if $(a^{-},a^{+}),(b^{-},b^{+})$ are
elements of either $NM^{+}({\bf A})$ or $NM^{-}({\bf A})$, then
$(a^{-},a^{+})\leq(b^{-},b^{+})$ iff
$(a^{-},a^{+})=(a^{-},a^{+})\wedge(b^{-},b^{+})=(a^{-}\vee b^{-},a^{+}\wedge
b^{+})$ iff $a^{-}\geq b^{-}$ and $a^{+}\leq b^{+}$.

Let us now start from any NM+ or NM--algebra $\bf B$ and define

$G({\bf B})=\\{b\ast b\mid b\in B\\}.$

######  Proposition 2.5.

For every NM+ or $NM^{-}$-algebra ${\bf B}$, the structure ${\bf G}({\bf
B})=(G({\bf B}),\wedge,\to^{2},\bot,\top)$, where for every $x,y\in G({\bf
B})$, $x\to^{2}y=(x\to y)\ast(x\to y)$, is a Gödel algebra. Further, ${\bf B}$
is d.i. iff so is ${\bf G}({\bf B})$.

Let ${\bf A}$ be a d.i. Gödel algebra and let ${\bf B}$ be a d.i. NM-algebra.
Consider the maps $\gamma:{\bf A}\to{\bf G}({\bf NM}^{\pm}(\bf A))$ and
$\eta:{\bf B}\to{\bf NM^{\pm}(\bf G({\bf B}))}$ defined by the following
stipulations: for all $a\in A$ and $b\in B$,

(1) $\gamma(a)=(\bot,a)\ast(\bot,a),$

and

(2) $\eta(b)=\left\\{\begin{array}[]{ll}(\bot,b\ast b)&\mbox{ if }b>\neg b;\\\
(\neg b\ast\neg b,\bot)&\mbox{ if }b\leq\neg b.\end{array}\right.$

######  Theorem 2.6.

For every d.i. Gödel algebra ${\bf A}$ and for every d.i. NM-algebra ${\bf
B}$:

  1. (1)

The map $\gamma$ is an isomorphism between ${\bf A}$ and ${\bf G}({\bf
NM}^{\pm}(\bf A))$;

  2. (2)

The map $\eta$ is an isomorphism between ${\bf B}$ and ${\bf NM^{\pm}(\bf
G({\bf B}))}$.

######  Theorem 2.7.

A NM-algebra ${\bf B}$ is d.i. iff there exists a d.i. Gödel algebra ${\bf A}$
such that either ${\bf B}\cong{\bf NM}^{+}({\bf A})$ or ${\bf B}\cong{\bf
NM}^{-}({\bf A})$.

##  3\. Towards NM-algebras with modal operators

We first recall the notion of Gödel algebra with two operators from [12].

######  Definition 3.1.

A Gödel algebra with operators (GAO for short) is a triple $({\bf
A},\Box,\lozenge)$ where ${\bf A}$ is a Gödel algebra, $\Box$ and $\lozenge$
are unary operators on $A$ satisfying the following equations:

$\begin{array}[]{ll}(\Box 1)\;\Box\top=\top&(\lozenge 1)\;\lozenge\bot=\bot\\\
(\Box 2)\;\Box(x\wedge y)=\Box x\wedge\Box y\mbox{ }\mbox{ }\mbox{ }&(\lozenge
2)\;\lozenge(x\vee y)=\lozenge x\vee\lozenge y.\end{array}$

The class of Gödel algebras with operators is a variety that we denote by
$\mathbb{GAO}$. It is easy to check that every ${\bf A}\in\mathbb{GAO}$
satisfies the following equation (algebraic counterpart of the well-known
axiom K in modal logic) and inequations (monotonicity conditions for $\Box$
and $\Diamond$):

(K):

$\Box(x\to y)\to(\Box x\to\Box y)=\top$.

(Mon):

If $x\leq y$ then $\Box x\leq\Box y$ and $\Diamond x\leq\Diamond y$.

