arXiv subjects

Thomas Vetterlein

Publications and source records attributed to Thomas Vetterlein.

16 recordsLinked to original sources

arXiv2025

A characterisation of orthomodular spaces by Sasaki maps

Given a Hilbert space $H$, the set $P(H)$ of one-dimensional subspaces of $H$ becomes an orthoset when equipped with the orthogonality relation $\perp$ induced by the inner product on $H$. Here, an \emph{orthoset} is a pair $(X,\perp)$ of a set $X$ and a symmetric, irreflexive binary relation $\perp$ on $X$. In this contribution, we investigate what conditions on an orthoset $(X,\perp)$ are sufficient to conclude that the orthoset is isomorphic to $(P(H),\perp)$ for some orthomodular space $H$, where \emph{orthomodular spaces} are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce \emph{Sasaki maps} on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset $(X,\perp)$ with sufficiently many Sasaki maps is isomorphic to $(P(H),\perp)$ for some orthomodular space, and we give more conditions on $(X,\perp)$ to assure that $H$ is actually a Hilbert space over $\mathbb R$, $\mathbb C$ or $\mathbb H$.

math-ph↗

arXiv2025

Axiomatising the dagger category of complex Hilbert spaces

We axiomatise the dagger category of complex Hilbert spaces and bounded linear maps, using exclusively purely categorical conditions. Our axioms are chosen with the aim of an easy interpretability: two of them describe the composition of objecs, two further ones deal with the decomposition of objects, and a final axiom expresses a symmetry property. The categorical reconstruction of complex Hilbert spaces addresses foundational issues in quantum physics. We present a simplified alternative to recent characterisations.

math.CT↗

arXiv2025

Categories of orthosets and adjointable maps

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is orthogonal to every element. The collection of subspaces of a Hilbert space that are spanned by a single vector provides a motivating example. We say that a map $f \colon X \to Y$ between orthosets with 0 possesses the adjoint $g \colon Y \to X$ if, for any $x \in X$ and $y \in Y$, $f(x) \perp y$ if and only if $x \perp g(y)$. We call $f$ in this case adjointable. For instance, any bounded linear map between Hilbert spaces induces a map with this property. We discuss in this paper adjointability from several perspectives and we put a particular focus on maps preserving the orthogonality relation. We moreover investigate the category OS of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory iOS of irredundant orthosets with 0. iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of projective Hilbert spaces.

math.LO↗

arXiv2025

Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation

An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space $H$, we may associate the orthoset $(P(H),\perp)$, consisting of the set of one-dimensional subspaces of $H$ and the usual orthogonality relation. $(P(H),\perp)$ determines $H$ essentially uniquely. We characterise in this paper certain kinds of Hermitian spaces by imposing transitivity and minimality conditions on their associated orthosets. By gradually considering stricter conditions, we restrict the discussion to a more and more narrow class of Hermitian spaces. We are eventually interested in quadratic spaces over countable subfields of $\mathbb R$. A line of an orthoset is the orthoclosure of two distinct element. For an orthoset to be line-symmetric means roughly that its automorphism group acts transitively both on the collection of all lines as well as on each single line. Line-symmetric orthosets turn out to be in correspondence with transitive Hermitian spaces. Furthermore, quadratic orthosets are defined similarly, but are required to possess, for each line $\ell$, a group of automorphisms acting on $\ell$ transitively and commutatively. We show the correspondence of quadratic orthosets with transitive quadratic spaces over ordered fields. We finally specify those quadratic orthosets that are, in a natural sense, minimal: for a finite $n \geq 4$, the orthoset $(P(R^n),\perp)$, where $R$ is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank $n$.

math.RA↗

arXiv2025

Dagger categories of orthosets and the complex Hilbert spaces

An orthoset is a non-empty set $X$ together with a symmetric binary relation $\perp$ and a constant $0$ such that $x \not\perp x$ for any $x \neq 0$, and $0 \perp x$ for any $x$. Maps $f \colon X \to Y$ and $g \colon Y \to X$ between orthosets are said to form an adjoint pair if, for any $x \in X$ and $y \in Y$, $f(x) \perp g$ if and only if $x \perp g(x)$. Hilbert spaces, equipped with the usual orthogonality relation and the zero vector, provide the motivating examples of orthosets. The usual adjoints of bounded linear maps between Hilbert spaces are adjoints also in our sense. We investigate dagger categories of orthosets and maps between them, requiring that any morphism and its dagger form an adjoint pair. We indicate conditions under which such a category is unitarily dagger equivalent to the dagger category of complex Hilbert spaces and bounded linear maps.

