Modular Lattices Research Articles

Groups with finitely many isomorphism classes of non-modular subgroups

Groups in which the non-moduar subgroups fall into finitely many isomorphism classes are considered, and it is proved that a (generalized) soluble group with this property either has modular subgroup lattice or is a minimax group. The corresponding result for (generalized) soluble groups with finitely many isomorphism classes of non-permutable subgroups is also obtained.

Dual digraphs of finite meet-distributive and modular lattices

We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Ploščica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work in this journal on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing three open problems.

Lattice Algebraic Structures on LDMFS Domains

The premise of the soft set was designed to model vague ideas. In this kind of model, where the set of parameters exhibits some degree of order, lattice order theory is quite beneficial. For researchers studying uncertainty, the notion of soft sets, lattices, and fuzzy sets has proven essential. In this study, the postulation of the Lattice ordered Linear Diophantine Multi-Fuzzy Soft Set (LLDMFSS) is laid out. The fundamental objective of this study is the formation of hybrid algebraic structures for LLDMFSS. Particularly, the characteristics of complete lattice, modular lattice, and distributive lattice were inhibited in LLDMFSS, and some pertinent findings were obtained. Subsequently, the modular and distributive LLDMFSS characterization theorem was implemented. The notion of the direct product for two LLDMFSSs was then addressed.

Strongly Extending Modular Lattices

In this paper, our purpose is to initiate the study of the concept of strongly extending modular lattices based on the similar notion of strongly extending modules. We will prove some basic properties of strongly extending modular lattices and employ this results to give applications to the category of modules with a fixed hereditary torsion class and Grothendieck categories.

A study of complete lattices of covering spaces

Let C(X) denote the set of all covering spaces (X', x', p) of (X,x) where (X,x) are path connected,locally path connected and semilocally simply connected pointed topological spaces. In [1], it is shown that (C(X),≥) is a lattice with respect to the partial order relation ≥. Also (C(X),≥) is a modular, bounded and complete lattice when π(X,x) is abelian. In this paper, we will study some properties of the complete lattice (C(X), ≥).

Some non-standard quasivarieties of lattices

The questions of the standardness of quasivarieties have been investigated by many authors. The problem "Which finite lattices generate a standard topological prevariety?" was suggested by D.M. Clark, B.A. Davey, M.G. Jackson and J.G. Pitkethly in 2008. We continue to study the standardness problem for one specific finite modular lattice which does not satisfy all Tumanov’s conditions. We investigate the topological quasivariety generated by this lattice and we prove that the researched quasivariety is not standard, as well as is not finitely axiomatizable. We also show that there is an infinite number of lattices similar to the lattice mentioned above.

On the lattice of conatural classes of linear modular lattices

The collection of all cohereditary classes of modules over a ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are the conatural classes of R-modules. In this paper we extend some results about cohereditary classes in R-Mod to the category mathcal {L_{M}} of linear modular lattices, which has as objects all complete modular lattices and as morphisms all linear morphisms. We introduce the big lattice of conatural classes in mathcal {L_{M}}, and we obtain some results about it, paralleling the case of R-Mod and arriving at its being boolean. Finally, we prove some closure properties of conatural classes in mathcal {L_{M}}.

Composition series of arbitrary cardinality in modular lattices and abelian categories

For a certain family of complete modular lattices, we prove a “Jordan–Hölder–Schreier-like” theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices with are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5⁎). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.

A programmable modular lattice metamaterial with multiple deformations

This paper describes a class of lattice metamaterials with tunable mechanical properties controlled by the material modulus and structural parameters of the cellular units. The deformation shapes can also be changed by external control. We have developed 3D printed quadrilateral cellular TPU units with specially shaped edge walls. A simplified mechanical model of the different cell deformation processes is established and verified by numerical simulations and experiments. On this basis, by tessellating and matching these transformations together, we create assembly grids with adjustable stiffness and deformation shapes controlled by thermal and displacement constraints.

On torsion theories and open classes of linear modular lattices

Given a ring R, a bijection exists between torsion theories and idempotent radicals. Further, the class of hereditary torsion theories contains the skeleton of the open classes. In this work, we extend the notions and main results in R-Mod about torsion classes, torsion-free classes, and open classes to the category of linear lattices, whose objects are complete modular lattices and whose morphisms are linear morphisms.

Pseudo-normalized Hecke eigenform and its application to extremal 2-modular lattices

It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.

On quasi-identities of finite modular lattices. II

The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based.

