Related papers: Predicative theories of continuous lattices
The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sasaki operations were introduced by the second two authors.…
Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible…
For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, --The class of all finite…
Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric…
We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices i.e. those approximate subgroups that are discrete and have finite co-volume. We prove an approximate…
This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new…
A lifting of a semilattice S is an algebra A such that the semilattice of compact (=finitely generated) congruences of A is isomorphic to S. The aim of this work is to give a categorical theory of partial algebras endowed with a partial…
We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower…
IIn a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through \emph{intervals of arbitrary size}, by a 1955 result of J. Jakub\'ik. In this note, I introduce the concept of…
The aim of this article is to extend the notions of strongly hollow and completely strongly hollow ideals of commutative rings to multiplicative lattices. We investigate their basic structural properties and prove several characterizations…
Slim semimodular lattices (for short, SPS lattices) and slim rectangular lattices (for short, SR lattices) were introduced by G. Gr\"atzer and E. Knapp in 2007 and 2009. These lattices are necessarily finite and planar, and they have been…
We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.
Topological concepts have been applied to a wide range of fields in order to successfully describe the emergence of robust edge modes that are unaffected by scattering or disorder. In photonics, indications of lasing from topologically…
We investigate the set-theoretic properties of the lattice of projections in the Calkin algebra of a separable infinite-dimensional Hilbert space in relation to those of the Boolean algebra $P(\omega)/{\rm fin}$, which is isomorphic to the…
We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of…
A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$…
The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient…
We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only…
We prove the existence of lattice isomorphic line arrangements having $\pi_1$-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is…
In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra and the quasi-local algebra. We show that strongly quasi-local…