Related papers: The geometric lattice of embedded subsets
Quasi-lattices are introduced in terms of 'join' and 'meet' operations. It is observed that quasi-lattices become lattices when these operations are associative and when these operations satisfy 'modularity' conditions. A fundamental…
We propose an interpretation for the meets and joins in the lattice of experimental propositions of a physical theory, answering a question of Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is isomorphic to the lattice…
I propose a mathematical framework for embedding an unshaped discrete lattice $L$ on a smooth manifold $M$. This framework simplifies complex concepts in pure mathematics and physics by connecting discrete lattice structures with continuous…
Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence…
Lattice theoretical generalizations of some classical linear algebra results are formulated. A vector space is replaced by its subspace lattice and a linear map is replaced by the induced lattice map. This map is a complete join…
This paper explores the embedding of lattice structures $L \subseteq \mathbb{R}^n$ into smooth manifolds $M \subseteq \mathbb{R}^n$ through a rigorous mathematical framework. Building upon the foundational results established in "Embedding…
Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough…
A specialization semilattice is a join semilattice together with a coarser preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If $X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a…
A general result by Jackson (Flat algebras and the translation of universal Horn logic to equational logic, J. Symb. Log. 73(1) (2008) 90--128) implies that the lattice of all quasivarieties of groups of exponent dividing $n$ embeds into…
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every…
Lattices and periodic point sets are well known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces…
The structure of the coincidence symmetry group of an arbitrary $n$-dimensional lattice in the $n$-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry…
A rack is a set together with a self-distributive bijective binary operation. In this paper, we give a positive answer to a question due to Heckenberger, Shareshian and Welker. Indeed, we prove that the lattice of subracks of a rack is…
In this paper we discuss the properties of the biordered set obtained from a complemented modular lattice and defines an operation using the sandwich elements of the biordered set. Further we describe a biordered subset satisfying certain…
Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of…
In this paper, a question due to Heckenberger, Shareshian and Welker on racks in [7] is positively answered. A rack is a set together with a selfdistributive bijective binary operation. We show that the lattice of subracks of every finite…
An assosiahedron $\mathcal{K}^n$, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in $\mathbf{R}^{n}$, forming $(n-1)$-dimensional polytope. A…
Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow…
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…