In what follows we will present first steps towards expanding nilpotent
minimum algebras with modal operators. In particular we will show that the
construction we recalled in Section 2 involving d.i. Gödel and NM-algebras
extends to the case of their modal expansions.

###  3.1. The case of NM+-algebras with operators

Let us start by considering a GAO $({\bf A},\Box,\lozenge)$ such that ${\bf
A}$ is d.i. and $\Box$ satisfies the following equation:

(N1):

$\Box\bot=\bot$.

Let us now define the operators $\boxminus$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ on
$NM^{+}({\bf A})$ as follows: for all $(a^{-},a^{+})\in NM^{+}({\bf A})$,

(-):

$\boxminus(a^{-},a^{+})=(\lozenge a^{-},\Box a^{+})$;

(-):

$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})=(\Box
a^{-},\lozenge a^{+})$.

######  Lemma 3.2.

For every $({\bf A},\Box,\lozenge)$ as above, and for all $(a^{-},a^{+})\in
NM^{+}({\bf A})$,
$\boxminus(a^{-},a^{+}),\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})\in
NM^{+}({\bf A})$.

  * Proof.  Let us prove that $\boxminus(a^{-},a^{+})\in NM^{+}({\bf A})$, that is to say, $\lozenge a^{-}\wedge\Box a^{+}=\bot$ in ${\bf A}$. Since ${\bf A}$ is d.i. , $(a^{-},a^{+})\in NM^{+}({\bf A})$ iff either $a^{-}=\bot$ or $a^{+}=\bot$. If $a^{-}=\bot$, then $\lozenge a^{-}\wedge\Box a^{+}=\lozenge\bot\wedge\Box a^{+}=\bot\wedge\Box a^{+}=\bot$. Conversely, if $a^{+}=\bot$, by $(N1)$, $\Box a^{+}=\bot$, whence again $\lozenge a^{-}\wedge\Box a^{+}=\bot$. This settles the claim. $\dashv$

Therefore, the operators $\boxminus$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ are well-
defined.

######  Definition 3.3.

A NM+-algebra with operators (NMAO+) is a system $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is a NM+-algebra and
$\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,:B\to
B$ satisfy the following conditions:

$\begin{array}[]{ll}(\boxminus
1)\;\boxminus\top=\top;&(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,1)\;\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\bot=\bot\\\
(\boxminus 2)\;\boxminus(x\wedge y)=\boxminus x\wedge\boxminus
y;&(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,2)\;\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(x\vee
y)=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x\vee\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,y;\\\
(F)\;\boxminus
f=f;&(\boxminus\mbox{-}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)\;\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x=\neg\boxminus\neg
x.\end{array}$

The following proposition collects some properties of NMAO+s.

######  Proposition 3.4.

In every NMAO+ $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
the following properties hold:

  1. (1)

$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,f=f$;

  2. (2)

If $a\geq f$, $\boxminus a\geq f$; if $a\leq f$, $\boxminus a\leq f$. The
positive and the negative elements of ${\bf B}$ are closed under $\boxminus$;

  3. (3)

If $a\leq f$,
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,a\leq f$; if
$a\geq f$, $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,a\geq
f$. The positive and the negative elements of ${\bf B}$ are closed under
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$.

  * Proof.  (1) By $(F)$, $(\boxminus\mbox{-}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$ and the fact that $f$ is the negation fixpoint, $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,f=\neg\boxminus\neg f=\neg\boxminus f=\neg f=f$.

(2) If $a\geq f$ immediately follows form $(F)$ and the monotonicity property
of $\boxminus$. If $a\leq f$, then $a\wedge f=a$, whence $\boxminus
a=\boxminus(a\wedge f)=\boxminus a\wedge\boxminus f$. Thus, $\boxminus
a\leq\boxminus f=f$.

