2026-09-21T04:55:01Zhttps://iris.unica.it/oai/requestoai:iris.unica.it:11584/2663842022-10-15T19:58:10Zcom_11584_207615com_11584_111066col_11584_265854
Substructurality and residuation in logic and algebra
GIL FEREZ, JOSE'
directoids
direttoidi
linguaggi riconoscibili
logiche sottostrutturali
recognizable languages
residuated lattices
reticoli residuati
substructural logics
Settore MAT/01 - Logica Matematica
A 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.
2015-05-15
info:eu-repo/semantics/doctoralThesis
http://hdl.handle.net/11584/266384
eng
numberofpages:160
info:eu-repo/semantics/openAccess
Università degli Studi di Cagliari
license:Non specificato