math.RA↗

arXiv2025

Adjointable maps between linear orthosets

Given an (anisotropic) Hermitian space $H$, the collection $P(H)$ of at most one-dimensional subspaces of $H$, equipped with the orthogonal relation $\perp$ and the zero linear subspace $\{0\}$, is a linear orthoset and up to orthoisomorphism any linear orthoset of rank $\geq 4$ arises in this way. We investigate in this paper the correspondence of structure-preserving maps between Hermitian spaces on the one hand and between the associated linear orthosets on the other hand. Our particular focus is on adjointable maps. We show that, under a mild assumption, adjointable maps between linear orthosets are induced by quasilinear maps between Hermitian spaces and if the latter are linear, they are adjointable as well. Specialised versions of this correlation lead to Wigner-type theorems; we see, for instance, that orthoisomorphisms between the orthosets associated with at least $3$-dimensional Hermitian spaces are induced by quasiunitary maps. In addition, we point out that orthomodular spaces of dimension $\geq 4$ can be characterised as irreducible Fréchet orthosets such that the inclusion map of any subspace is adjointable. Together with a transitivity condition, we may in this way describe the infinite-dimensional classical Hilbert spaces.

math.RA↗

arXiv2021

Transitivity and homogeneity of orthosets and the real Hilbert spaces

An orthoset (also called an orthogonality space) is a set $X$ equipped with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. In quantum physics, orthosets play a central role. In fact, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. A complex Hilbert space can be seen as a real Hilbert space endowed with a complex structure. This fact motivates us to explore characteristic features of real Hilbert spaces by means of the abelian groups of rotations of a plane. Accordingly, we consider orthosets together with the groups of automorphisms that keep the orthogonal complement of a given pair of distinct elements fixed. We establish that, under a transitivity and a homogeneity assumption, an orthoset arises from a projective (anisotropic) Hermitian space. To find conditions under which the latter's scalar division ring is $\mathbb R$ is difficult in the present framework. However, restricting considerations to divisible automorphisms, we can narrow down the possibilities to positive definite quadratic spaces over an ordered field. The further requirement that the action of these automorphisms is quasiprimitive implies that the scalar field is a subfield of $\mathbb R$.

math.RA↗

arXiv2021

Linear orthogonality spaces as a new approach to quantum logic

The notion of an orthogonality space was recently rediscovered as an effective means to characterise the essential properties of quantum logic. The approach can be considered as minimalistic; solely the aspect of mutual exclusiveness is taken into account. In fact, an orthogonality space is simply a set endowed with a symmetric and irreflexive binary relation. If the rank is at least $4$ and if a certain combinatorial condition holds, these relational structures can be shown to give rise in a unique way to Hermitian spaces. In this paper, we focus on the finite case. In particular, we investigate orthogonality spaces of rank at most $3$.

math.LO↗

arXiv2020

A geometric approach to Wigner-type theorems

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of ${\mathcal P}(H)$ to itself is induced by a linear or conjugate-linear isometry ($H$ is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