(3) If $a\leq f$ the claim follows from (1) and the monotonicity property of
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$. Conversely,
if $a\geq f$, then $a\vee f=a$, whence
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,a=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a\vee
f)=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,a\vee\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,f$
which shows that
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,a\geq\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,f=f$.
$\dashv$

Now we can show that, if we start with a GAO $({\bf A},\Box,\lozenge)$ with
$\bf A$ being d.i. and such that $\Box\bot=\bot$, then the operators
$\boxminus$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ endow ${\bf
NM}^{+}({\bf A})$ with the structure of a NMAO.

######  Proposition 3.5.

For every GAO $({\bf A},\Box,\lozenge)$ which satisfies (N1) and such that
${\bf A}$ is d.i. , $({\bf NM}^{+}({\bf
A}),\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
is a NMAO+. Further, ${\bf NM}^{+}({\bf A})$ is d.i. as NM-algebra.

  * Proof.  It suffices to prove that $\boxminus$ and $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ satisfy the properties of Definition 3.10. First of all, recall that the top element of ${\bf NM}^{+}({\bf A})$ is the pair $(\bot,\top)$. Thus, $\boxminus(\bot,\top)=(\lozenge\bot,\Box\top)=(\bot,\top)$ and $(\boxminus 1)$ holds. Analogously, it is easy to see that $(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,1)$ holds as well.

As for $(\boxminus 2)$, let $(a^{-},a^{+}),(b^{-},b^{+})\in NM^{+}({\bf A})$.
Then, recalling the definition of $\wedge$ in ${\bf NM}^{+}({\bf A})$,
$\boxminus((a^{-},a^{+})\wedge(b^{-},b^{+}))=\boxminus(a^{-}\vee
b^{-},a^{+}\wedge a^{+})=(\lozenge(a^{-}\vee b^{-}),\Box(a^{+}\wedge
a^{+}))=(\lozenge a^{-}\vee\lozenge b^{-},\Box a^{+}\wedge\Box
b^{+})=\boxminus(a^{-},a^{+})\wedge\boxminus(b^{-},b^{+})$. In a similar way
one can prove that
$(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,2)$ also holds.

Now, let $(a^{-},a^{+})\in NM^{+}({\bf A})$. Then,
$\neg\boxminus\neg(a^{-},a^{+})=\neg\boxminus(a^{+},a^{-})=\neg(\lozenge
a^{+},\Box a^{-})=(\Box a^{-},\lozenge
a^{+})=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})$
and
$(\boxminus\mbox{-}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
holds.

Let us finally prove $(F)$. The negation fixpoint of ${\bf NM}^{+}({\bf A})$
is $(\bot,\bot)$ and hence $\boxminus(\bot,\bot)=(\lozenge\bot,\Box\bot)$.
Since $\lozenge\bot=\bot$ and, by (N1), $\Box\bot=\bot$, by the order relation
of ${\bf NM}^{+}({\bf A})$, $\boxminus(\bot,\bot)=(\bot,\bot)$.

That ${\bf NM}^{+}({\bf A})$ is a d.i. NM+-algebra follows directly from
Theorem 2.7 together with the hypothesis that ${\bf A}$ is d.i. $\dashv$

Conversely, let us start with a NMAO+ of the form $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is d.i. Then, let us define ${\bf G}({\bf B})$ as in
Proposition 2.5 and unary operators $\Box,\lozenge:G({\bf B})\to G({\bf B})$
as: $\Box x=\boxminus x\ast\boxminus x$ and $\lozenge
x=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x\ast\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x$.
Then the following holds.

######  Proposition 3.6.

For every NMAO+ $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is d.i., $({\bf G}({\bf B}),\Box,\lozenge)$ is a GAO in which
${\bf G}({\bf B})$ is d.i. and $\Box$ satisfies (N1).

