Related papers: Convergence without points
We define a point-free construction of real exponentiation and logarithms, i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0,…
The theory of condensed mathematics by Dustin Clausen and Peter Scholze claims that topological spaces should be replaced by the definition of condensed sets. The main purpose of this paper is to investigate in which way the theory of…
We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that…
Given a bounded lattice $L$ with bounds $0$ and $1$, it is well known that the set $\mathsf{Pol}_{0,1}(L)$ of all $0,1$-preserving polynomials of $L$ forms a natural subclass of the set $\mathsf{C}(L)$ of aggregation functions on $L$. The…
By a 1941 result of Ph. M. Whitman, the free lattice FL(3) on three generators includes a sublattice $S$ that is isomorphic to the lattice FL($\omega$)=FL($\aleph_0$) generated freely by denumerably many elements. The first author has…
The geometric properties of a lattice can have profound consequences on its band spectrum. For example, symmetry constraints and geometric frustration can give rise to topologicially nontrivial and dispersionless bands, respectively.…
We present a contravariant reflection of the compact $T_1$-spaces with arrows given by closed continuous functions into the category of bounded distributive lattices with arrows given by closed subfit morphisms. This reflection extends both…
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…
Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in…
Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a…
A net $(x_\gamma)_{\gamma\in\Gamma}$ in a locally solid Riesz space $(X,\tau)$ is said to be unbounded $\tau$-convergent to $x$ if $|x_\gamma-x|\wedge u\mathop{\overset{\tau}{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is…
We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods, and formulate and…
Single-ended circuit topologies, and a theorem for the development thereof, are presented with which one may realize constant-resistance (or reflectionless) filters, having ideally zero reflection coefficient at all frequencies and from all…
Let (L_i : i\in I) be a family of lattices in a nontrivial lattice variety V, and let \phi_i: L_i --> M, for i\in I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the…
We study light-front physics and conformal symmetry, and their interplay both on and off the light cone. The full symmetry of the light cone is conformal symmetry not just Lorentz symmetry. Spontaneously breaking conformal symmetry gives…
Sampling theory is a discipline in communications engineering involved with the exact reconstruction of continuous signals from discrete sets of sample points. From a physics perspective, this is interesting in relation to the question of…
A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two…
We give a definition of compactness in L-fuzzy topological spaces and provide a characterization of compact L-fuzzy topological spaces, where L is a complete quasi-monoidal lattice with some additional structures, and we present a version…
A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a…
We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the…