math-ph↗

arXiv2020

Categories of orthogonality spaces

An orthogonality space is a set equipped with a symmetric and irreflexive binary relation. We consider orthogonality spaces with the additional property that any collection of mutually orthogonal elements gives rise to the structure of a Boolean algebra. Together with the maps that preserve the Boolean structures, we are led to the category ${\mathcal N}{\mathcal O}{\mathcal S}$ of normal orthogonality spaces. Moreover, an orthogonality space of finite rank is called linear if for any two distinct elements $e$ and $f$ there is a third one $g$ such that exactly one of $f$ and $g$ is orthogonal to $e$ and the pairs $e, f$ and $e, g$ have the same orthogonal complement. Linear orthogonality spaces arise from finite-dimensional Hermitian spaces. We are led to the full subcategory ${\mathcal L}{\mathcal O}{\mathcal S}$ of ${\mathcal N}{\mathcal O}{\mathcal S}$ and we show that the morphisms are the orthogonality-preserving lineations. Finally, we consider the full subcategory ${\mathcal E}{\mathcal O}{\mathcal S}$ of ${\mathcal L}{\mathcal O}{\mathcal S}$ whose members arise from positive definite Hermitian spaces over Baer ordered $\star$-fields with a Euclidean fixed field. We establish that the morphisms of ${\mathcal E}{\mathcal O}{\mathcal S}$ are induced by generalised semiunitary mappings.

math.RA↗

arXiv2020

Gradual transitivity in orthogonality spaces of finite rank

An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements $e$ and $f$, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps $e$ to $f$, and any of the automorphisms leaves the elements orthogonal to $e$ and $f$ fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

math-ph↗

arXiv2018

Logic for approximate entailment in ordered universes of discourse

The Logic of Approximate Entailment (LAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAE where, at the semantic level, the underlying set of worlds is moreover endowed with an order structure. The chosen framework is designed in view of possible applications.

math.LO↗

arXiv2018

Rees coextensions of finite, negative tomonoids

A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We deal in this paper with tomonoids that are finite and negative, where negativity means that the monoidal identity is the top element. Examples can be found, for instance, in the context of finite-valued fuzzy logic. By a Rees coextension of a negative tomonoid $S$, we mean a negative tomonoid $T$ such that a Rees quotient of $T$ is isomorphic to $S$. We characterise the set of all those Rees coextensions of a finite, negative tomonoid that are by one element larger. We thereby define a method of generating all such tomonoids in a stepwise fashion. Our description relies on the level-set representation of tomonoids, which allows us to identify the structures in question with partitions of a certain type.

math.RA↗

arXiv2018

Logic of Approximate Entailment in quasimetric spaces

The logic LAE discussed in this paper is based on an approximate entailment relation. LAE generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance. We adopt the conceptual framework defined by E. Ruspini; our work is towards a contribution to the investigation of suitable logical calculi. LAE is based on the assumption that the distance function is a quasimetric. We provide a proof calculus for LAE and we show its soundness and completeness for finite theories. As our main tool for showing completeness, we use a representation of proofs by means of weighted directed graphs.

math.LO↗

arXiv2018

Real coextensions as a tool for constructing triangular norms

We present in this paper a universal method of constructing left-continuous triangular norms (l.-c. t-norms). The starting point is an arbitrary, possibly finite, totally ordered monoid fulfilling the conditions that are characteristic for l.-c. t-norms: commutativity, negativity, and quanticity. We show that, under suitable conditions, we can extend this structure by substituting each element for a real interval. The process can be iterated and if the final structure obtained in this way is order-isomorphic to a closed real interval, its monoidal operation can, up to isomorphism, be identified with a l.-c. t-norm. We specify the constituents needed for the construction in an explicit way. We furthermore illustrate the method on the basis of a number of examples.

math.LO↗

arXiv2018

Reasoning with graded information: the case of diagnostic rating scales in healthcare

In medicine one frequently deals with vague information. As a tool for reasoning in this area, fuzzy logic suggests itself. In this paper we explore the applicability of the basic ideas of fuzzy set theory in the context of medical assessment questionnaires, which are commonly used, for instance, to support the diagnosis of psychological disorders. The items of a questionnaire are answered in a graded form; patients are asked to choose an element on a linear scale. The derived diagnostic hypotheses are graded as well. This leads to the question whether there is a logical formalism that is suitable to capture the score calculation of medical assessment questionnaires and thereby provides a mathematical justification of the way in which the calculation is typically done. We elaborate two alternative approaches to this problem. First, we follow the lines of mathematical fuzzy logic. For the proposed logic, which can deal with the formation of mean values, we present a Hilbert-style deduction system. In addition, we consider a variant of the prototype approach to vagueness. In this case we are led to a framework for which to obtain a logical calculus turns out to be difficult, yet our gain is a model that is conceptually comparably well-justifiable.

math.LO↗