  * Proof.  First of all ${\bf G}({\bf B})$ is d.i. because of Proposition 2.5. Moreover (N1) easily holds because, in ${\bf B}$, $b\ast b=\bot$ iff $b\leq f$ and for all such $b$’s, $\boxminus b\leq f$ because of Proposition 3.4 (2). Therefore, $\Box\bot=\boxminus\bot\ast\boxminus\bot=\bot$. Let hence prove that ${\bf G}({\bf B})$ is a GAO. The equations ($\Box 1$) and ($\lozenge 1$) are easily satisfied. As for the other equations, let $x,y\in G({\bf B})$, that is, there are $b_{x},b_{y}\in B$ such that $x=b_{x}\ast b_{x}$ and $y=b_{y}\ast b_{y}$. Let us prove ($\Box 2$) distinguishing the following cases:

(-) If $b_{x},b_{y}>f$, $x=b_{x}\ast b_{x}=b_{x}\wedge b_{x}=b_{x}$ and
$y=b_{y}\ast b_{y}=b_{y}\wedge b_{y}=b_{y}$. Thus the claim follows from
($\boxminus 2$).

(-) If $b_{x},b_{y}\leq f$, $x=b_{x}\ast b_{x}=\bot=b_{y}\ast b_{y}=y$ and the
claim trivially holds.

(-) If $b_{x}>f$ and $b_{y}\leq f$ or $b_{x}\leq f$ and $b_{y}>f$, then the
claims follows from the previous cases and the observation that, since ${\bf
B}$ is d.i. , for all $b_{1}>f$ and $b_{2}\leq f$, $b_{1}\wedge b_{2}=b_{2}$.

Finally, ($\lozenge 2$) also holds arguing as above. For instance, if
$b_{x}>f$ and $b_{y}\leq f$, then $x=b_{x}$ and $y=\bot$, $\lozenge(x\vee
y)=\lozenge x=\lozenge x\vee\lozenge y$ because, by ($\lozenge 1$),
$\lozenge\bot=\bot$. $\dashv$

######  Theorem 3.7.

For every GAO $({\bf A},\Box,\lozenge)$ which satisfies (N1) and such that
${\bf A}$ is d.i. and for every NMAO+ $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is d.i., the following claims hold:

  1. (1)

$({\bf A},\Box,\lozenge)\cong({\bf G}({\bf NM}^{+}(\bf A)),\Box,\lozenge)$;

  2. (2)

$({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)\cong({\bf
NM}^{+}({\bf G}({\bf
B})),\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$.

  * Proof.  In the light of Propositions 3.5, 3.6 and Theorem 2.6, it suffices to prove that the maps $\gamma$ and $\eta$ defined in (1) and (2) respectively, preserve the modal operators.

(1) Let $a\in A$. If $a=\bot$ the claim trivially follows from (N1). Thus, let
$a>\bot$ and assume $\Box a>\bot$. Then, $\gamma(\Box a)=(\bot,\Box
a)\ast(\bot,\Box a)=(\lozenge\bot,\Box a)\ast(\lozenge\bot,\Box a)=((\Box
a\vee\Box a)\to(\lozenge\bot\vee\lozenge\bot),\Box a\wedge\Box a)=(\neg\Box
a,\Box a)=(\bot,\Box a)=(\lozenge\bot,\Box a)=(\lozenge\neg a,\Box
a)=\boxminus(\neg a,a)=\boxminus(a\to\bot,a)=\boxminus((a\vee
a)\to(\bot\vee\bot),a\wedge
a)=\boxminus((\bot,a)\ast(\bot,a))=\Box(\gamma(a))$, where in the 5th and the
7th equalities we used Lemma 2.2 together with the fact that $\Box a>\bot$. If
$\Box a=\bot$, $\Box(\gamma(a))=\Box((\bot,a)\ast(\bot,a))=\boxminus((a\vee
a)\to(\bot\vee\bot),a\wedge a)=\boxminus(\neg
a,a)=\boxminus(\bot,a)=(\lozenge\bot,\Box
a)=(\bot,\bot)=\gamma(\bot)=\gamma(\Box a)$.

