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Substructurality and residuation in logic and algebra
Get PDFA very and natural way of introducing a logic is by using a sequent calculus, or Gentzen
system. These systems are determined by specifying a set of axioms and a set of rules.
Axioms are then starting points from which we can derive new consequences by using
the rules. Hilbert systems consist also on a set of axioms and a set of rules that are used
to deduce consequences. The main difference is that, whereas the axioms in Hilbert
systems are formulas, and the rules allow to deduce certain formulas from other sets of
formulas, in the case of Gentzen systems the axioms are sequents and the rules indicate
which sequents can be inferred from other sets of sequents. By a sequent we understand
a pair hG, Si, where G and S are finite sequences of formulas. We denote the sequent
hG, Si by G . S.1 The sequent G . S intends to formalize – at least in its origin – the
concept “the conjunction of all the formulas of G implies the disjunction of all the
formulas of S.”
The notion of a sequent calculus was invented by G. Gentzen in order to give axiomatizations
for Classical and Intuitionistic Propositional Logics. And the rules he
gave in both cases can be grouped in different categories: because of its character, the
Cut rule deserves a special category for itself; then we have the rules of introduction
and elimination of each one of the connectives, both on the left and on the right – of
the symbol . –; and finally a set of rules that do not involve any particular connective.
These rules are necessary in Classical and Intuitionistic logics because in these logics
1Traditional notations for sequents are G ) S and G ` S, but since both the symbols ) and ` have
many other meanings, we prefer to denote sequents by using the less overloaded symbol ., which can also
be found in literature with this use.
the order in which we are given the premises, or if we have them repeated, is irrelevant,
and we do not loose consequences if we extend the set of hypotheses. But there are other
logics that do not satisfy all these rules: for instance, relevance logics and linear logic.
At first, these logics were studied separately, and different theories were developed for
their investigation. But later on, researches arrived to the conclusion that all of them
share a common feature, which became more apparent after the work of W. Blok and
D. Pigozzi. It was discovered that (pointed) residuated lattices – or FL algebras – are
the algebraic counterpart of substructural logics.
In the XIX century, Boole noticed a close connection between “the laws of thought,”
as he put it, and algebra. After him, other mathematicians put together all the pieces
and described a sort of algebras, named Boole algebras after him, and shed light on the
connection anticipated by Boole: Boole algebras are the “natural” semantics for Classical
Propositional Logic. More connections were discovered between other logics and
other sorts of algebras: for instance, Heyting algebras are the “natural” semantics for
Intuitionistic Propositional Logic, and MV algebras for Łukasievicz Multivalued Logic.
But it was not until 1989, when Blok and Pigozzi published their book Algebraizable
Logics, that for the first time the connections between these logics and classes of algebras
were finally described with absolute precision. According to their definitions,
these classes of algebras are the equivalent algebraic sematics of the corresponding logics.
That is, these classes of algebras are the algebraic counterparts of the corresponding
logics. Their ideas paved the way to a new branch of mathematics called Abstract Algebraic
Logic, which investigates the connections between logics and classes of algebras,
and the so-called bridge theorems: that is, theorems that establish bridges between some
property of one realm (logic or algebra) with another property of the other realm.
The core of the connection between substructural logics and residuated lattices is
that in all these logics, some theorem of the following form could always be proven.
Thus, we could think that the metalogical symbol ’,’ is acting as a real connective. More
precisely, we could introduce a new connective , called fusion, and impose the following
rule. Given an algebraic model with a lattice reduct, it is usually the case that the meet and
join operations serve as the interpretations of the conjunction and disjunction connectives.
What should be then the interpretation of the fusion? Usually, the elements of the
lattice are thought as different degrees of truth, and “a . b is provable” is interpreted as “for every assignment, the degree of truth of a is less than that of b.” Under this
natural interpretation, the condition (1) becomes:
That is, the fusion is interpreted as a residuated operation on the lattice.
