Spring 2024

The theory of residuated lattices was developed to uniformly treat the common features of a number of key motivating examples, two of the most prominent ones being the varieties of Heyting algebras and of lattice-ordered groups (l-groups). One fruitful perspective on residuated lattices is to place l-groups at its center and to think of other varieties of residuated lattices as obtained from l-groups by applying two simple constructions, namely so-called nuclear and conuclear images.

A variety of residuated lattices which still preserves much group-like behavior is the variety of GMV-algebras, which subsumes both l-groups and MV-algebras. This makes the variety a reasonable target for extending classical results about l-groups. In particular, Ledda, Paoli and Tsinakis have successfully extended Hölder's theorem, one of the early classical results about ordered groups, to GMV-algebras. The original theorem describes the totally ordered groups which embed into the ordered additive group of reals as precisely the ones which enjoy the Archimedean property (informally, no positive quantity is infinitely smaller than another). The extension similarly describes the GMV-algebras embeddable as residuated lattices into either the reals, the negative reals, or the standard MV-chain [0, 1].

In this talk I will review the above line of research and aim for an extension of Hölder's theorem to lattice- and semilattice-ordered monoids. Somewhat surprisingly, the appropriate substitute for the Archimedean property is the property that every congruence arises from an ideal. In particular, I will confirm a conjecture of Peter Jipsen and Luca Spada (where this key property was first identified) that the finite semilattice-ordered monoids with a zero which satisfy this property are exactly the reducts of finite MV-algebras in this signature.

The study of integrality in residuated structures plays an important role in algebraic logic since residuated lattices are the algebraic semantics of substructural logics, and integrality (x ≤ 1) is equivalent to the proof-theoretical rule called weakening.

An involutive partially-ordered semigroup (ipo-semigroup) consists of a partial order, an associative binary operation, and a pair of order-reversing involutions, –  and ∼ that satisfy double negation (∼ – x = x = – ∼ x) and residuation. If the semigroup has an identity, we refer to the structure as an ipo-monoid.

In this talk, we investigate the class of locally integral ipo-semigroups, in which the elements of the form x/x act locally as identities ((x/x)∙x = x). We show that every locally integral ipo-semigroup decomposes uniquely into a Płonka sum over a semilattice directed system of integral ipo-monoids. We also solve the reverse problem by providing necessary and sufficient conditions so that the glueing of a system of integral ipo-monoids becomes an ipo-semigroup. Finally, generalizing the work of Jipsen, Tuyt, and Valota (2021), we provide a construction by which a glueing of locally integral lattice-ordered monoids will also be lattice-ordered.

"Interpolation" refers to a cluster of metalogical properties, intuitively corresponding to a range of different ways that the validity of given inferences in a logic may be explained in the object language. While interpolation has been studied quite thoroughly in different logical contexts, until recently there has been little in the way of a coherent picture. The deep connection between interpolation and various amalgamation properties (given in suitable categories of model structures) provides a route to developing a more systematic understanding. This talk surveys recent progress under this research program, with applications drawn from various substructural logics (including fuzzy, relevant, intuitionistic, and linear logics) as well as their modal extensions.

If one really believes that mathematics is a human construction, the result of a dynamic-evolutionary process, then it can be seen as a conquered and local truth, which means certain and reliable information, and not as a given and universal truth. It becomes crucial to avoid principles such as the Law of the Excluded Middle, the Power Set Axiom and the Axiom of Choice, which destroy the informational difference between 'exists' and 'not-forall-not', inductively generated domains and not, operations and functions. The resulting foundation does not suffer from the problems of the past.

 This talk gives an overview of what mathematics, in particular topology, has been developed on this 'weak' foundation. Most importantly, it illustrates the vast new areas of mathematical thought that emerge, in particular the co-presence and connection between real-effective and ideal-infinitary aspects.

The challenge now is to show that this could lead to a new Kuhnian paradigm in mathematics, a prospect that clearly points to much future work.

We know the category Pos of partially ordered sets (posets) and monotonic functions is dually equivalent to the category Stone(DL) of Stone distributive lattices and continuous lattice homomorphisms. Likewise, Stone(DL) is equivalent to the category of completely distributive lattices and complete lattice homomorphisms

We present an apparently novel duality for Pos that arises from considering the category Pos*, of posets (or equivalently, Qos* of preordered sets) and weakening relations. In Pos* (or Qos*), we define a monad D — the “downset” monad, and show that (a) every poset is equipped uniquely with a D-algebra, (b), there are no other D-algebras, and (c) the D-algebra morphisms are precisely dual to monotonic functions between the carriers of these algebras. Thus Pos is dually equivalent to the category of D-algebras on Pos*.

Residuated structures play an important role in the field of algebraic logic since they constitute the equivalent algebraic semantics, in the sense of Blok and Pigozzi, of substructural logics. The algebraic investigation of residuated lattices is a powerful tool in the systematic and comparative study of such logics. Thus, the development of constructions that allow one to obtain new structures from known ones is of utter importance for the understanding of both residuated lattices and substructural logics.