As for the $\lozenge$ let again $a\in A$, $a>\bot$, $\lozenge a>\bot$. Then
$\gamma(\lozenge(a))=(\bot,\lozenge a)\ast(\bot,\lozenge a)=(\lozenge
a\to\bot,\lozenge a)=(\neg\lozenge a,\lozenge a)=(\bot,\lozenge
a)=(\Box\bot,\lozenge a)=(\Box\neg a,\lozenge
a)=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,((a\vee
a)\to(\bot\vee\bot),a\wedge
a)=\lozenge((\bot,a)\ast(\bot,a))=\lozenge(\gamma(a))$. Again we used Lemma
2.2 together with the fact that $\lozenge a>\bot$, and (N1). The case
$\lozenge a=\bot$ is analogous to the above and omitted.

(2) Consider a $b\in B$. If $b=f$, $\eta(\boxminus f)=\eta(f)=f=\boxminus(f)$.
Then, let us take into account the following cases:

(-) $b>f$ and hence $b>\neg b$. Thus, $\eta(\boxminus b)=(\bot,\boxminus
b\ast\boxminus b)$. The positive elements of ${\bf B}$ are closed under
$\boxminus$ (Proposition 3.4 (2)), hence $(\bot,\boxminus b\ast\boxminus
b)=(\bot,\boxminus
b)=(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\bot,\boxminus
b)=\boxminus(\eta(b))$.

(-) $b<f$ and hence $b<\neg b$. In this case $\eta(\boxminus b)=(\neg\boxminus
b,\bot)=(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b,\bot)$. By Proposition 3.4 (3),
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg b>f$ and
hence $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b\wedge\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b\ast\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\neg
b=\lozenge\neg b$. Thus, $\eta(\boxminus b)=(\lozenge\neg
b,\bot)=(\lozenge\neg b,\Box\bot)=\boxminus\eta(b)$.

As for the $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$,
the proof is similar. Let us sketch, for example, the case $b<f$. Again, $\neg
b>f$ and $\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,b\leq
f$, whence
$\eta(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,b)=(\neg\lozenge
b,\bot)=(\boxminus\neg b,\bot)=(\Box\neg
b,\lozenge\bot)=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(\neg
b,\bot)=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,\eta(b)$.

The claim is hence settled. $\dashv$

######  Example 3.8.

Let ${\bf A}$ be the Gödel algebra whose Hasse diagram is depicted in both (a)
and (b) in Figure 1 by solid lines, and let $\Box,\lozenge:A\to A$ be as
defined by the dashed arrows in (a) and (b) respectively. Notice that $({\bf
A},\Box,\lozenge)$ satisfies (N1).

${\bot}$${a}$${b}$${c}$${\top}$

${\bot}$${a}$${b}$${c}$${\top}$

${\bot}$${x}$${y}$${z}$${f}$${k}$${t}$${s}$${\top}$

${\bot}$${x}$${y}$${z}$${f}$${k}$${t}$${s}$${\top}$

(a) (b) (c) (d)

Figure 1. A d.i. Gödel algebra with a $\Box$ operator satisfying (N1) (a) and
a $\lozenge$ operator (b), and the corresponding d.i. NM+-algebra with the
operators $\boxminus$ (c) and $\boxminus$ (d).

The Hasse diagram of the algebra ${\bf NM}^{+}({\bf A})$ is the one depicted
in (c) and (d) in Figure 1 with solid lines, where: $\bot=(\top,\bot)$,
$x=(c,\bot)$, $y=(b,\bot)$, $z=(a,\bot)$, $f=(\bot,\bot)$, $k=(\bot,a)$,
$t=(\bot,b)$, $s=(\bot,c)$, $\top=(\bot,\top)$. The operators $\boxminus$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ on
$NM^{+}({\bf A})$ defined as above correspond to the dashed arrows depicted in
(c) and (d) of Figure 1 respectively. For instance, $\boxminus
z=\boxminus(a,\bot)=(\lozenge a,\Box\bot)=(\bot,\bot)=f$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,t=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(\bot,b)=(\Box\bot,\lozenge
b)=(\bot,a)=k$.