Being the algebraic semantics of substructural logics and containing many interesting
subvarieties such as Heyting algebras, MV algebras, and lattice-ordered groups,
to name a few, the variety of residuated lattices is of utmost importance to the studies
of Logic and Algebra, hence our interest. In this dissertation we carry out some
investigations on different problems concerning residuated lattices.
In what follows we give a brief description of the contents and organization of this
dissertation. Every chapter – except for the first one, which is devoted to setting the
preliminaries – starts with an introduction in which the reader will find a lengthier
explanation of the subject of the chapter, the way the material is organized, and references.
We start by compiling in Chapter 1 all the essential well-known results about residuated
lattices that we will need in the subsequent chapters. We present here the definitions
of those concepts that are not specific to some particular chapter, but general.
We define the variety of residuated lattices, and some of its more significant subvarieties.
We also introduce nuclei, and nucleus retracts. As it is widely known, the lattice
of normal convex subalgebras of a residuated lattice is isomorphic to its congruence
lattice, and hence its importance. But it turns out that also the lattice of convex (not
necessarily normal) subalgebras is of great significance, specially in the case of e-cyclic
residuated lattices. Many of its properties depend on the fact that it is a pseudo-complemented
lattice. Actually, it is a Heyting algebra. For instance, polars are special
sets usually defined in terms of a certain notion of orthogonality; in the case of e-cyclic
residuated lattices, polars are the pseudo-complements of the convex subalgebras. We
end the chapter by briefly explaining the notions of semilinearity and projectability for
residuated lattices.
In the 1960’s, P. F. Conrad and other authors set in motion a general program for the
investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties
of these algebras by inquiring into the structure of their lattices of convex `-subgroups.
This approach can be naturally extended to residuated lattices and their convex
subalgebras. We devote Chapters 2 and 3 to two different problems that can be framed
within Conrad’s program for residuated lattices. More specifically, in Chapter 2 we
revisit the Galatos-Tsinakis categorical equivalence between integral GMV algebras and negative cones of `-groups with a nucleus, showing that it restricts to an equivalence
of the full subcategories whose objects are the projectable members of these classes.
Afterwards, we introduce the notion of Gödel GMV algebras, which are expansions
of projectable integral GMV algebras by a binary term that realizes a positive Gödel
implication in every such algebra. We see that Gödel GMV algebras and projectable integral
GMV algebras are essentially the same thing. Analogously, Gödel negative cones
are those Gödel GMV algebras whose residuated lattice reducts are negative cones of
`-groups. Thus, we turn projectable integral GMV algebras and negative cones of projectable
`-groups into varieties by including this implication in their signature. We
prove that there is an adjunction between the categories whose objects are the members
of these varieties and whose morphisms are required to preserve implications.
We devote Chapter 3 to the study of certain kinds of completions of semilinear
residuated lattices. We can find in the literature different notions of completions for
residuated lattices, like for example Dedekind-McNeil completions, regular completions,
complete ideal completions, . . . Very often it happens that for a certain algebra in
a variety of residuated lattices, those completions exists but do not belong to the same
variety. That is, varieties are not closed, in general, under the operations of taking these
kinds of completions. But there are other notions of completions that might have better
properties in this regard. Conrad and other authors proved the existence of lateral completions,
projectable completions, and orthocompletions for representable `-groups, and
moreover, that the varieties of representable `-groups are closed under these completions.
Our goal in this chapter is to prove the existence of lateral completions, (strongly)
projectable completions, and orthocompletions for semilinear e-cyclic residuated lattices,
as they are a natural generalization of representable `-groups. We introduce all
these concepts along the chapter, and prove first that every semilinear e-cyclic residuated
lattice can be densely embedded into another residuated lattice which is latterly
complete and strongly projectable. We obtain this lattice as a direct limit of a certain
family of algebras obtained from the original lattice by taking quotients and products,
so the direct limit stays in the same variety where the original algebra lives. Finally,
we prove that for semilinear GMV algebras, we can find minimal dense extensions
satisfying all the required properties.