One of the most well known construction that allows one to get a new algebra starting from two known ones is the ordinal sum construction. Intuitively, we stack one algebra on top of the other gluing the top element. This construction preserves both products and divisions of the initial algebras. This is the main difference with the gluing construction introduced by Galatos and Ugolini, where the products are preserved but some of the divisions are redefined. In this work, we propose two possible iterations of this construction, which allows us to understand how some residuated chains can be decomposed and how their different parts interact. Moreover, we provide an axiomatization for the varieties generated by n-potent, archimedean chains that can be constructed via such iterated partial gluing.

Algebraic constructions of this kind are also important for the study of logical properties of the associated substructural logics. One of the most well-known bridge theorems is the connection between the amalgamation property in a variety of algebras and the Robinson property, or in some relevant cases the deductive interpolation property, of the corresponding logic.

In this work we focus on the study of amalgamation in classes of totally ordered residuated lattices, solving some open problems: most importantly, we establish that semilinear commutative (integral) residuated lattices and their pointed versions do not have the amalgamation property.

A special class of semirings, often referred to as rigs, are those semirings R=(R,+,*,0,1) where both operations are commutative, where multiplication distributes over addition, and in which 0 is multiplicatively absorbing; the name coming from the pun "ring" to "rng" (ring without Identity), where a "rig" is a "ring" without iNverses. As they encompass (commutative) rings (by taking the appropriate fragment), rigs are pervasive structures in mathematics, finding applications in fields as diverse as algebraic geometry, computer science, and mathematical logic. 

Of special importance are the classes of additively-idempotent rigs and integral rigs (i.e., satisfying 1+x=x), often referred to as 2-rigs and irigs, respectively, both of which define varieties of algebras and, moreover, the former subsuming the latter. Since addition is idempotent in these structures, the additive fragment is a semilattice inducing a partial order in which 0 is the least element, and in the case of irigs, also has 1 as largest element. While the names "2-rigs" and "irigs" originate from algebraic-geometric considerations, they also constitute the {v,*,1,0}-fragments of (commutative) residuated structures, namely FLe and FLew-algebras, respectively. In particular, bounded distributive lattices are exactly those irigs in which multiplication is (also) idempotent.

While the literature of semirings is vast, due to their diverse application, a serious study of basic universal algebraic properties, specifically for 2-rigs and irigs, seems to be lacking. This talk will therefore be concerned with exactly this. We will investigate subdirectly irreducibles, free algebras, the finite model property, etc.

Residuated structures play an important role in the field of algebraic logic since they constitute the equivalent algebraic semantics of a large class of logics, including classical logic and many of the most well-known systems of nonclassical logics, e.g. intuitionistic logic, intermediate logics, fuzzy logics, relevance logics and more. While many deep results have been obtained in the last decades, at the present moment large classes of residuated lattices lack a structural description; because of this, we are interested in constructions that allow one to obtain new structures from known ones. We start this talk by discussing a well-understood construction, the ordinal sum (also called 1-sum), that glues together two integral residuated lattices intersecting at their top; this construction has been successfully used to represent totally-ordered divisible integral residuated lattices. We will discuss the relevance and the limits of the ordinal sum with respect to the description of (non-divisible) integral chains, and show that if divisibility is dropped altogether, one cannot obtain the same kind of results. We will then explore two directions: on the one hand, we introduce a hierarchy of (non-divisible) varieties that can be effectively described by ordinal sums, and on the other hand, we generalize the ordinal sum construction by allowing one to glue residuated structures that intersect at a deductive filter.

A variety of algebras is called regular when it does satisfy regular identities only, where an identity σ ≈ τ (in a fixed algebraic language) is regular provided that exactly the same variables actually occur in both the terms σ and τ. Examples of regular varieties include semigroups, (commutative) monoids and semilattices. A variety is irregular if it is not regular. In particular, a variety V is strongly irregular if it possesses a term-definable (binary) operation f(x,y) such that V satisfies f(x,y) ≈ x, i.e. the operation f behaves in V as a projection in the first component. Strongly irregular varieties are very common, as they include, for instance, groups, rings and any variety having a lattice (or group) reduct.

Given a variety V, one can consider its regularization R(V), namely the variety satisfying only the regular identities holding in V. In case V is strongly irregular, then the elements of R(V) admit a well-behaved structure theory: any algebra in R(V) can be represented as a Płonka sum over a semilattice direct system of algebras in V, a construction introduced by the Polish algebraist J. Płonka. 

After formally introducing the construction of the Płonka sum, in this seminar, I will provide the details of the Płonka representation theorem for regularized varieties and show some of its applications, such as the description of the lattice of the subvarieties of R(V), of the subdirectly irreducible members in R(V) and of the equational basis (for R(V) from V, and vice versa). Finally, I will briefly explain how the theory of Płonka sums has been applied in algebraic logic.

In pointfree topology, one abstracts the poset of open subsets of a topological space, by replacing it with a frame (a complete lattice, where finite meet distributes over arbitrary joins). In this talk, we propose an analogous abstraction of the poset of connected subsets of a topological space. To motivate this abstraction, we use it to characterise posets of connected elements in a locally connected frame, and establish the intuition underlying the proof of this result.