###  3.2. The case of NM--algebras with operators

Let $({\bf A},\Box,\lozenge)$ be a GAO in which ${\bf A}$ is d.i. and
satisfying the condition (N1) and the following additional ones:

$\begin{array}[]{ll}(SM\Box)\,\mbox{ If }a>\bot\mbox{, then }\Box
a>\bot;&\hskip 8.5359pt(SM\lozenge)\,\mbox{ If }a>\bot\mbox{, then }\lozenge
a>\bot.\end{array}$

Let us define $\boxminus$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ on
$NM^{-}({\bf A})$ as above: for all $(a^{-},a^{+})\in NM^{-}({\bf A})$,
$\boxminus(a^{-},a^{+})=(\lozenge a^{-},\Box a^{+})$ and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})=(\Box
a^{-},\lozenge a^{+})$.

######  Lemma 3.9.

For every $({\bf A},\Box,\lozenge)$ as above, and for all $(a^{-},a^{+})\in
NM^{-}({\bf A})$,
$\boxminus(a^{-},a^{+}),\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})\in
NM^{-}({\bf A})$.

  * Proof.  Let us prove the claim for the case of $\boxminus$ and in particular that for all $(a^{-},a^{+})$, if it satisfies $(a^{-}\wedge a^{+})\vee\neg(a^{-}\vee a^{+})=\bot$, then $(\lozenge a^{-}\wedge\Box a^{+})\vee\neg(\lozenge a^{-}\vee\Box a^{+})=\bot$. Since ${\bf A}$ is directly indecomposable, as Gödel algebra, then so is $NM^{-}({\bf A})$ as NM-algebra. Therefore, $(a^{-}\wedge a^{+})\vee\neg(a^{-}\vee a^{+})=\bot$ iff either $a^{-}=\bot$ and $a^{+}>\bot$, or $a^{-}>\bot$ and $a^{+}=\bot$. If the former is the case, then $\lozenge a^{-}=\bot$ by $(\lozenge 1)$ and $\Box a^{+}>\bot$ by (SM$\Box$). If the latter is the case, similarly, $\lozenge a^{-}>\bot$ by (SM$\lozenge$) and $\Box a^{+}=\bot$ because of (N1). Thus in both cases $(\lozenge a^{-}\wedge\Box a^{+})\vee\neg(\lozenge a^{-}\vee\Box a^{+})=\bot$, which settles the claim. $\dashv$

For the following, recall the equations of Definition 3.10.

######  Definition 3.10.

An NM--algebra with operators (NMAO-) is a system $({\bf B},\Box,\lozenge)$
where ${\bf B}=(B,\ast,\wedge,\neg,\bot)$ be a NM--algebra, and
$\Box,\lozenge:B\to B$ satisfy the following conditions: $(\boxminus 1)$;
$(\boxminus 2)$;
$(\boxminus\mbox{-}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$;
$(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,1)$;
$(\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,2)$; and

$(P):\;\boxminus(x\vee\neg x)=\boxminus x\vee\neg\boxminus x$; $(N):\;\mbox{
if }x\leq\neg x\mbox{, then
}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x\leq\neg\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,x.$

######  Proposition 3.11.

The following properties hold in every NMAO-:

  1. (1)

The positive and the negative elements of ${\bf B}$ are closed under
$\boxminus$;

  2. (2)

The positive and the negative elements of ${\bf B}$ are closed under
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$.

  * Proof.  (1) Suppose $x\geq\neg x$. Then $x\vee\neg x=x$ and hence, by (P), $\boxminus x=\boxminus(x\vee\neg x)=\boxminus x\vee\neg\boxminus x$. Thus, $\boxminus x\geq\neg\boxminus x$. Conversely, if $x\leq\neg x$, by (P), $\boxminus\neg x=\boxminus(x\vee\neg x)=\boxminus x\vee\neg\boxminus x$, whence $\boxminus x\leq\boxminus\neg x$.

(2) The first part of the claim follows from (N), the second by the order-
reversing property of $\neg$. $\dashv$

Now we prove that, if A is a d.i. Gödel algebra and $({\bf A},\Box,\lozenge)$
is a GAO satisfying (N1), (SM$\Box$) and (SM$\lozenge$), then ${\bf
NM}^{-}({\bf A})$ is a NM--algebra with operators.

