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We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
We extend the notions of complete intersection dimension and lower complete intersection dimension to the category of complexes with finite homology and verify basic properties analogous to those holding for modules. We also discuss the…
We generalize the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. A key property of this invariant is its semi-additivity on short exact sequences. We show how to…
Language is typically modelled with discrete sequences. However, the most successful approaches to language modelling, namely neural networks, are continuous and smooth function approximators. In this work, we show that Transformer-based…
We establish a relation between fully extended $2$-dimensional TQFTs and recognisable weighted formal languages, rational biprefix codes and lattice TFTs. We show the equivalence of $2D$ closed TFTs and rational exchangeable series and we…
Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.
The widespread relevance of complex networks is a valuable tool in the analysis of a broad range of systems. There is a demand for tools which enable the extraction of meaningful information and allow the comparison between different…
In this note, we extend the quasi-projective dimension of finite (that is, finitely generated) modules to homologically finite complexes, and we investigate some of homological properties of this dimension.
In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of…
In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely…
A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed version of…
We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
An ordinal-valued metric taking its values in the set of all countable ordinals can be assigned to a metrizable set of nodes in a transfinite graph. Then, a variety of results concerning nodal eccentricities, radii, diameters, centers,…
Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\boxtimes$ by generators and relations, and…
Limits and colimits of diagrams, defined by maps between sets, are universal constructions fundamental in different mathematical domains and key concepts in theoretical computer science. Its importance in semantic modeling is described by…
We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…
We present some identities dealing with reflexive and admissible relations and which, through a variety, are equivalent to congruence modularity.
The paper is devoted to the study of the boundary behavior of mappings. We consider mappings that satisfy inverse moduli inequalities of Poletskii type, under which the images of the domain under the mappings may change. It is proved that a…
We define extension maps as maps that extend a system (through adding ancillary systems) without changing the state in the original system. We show, using extension maps, why a completely positive operation on an initially entangled system…