Related papers: Join-continuity + Hypercontinuity = Prime continui…
There are well-known constructions of integrable systems which are chains of infinitely many copies of the equations of the KP hierarchy ``glued'' together with some additional variables, e.g., the modified KP hierarchy. Another…
In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in \cite{Celani2020} for semilattices together with a topological description of their canonical extension. As an…
For a finite distributive lattice $D$, let us call $Q \subseteq D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$…
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of…
The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously…
This paper focuses on semilattices with adjunctions (SLatas), which are semilattices with a greatest element enriched with a pair of adjoint maps. We develop a spectral-style duality for SLatas, building on prior topological dualities for…
We investigate two approximation relations on a T0 topological space, the n-approximation, and the d-approximation, which are generalizations of the way-below relation on a dcpo. Different kinds of continuous spaces are defined by the two…
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of…
The topologies permitted in joint ocular dominance (OD), orientation preference (OP), and direction preference (DP) maps in the primary visual cortex (V1) are considered, with the aim of finding a maximally symmetric periodic case that can…
For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is…
Let $k$ be a perfect field of characteristic $p \geq 3$, and let $K$ be a finite totally ramified extension of $K_0 = W(k)[p^{-1}]$. Let $L_0$ be a complete discrete valuation field over $K_0$ whose residue field has a finite $p$-basis, and…
The existence and stability of defect solitons in parity-time (PT) symmetric optical lattices with nonlocal nonlinearity are reported. It is found that nonlocality can expand the stability region of defect solitons. For positive or zero…
M.S. Rao recently investigated some sorts of special filters in distributive pseudocomplemented lattices. In our paper we extend this study to lattices which need neither be distributive nor pseudocomplemented. For this sake we define a…
A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible,…
For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that Id(P) is always,…
Let $O$ be a complete d.v.r. and $G$ a finite group. We give two applications of an adjunction in the stable category of $OG$. The first application gives necessary and sufficient conditions for the middle term of an almost split sequence…
A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of…
Lower semi-continuity (\texttt{LSC}) is a critical assumption in many foundational optimisation theory results; however, in many cases, \texttt{LSC} is stronger than necessary. This has led to the introduction of numerous weaker continuity…
Let $X$ and $Y$ be topological spaces, let $Z$ be a metric space, and let $f: X\times Y\to Z$ be a mapping. It is shown that when $Y$ has a countable base $\mathcal B$, then under a rather general condition on the set-valued mappings $X\ni…
We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.