######  Proposition 3.12.

For any GAO $({\bf A},\Box,\lozenge)$ satisfying (N1), (SM$\Box$) and
(SM$\lozenge$) such that ${\bf A}$ is d.i., $({\bf NM}^{-}({\bf
A}),\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
is a NMAO- and ${\bf NM}^{-}({\bf A})$ is d.i. as NM-algebra.

  * Proof.  The equations ($\boxminus 1$), ($\boxminus 2$), ($\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,1$), ($\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,2$) and ($\boxminus\mbox{-}\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$) hold with the same proof of Proposition 3.5. Let us hence prove (P) and (N).

If $(a^{-},a^{+})\in NM^{-}({\bf A})$, then either $a^{-}=\bot$ or
$a^{+}=\bot$. We assume that $a^{-}=\bot$ without loss of generality (the case
$a^{+}=\bot$ is symmetric and omitted). Then,
$\boxminus((a^{-},a^{+})\vee\neg(a^{-},a^{+}))=\boxminus(a^{-}\wedge
a^{+},a^{+}\vee a^{-})=\boxminus(\bot,a^{+})=(\lozenge\bot,\Box
a^{+})=(\bot,\Box a^{+})$. On the other hand,
$\boxminus(a^{-},a^{+})\vee\neg\boxminus(a^{-},a^{+})=(\lozenge\bot,\Box
a^{+})\vee\neg(\lozenge\bot,\Box a^{+})=(\bot,\Box a^{+})\vee(\Box
a^{+},\bot)=(\bot\wedge\Box a^{+},\bot\vee\Box a^{+})=(\bot,\Box a^{+})$. Thus
(P) holds.

As for (N), if $(a^{-},a^{+})\leq\neg(a^{-},a^{+})$, then $a^{-}\geq\bot$ and
$a^{+}=\bot$. Therefore,
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{-},a^{+})=(\Box
a^{-},\lozenge a^{+})=(\Box a^{-},\bot)$. On the other hand
$\neg\boxminus(a^{-},a^{+})=\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,(a^{+},a^{-})=(\Box
a^{+},\lozenge a^{-})=(\bot,\lozenge a^{-})$ and $(\Box
a^{-},\bot)\leq(\bot,\lozenge a^{-})$. $\dashv$

And this is the converse direction, from NMAO-s to GAOs.

######  Proposition 3.13.

For every NMAO- $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is d.i. , $({\bf G}({\bf B}),\Box,\lozenge)$ is a GAO where
${\bf G}({\bf B})$ is d.i. and $\Box$ satisfies (N1), (SM$\Box$) and
(SM$\lozenge$).

  * Proof.  For every d.i. NM--algebra ${\bf B}$, Proposition 2.5 tells us that ${\bf G}({\bf B})$ is a d.i. Gödel algebra. The proofs of ($\Box 1$), ($\Box 2$), ($\lozenge 1$), ($\lozenge 2$) and (N1) are as in Proposition 3.6, hence let us prove (SM$\Box$) and (SM$\lozenge$).

Take $a>\bot$ in ${\bf G}({\bf B})$ and let $b\in B$ such that $a=b\ast
b>\bot$. Thus $b>\neg b$ and $a=b\wedge b=b$. It follows that $\Box
a=\boxminus a\ast\boxminus a=\boxminus b\ast\boxminus b$. By Prop. 3.12,
$\boxminus b>\neg\boxminus b$, whence $\Box a=\boxminus b\ast\boxminus
b=\boxminus b>\neg\boxminus b$ and hence, in ${\bf G}({\bf B})$, $\Box
a>\bot$. Similarly, $\lozenge a=\lozenge b\ast\lozenge b=\lozenge b$ (again by
Prop. 3.12) whence, in ${\bf G}({\bf B})$, $\lozenge a>\bot$. $\dashv$

Essentially the same proof of Theorem 3.7, together with the above
Propositions 3.12 and 3.13, proves the following representation theorem.