In Chapter 4 we study the failure of the Amalgamation Property on several varieties
of residuated lattices. The Amalgamation Property is of particular interest in the study
of residuated lattices due to its relation with various syntactic interpolation properties
of substructural logics. There are no examples to date of non-commutative varieties of
residuated lattices that satisfy the Amalgamation Property. The variety of semilinear
Abstract 5
residuated lattices is a natural candidate for enjoying this property, since most varieties
that have a manageable representation theory and satisfy the Amalgamation Property
are semilinear. However, we prove that this is not the case, and in the process we
establish that the same happens for the variety of semilinear cancellative residuated
lattices, that is, it also lacks the Amalgamation Property. In addition, we prove that
the variety whose members have a distributive lattice reduct and satisfy the identity
x(y ^ z)w xyw ^ xzw also fails the Amalgamation Property.
In Chapter 5 we show how some well-known results of the theory of automata, in
particular those related to regular languages, can be viewed within a wider framework.
In order to do so, we introduce the concept of module over a residuated lattice, and
show that modules over a fixed residuated lattice – that is, partially ordered sets acted
upon by a residuated lattice – provide a suitable algebraic framework for extending
the concept of a recognizable language as defined by Kleene. More specifically, we introduce
the notion of a recognizable element of a residuated lattice by a finite module
and provide a characterization of such an element in the spirit of Myhill’s characterization
of recognizable languages. Further, we investigate the structure of the set of
recognizagle elements of a residuated lattice, and also provide sufficient conditions for
a recognizable element to be recognized by a Boolean module.
We summarize in Chapter 6 the main results of this dissertation and propose some
of the problems that still remain open. We end this dissertation with an appendix
on directoids. These structures were introduced independently three times, and their
aim is to study directed ordered sets from an algebraic perspective. The structures
that we have studied in this dissertations have an underlying order, but moreover they
have a lattice reduct. That is not always the case for directed ordered sets. Hence
the importance of the study of directoids. We prove some properties of directoids and
their expansions by additional and complemented directoids. Among other results,
we provide a shorter proof of the direct decomposition theorem for bounded involute
directoids. We present a description of central elements of complemented directoids.
And finally we show that the variety of directoids, as well as its expansions mentioned
above, all have the strong amalgamation property
L'economia dell'alta formazione
Get PDFThe economic literature has devoted surprisingly little attention to the relationship between ’afterfull-
degree’ studies and labour market outcomes, compared to the case of secondary and postsecondary
education. The aim of this thesis is to shade a light on the characteristics and returns
associated to investing in further years of education after a full degree. In fact, the practice of
enrolling in extra years of higher education is common in Italy; furthermore it has been strongly
incentivised by both national and regional governments. For the analysis, I exploit an unique dataset
and the administrative documents relative to the biggest scholarship programs implemented in
the regions of Southern Italy. The research question is whether these investments translated into
short-term benefits in terms of labour market performance for the funding recipients or whether
they simply foster the brain drain. Results - OLS and IV - seem to suggest that both phenomena
occurred, even if the latter estimates are weaker. Also, I try to put the emphasis on the ’quality
side’, by exploiting a policy twist occurred in one of the regional programs: I show that students
react to incentive towards attending better institutions, but that they are also responsive to the
feature of the selection process itself
The Municipal Bond Market in Italy: an empirical analysis of the determinants of yields and credit ratings
Get PDFThis PhD thesis investigates the sub-sovereign financial market in Italy,
specifically focusing on the municipal bond market and on local government
credit ratings. This research aims to identify empirically the economic and
financial variables which affect bond yields and credit ratings of Italian local
governments, and to understand the role of such variables in explaining the
differences in interest costs paid by different bond issuers (or paid by the same
issuer at different times) and the differences in the rating judgments assigned
by the major international rating agencies.
The Italian local government finance system is characterized by the several
tiers of sub-sovereign governments (regions, provinces and cities) being
differently linked to the central government finances, and by the central
government itself conveying a non-negligible sovereign financial risk. The three
local government tiers have a different institutional nature, while cities and
provinces (jointly defined as local authorities) possess a similar status and are
subject to the same legislation, regions are characterized by a greater
independence and by a significantly higher degree of financial autonomy. These
facts should affect the investors’ evaluation of local government credit risk and,
therefore, one of the aims of this research is to analyze the determinants of subsovereign
bond yields and credit ratings in this institutional framework.
Credit ratings are investigated under a dual perspective, beyond the analysis on
the factors driving the determination of credit ratings, this research also aims to
study the role played by ratings in affecting local bond yields. First, testing if the
purchase of a rating is rewarded by lower yields, i.e. if rated issuers pay lower
yields than their unrated peers, and then if the specific creditworthiness
assessment assigned by rating agencies is considered by investors when they
price a municipal bond, i.e. if a AAA-rated issuer pays less than a BBB-rated
one.
This work is organized in two parts. Part I provides the research framework,
rationale and background, including the research planning, a description of the
local government finance system in Italy and the review of the literature. Part II
is dedicated to the empirical analysis, presenting the empirical methodology and results
Deep coral forests from the Island of Sardinia
Get PDFForests are major ecological structures of the terrestrial environment that enhance biodiversity through different kinds of specialized relations that exist between the vegetal canopy and organisms from all animal reigns. The concept of forest has recently been transplanted in the marine environment; marine forests share all ecological features of terrestrial forests. What is actually peculiar in the marine environment is the shift from the ‘vegetal’ forest in shallow depths (within the photic zone) to the ‘animal’ forest (dwelling in the circalittoral zone) throughout the depth gradient. ‘Animal forest’ concept refers to forests entirely constituted by animals, that substitute to the vegetal canopy in the three dimensional development of the habitat. Benthic suspension feeders are, in fact, the main contributors to the habitat biomass and create important three-dimensional structures that enhance the ecosystem’s overall complexity. Among factors controlling the horizontal distribution of such ecosystems, the geomorphology of the sea bottom is thought to be a key factor in the distribution of deep-sea coral as many coral species require hard substrates, particular light and current conditions. Through the combination of non-invasive techniques such as ROV footage coupled with multi-beam surveys in contrasting geological settings, this thesis aims to test (both at species and community level) if and how distribution and composition of coral forests dwelling in the mesophotic environment (from 50 to 200m depth) could be influenced by contrasting geological settings. Uni- and multivariate as well as multiple regression analyses were used to test for differences in alpha and beta biodiversity in roche du large ecosystems and canyons, which are the most common geological features along continental margins worldwide. Results revealed that, in contrast to what was previously reported in scientific literature, red coral populations are not structured along a bathymetric gradient but other intraspecific, interspecific and environmental interactions may shape the population structure. Investigations on communities emphasized a high variability at mesoscale (10s of km) that overcomes the macroscale (100s of km). In particular, tested environmental variables explain patterns of observed diversity, that is higher within the same geological feature rather than among contrasting geological settings. Finally, a ROV survey conducted over a roche du large ecosystem off Carloforte (SW Sardinia) revealed the presence of a pristine millennial black coral forest of Leiopathes glaberrima. The investigated site is a millennial deep coral oases as well as a nursery area for the spotted catshark Schyliorhinus canicula, which deserves special protection.