######  Theorem 3.14.

For all GAO $({\bf A},\Box,\lozenge)$ which satisfies (N1), (SM$\Box$) and
(SM$\lozenge$) and ${\bf A}$ is d.i. and for all NMAO- $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$
where ${\bf B}$ is d.i., we have that $({\bf A},\Box,\lozenge)\cong({\bf
G}({\bf NM}^{-}(\bf A)),\Box,\lozenge)$ and $({\bf
B},\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)\cong({\bf
NM}^{-}({\bf G}({\bf
B})),\boxminus,\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,)$.

Figure 2 shows the effect of the construction NM- to the same GAO we already
discussed in Example 3.8 and whose Gödel algebra reduct is directly
indecomposable.

${\bot}$${a}$${b}$${c}$${\top}$

${\bot}$${a}$${b}$${c}$${\top}$

${\bot}$${x}$${y}$${z}$${k}$${t}$${s}$${\top}$

${\bot}$${x}$${y}$${z}$${k}$${t}$${s}$${\top}$

(a) (b) (c) (d)

Figure 2. A d.i. Gödel algebra endowed with a $\Box$ operator (a) and a
$\lozenge$ operator (b) satisfying (N1), (SM$\Box$) and (SM$\lozenge$), and
the corresponding d.i. NM--algebra with the operators $\boxminus$ (c) and
$\mathord{\lozenge\footnotesize{\rule[2.7pt]{5.0pt}{0.4pt}}}\,$ (d).

##  4\. Conclusion and future work

In this paper we have studied the effect of taking connected and disconnected
rotations of Gödel algebras with operators whose Gödel reduct is directly
indecomposable. By doing so, we have introduced the varieties of NM+ and NM--
algebras with modal operators and we have characterized the members of these
varieties whose NM-reduct is directly indecomposable.

Although the aim of the present paper is to put forward an algebraic analysis
of the modal structures we took into account, in [11, 12] we also gave a
description of the dual relational frames that arise from finite Gödel
algebras with operators (forest frames). Those are the prelinear version of
the models, based on posets, that are a semantics for intuitionistic modal
logic (see e.g., [21, 22]). In our future work, we plan to extend the present
approach essentially in thee directions:

1\. From the point of view of forest frames, taking into account that for
every finite d.i. Gödel algebra ${\bf A}$, ${\bf NM}^{+}({\bf A})$ and ${\bf
NM}^{-}({\bf A})$ have the same prime filters, we plan to investigate how
NMAO+ and NMAO--algebras relate to forest frames and to extend to the NM-case
the isomorphic representation theorem from [11].

2\. From an algebraic perspective, besides connected and disconnected
rotations, NM-algebras can be also seen as twist-structures obtained from
Gödel algebras. It is hence interesting to deepen this latter construction for
modal Gödel algebras and compare the modal NM-algebras obtained in such a way
with the modal NM+\- and NM--algebras defined in the present paper, also in
light of the general results proved in [19].

3\. From a more general point of view, NM-algebras can be also seen as a
subvariety of Nelson lattices. In [20], the authors introduce a definition of
algebra with operators more general than the considered one in the current
paper in the sense that they are not required to satisfy Mon (monotony rules
for modal operators). It will be interesting to compare this approach to ours.

Acknowledgments. The authors thank the anonymous referees for their comments.
Authors acknowledge partial support by the MOSAIC project (EU H2020-MSCA-
RISE-2020 Project 101007627). Flaminio and Godo also acknowledge partial
support by the Spanish project PID2019-111544GB-C21 funded by
MCIN/AEI/10.13039/501100011033. Menchon acknowledge partial support by
argentinean projects PIP 112-20200101301CO (CONICET) and PICT-2019-2019-00882
(ANPCyT). The fourth author wants to acknowledge partial support by the
following argentinean projects: PIP 112-20150100412CO (CONICET) and UBA-
CyT-20020190100021BA.

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