Our results enhance knowledge on coral forests dwelling in the mesophotic environment and provide useful insights for recasting conservation strategies, both at species and community level
A multidisciplinary approach for puzzling over fish connectivity in the Mediterranean Sea: The role of early life history stages of red mullet (Mullus barbatus)
Get PDFIntegrating connectivity patterns into marine ecosystem management is a fundamental step, specially for stock subjected to the combined impacts of human activities (overfishing, habitat degradation, etc.) and climate changes. Thus, management of marine resources must incorporates the spatial scales over which the populations are connected. Notwithstanding, studying these dynamics remains a crucial and hard task and the predictions of the temporal and spatial patterns of these mechanisms are still particularly challenging. This thesis aims to puzzle over the red mullet Mullus barbatus population connectivity in the Western Mediterranean Sea, by implementing a multidisciplinary approach. Otolith sclerochronology, larval dispersal modelling and genetic techniques were gathered in this study. More particularly, this research project focused on early life history stages of red mullet and their role in the characterization of connectivity dynamics. The results show that M. barbatus larval dispersal distances can reach a range of 200 km. The differences in early life traits (i.e. PLD, spawning and settlement dates) observed between various areas of the Western Mediterranean Sea suggest a certain level of larval patchiness, likely due to the occurrence of different spawning pulses during the reproductive period. The dispersal of individuals across distant areas, even not significant in demographic terms, is accountable for the maintenance of the genetic flow among different demes. Fluctuations in the level of exchange among different areas, due to the variability of the source-sink dynamics, could have major implications in the population connectivity patterns. These findings highlight the reliability of combining several approaches and represent a benchmark for the definition of a proper resource management, with considerable engagements in effectively assuring the beneficial effects of the existent and future conservation strategies
Thermodynamical, structural, and dynamical characterization of solvent mixtures of technological interest: experimental and computational studies
Get PDFN-methyl-2-pyrrolidone(NMP) and ionic liquids(ILs)are solvents with “green” characteristics, such as low volatility and low toxicity. They can be
used as a valid alternative to the common organic solvents with a high
environmental, health and safety impact. The aim of my research was to
investigate their thermodynamical, structural, and dynamical properties,
both in the neat state and in mixture, by using a combined approach of different experimental techniques together with computational ones.
In PAPER I, the structural effect of water on NMP over the whole
concentration range were studied by using molecular) dynamics (MD)
simulations, wide angle X ray scattering experiments, and density
measurements. The reason of why a density maximum is observed
experimentally is explained. As an extension of the previous study we
further investigated NMP Water mixtures by a combined use of NMR
spectroscopy, calorimetric measurements, and puckering analysis of MD
simulations (PAPER II).These results provided additional information on
the structural and dynamics changes of NMP taking place upon dilution.
In PAPER III and IV the results of volumetric and calorimetric
measurements concerning some alkylammonium nitrate (XAN) ionic
liquids + NMP and ethylammonium,alkanoate,(EAX),ionic,liquids + Water,
binary mixtures are reported. Negative VE and HE values were obtained for
all of the investigated systems in PAPER III and IV, indicating the presence
of strong XAN + NMP and EAX + Water interactions
The structure of nano sized poorly-crystalline iron oxy-hydroxides
Get PDFIn this thesis the structures of three nano sized poorly-crystalline
iron oxy-hydroxides, feroxyhyte, ferrihydrite and schwertmannite, are
studied and analyzed with the aim of clarifying some dubious structural features. The widely used Rietveld re�nement and EXAFS analysis are employed in order to address the consistency of the structural models proposed for these materials, one of which, feroxyhyte,
is suggested in this thesis. Furthermore, a new computer program
exploiting the Reverse Monte Carlo algorithm and the Debye Scattering Equation is presented and used in order to analyze the elusive
structures of these iron oxy-hydroxides
Il comportamento termoigrometrico degli edifici in legno nel clima Mediterraneo
Get PDFIn Sardinia timber and particulary cross laminated timber (CLT) are starting to be used for buildings due to their excellent properties and speed of execution.
But timber is not a traditional material for wall in Sardinia and the climate (very wet but also very hot in summer) brought about many questions in designers.
In this work dynamic hygrothermal simulations are performed by the use of WUFI software (Wärme und Feuchte Instationär – transient heat and moisture) developed by Fraunhofer Institute for Building Physics (IBP) and demonstrated that CLT is suitable in weather conditions of Sardinia. The commercial research software (WUFI PRO 5.3) was used to calculate heat and moisture transfer through 7 different walls and 7 types of roof in Cagliari, Oristano, Nuoro and Sassari due to their climatic conditions. The results show that there are no risks of surface or interstitial condensation and hygrothermal conditions inhibit mould growth.
In the second part the research shows the project of a family house and townhouses in Cagliari using ProCasaClima software.
ProCasaClima 2013 software was used to calculate heating and cooling in the buildings and thermal comfort for the occupants.
The buildings observe Italian regulations on energy efficiency but in order to obtain a “zero energy building” and to increase thermal comfort passive design strategies must be applied.
These strategies are not the same for all the Italian climates.
In Cagliari the most effective strategies to reduce the need for cooling are: shading elements to avoid summer sun, night ventilation of thermal mass, areal heat capacity and light colours for building envelope.
The research shows that also in Sardinia a building in CLT can reach high thermal performance and can provide comfort to its occupants
Soft pneumatic devices for blood circulation improvement
Get PDFThe research activity I am presenting in this thesis lies within the framework of a cooperation between the University of Cagliari (Applied Mechanics and Robotics lab, headed by professor Andrea Manuello Bertetto, and the research group of physicians referencing to professor Alberto Concu at the Laboratory of Sports Physiology, Department of Medical Sciences), and the Polytechnic of Turin (professor Carlo Ferraresi and his equipe at the Group of Automation and Robotics, Department of Mechanical and Aerospace Engineering) This research was also funded by the Italian Ministry of Research (MIUR – PRIN 2009). My activity has been mainly carried on at the Department of Mechanics, Robotics lab under the supervision of prof. Manuello; I have also spent one year at the Control Lab of the School of Electrical Engineering at Aalto University (Helsinki, Finland). The tests on the patients were taken at the Laboratory of Sports Physiology, Cagliari. I will be describing the design, development and testing of some soft pneumatic flexible devices meant to apply an intermittent massage and to restore blood circulation in lower limbs in order to improve cardiac output and wellness in general. The choice of the actuators, as well as the pneumatic circuits and air distribution system and PLC control patterns will be outlined. The trial run of the devices have been field--‐tested as soon a prototype was ready, so as to tune its features step--‐by--‐ step. I am also giving a characterization of a commercial thin force sensor after briefly reviewing some other type of thin pressure transducer. It has been used to gauge the contact pressure between the actuator and the subject’s skin in order to correlate the level of discomfort to the supply pressure, and to feed this value back to regulate the supply air flow. In order for the massage to be still effective without causing pain or distress or any cutoff to the blood flow, some control objective have been set, consisting in the regulation of the contact force so that it comes to the constant set point smoothly and its value holds constant until unloading occurs. The targets of such mechatronic devices range from paraplegic patients lacking of muscle tone because of their spinal cord damage, to elite endurance athletes needing a circulation booster when resting from practicing after serious injuries leading to bed rest. Encouraging results have been attained for both these two categories, based on the monitored hemodynamic variables
A.P.M.S. – Airport Pavement Management System – specificamente contestualizzato in aeroporti a traffico stagionale
Get PDFThe Pavement Management System is a useful tool for operators and managers providing a
systematic and objective method for pavement condition evaluation, maintenance planning decisions
and budget allocation. Also for Airports, it represents a support to improve the decision-making
process allowing the prediction of the pavements’ condition in future years depending on the
evolution of the traffic and allowing a feedback of past applications. The high seasonality airports
peculiarity is that there are ten times and over more movements in the peak season. This aspect
consist in a severe time reduction due to the strong increase of traffic in the peak season with only
strictly necessary and no longer be postponed maintenance actions allowed, then bringing up the
question of construction techniques and material selection, maintenance planning and monitoring
techniques. In this research this aspects were analyzed and some applications aimed to solve the
issues encountered are proposed. The work also focused the attention on the pavement condition
evaluation, especially at Network Level, with operators faced with low budget availability and
looking for easy and rapid management and comparison of available data. An application of the
developed framework was conducted on a Sardinian airport with the peak of traffic in the summer
season, from June to September