Robotic arm three-dimensional printing and modular construction of a meter-scale lattice façade structure

Recent advances in additive manufacturing have made the construction of large-scale 3D-printed structures possible. However, most case studies so far have primarily used concrete as the building material and regular geometries as structural forms. In this work, we present a recently built structural art installation located in Shanghai that is composed of lattice panels with regular and irregular shapes enabled by robotic arm 3D printing. To fabricate the spatial lattice structure, we developed a customized end effector for extrusion-based additive manufacturing using a robotic arm. With the developed printing head and printing path control algorithm, the modular lattice structural panels were printed using ABS material via an industrialized robotic arm and then assembled on the site as a facade. This study showcases the potential to use such modular 3D-printed lattice structures with more freedom in form and partially replace labor with robotic arms for accelerated construction and reduced cost.

Modular Multi-Copter Structure Control for Cooperative Aerial Cargo Transportation

The control problem of a multi-copter swarm, mechanically coupled through a modular lattice structure of connecting rods, is considered in this article. The system’s structural elasticity is considered in deriving the system’s dynamics. The devised controller is robust against the induced flexibilities, while an inherent adaptation scheme allows for the control of asymmetrical configurations and the transportation of unknown payloads. Certain optimization metrics are introduced for solving the individual agent thrust allocation problem while achieving maximum system flight time, resulting in a platform-independent control implementation. Experimental studies are offered to illustrate the efficiency of the suggested controller under typical flight conditions, increased rod elasticities and payload transportation.

The maximum pre-period property of the direct product of chains

The pre-period of a finite algebra is the maximum pre-period of its endomorphisms. We know that the pre-period of any finite modular lattice is less than or equal to the length of the lattice. A finite modular lattice is said to have the maximum pre-period property (MPP) if its pre-period and its length are equal. In this paper, we study MPP of the direct product of chains.

On properties of the lattice of all τ-closed n-multiply σ-local formations

Let be some partition of the set of all primes , G a finite group, a class of finite groups, and A function f of the form is called a formation σ-function. For any formation σ-function f the class is defined as follows: If for some formation σ-function f we have then the class is called σ-local and f is called a σ-local definition of Every formation is called 0-multiply σ-local. For a formation is called n-multiply σ-local provided either is the class of all identity groups or where is -multiply σ-local for all Let be a set of subgroups of G such that . Then τ is called a subgroup functor if for every epimorphism : and any groups and we have and . A class of groups is called τ-closed if for all . In this paper we prove that the lattice of all τ-closed n-multiply σ-local formations is an algebraic modular lattice of formations.

Variable stiffness plate tensegrity structures inspired with topology optimization

The present study focuses on the research on modular tensegrity-based plate lattices, which are reinforced with an internal skeleton consisting of enhanced tensegrity cells. The enhanced cells that create the skeleton should have different mechanical properties than other cells of the lattice. The novel idea described in this study consists in using the results obtained from the topology optimization of various plate structures and to simulate the topologically optimal material layout in the corresponding lattice plates by using the reinforced skeleton. Three possible ways of adjusting the properties of tensegrity cells within the lattices are described: adjustment of geometric parameters, i.e. cable-to-strut stiffness ratio, adjustment of the self-equilibrated system of normal forces, and the hybrid solution. The effectiveness of the proposed techniques is demonstrated for two in-plane plate deformation tasks. Although this study focuses on large-scale plate lattices, a similar approach can also be applied to smaller scales.

Neutrosophic N−Ideals on Sheffer Stroke BCK-Algebras

In this study, a neutrosophic N −subalgebra and neutrosophic N−ideal of a Sheffer stroke BCK-algebras are defined. It is shown that the level-set of a neutrosophic N−subalgebra (ideal) of a Sheffer stroke BCK-algebra is a subalgebra (ideal) of this algebra and vice versa. Then we present that the family of all neutrosophic N−subalgebras of a Sheffer stroke BCK-algebra forms a complete distributive modular lattice and that every neutrosophic N−ideal of a Sheffer stroke BCK-algebra is the neutrosophic N −subalgebra but the inverse does not usually hold. Also, relationships between neutrosophic N−ideals of Sheffer stroke BCK-algebras and homomorphisms are analyzed. Finally, we determine special subsets of a Sheffer stroke BCK-algebra by means of N−functions on this algebraic structure and examine the cases in which these subsets are its ideals.

The lattice coordinatized by a ring

A coordinatization functor L ( − , 1 ) : Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R ↦ L ( R ) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective ( − ) op endofunctors on Ring and Latt ; and 3) it admits localization at a left R -module. The complemented elements of L ( R , 1 ) form a partially ordered set S ( R ) isomorphic to the space of direct summands of R R . The right nonsingular rings for which the embedding of S ( R ) into the localization L ( R , 1 ) Q at the right maximal ring of quotients Q R is an isomorphism are characterized by a property that every finite matrix subgroup φ ( R R ) of the left R -module R R is essential in an element of S ( R ) . In that case, the space S ( R ) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

A STUDY OF COVERING SPACES THROUGH LATTICES

Let $C(X)$ denote the set of all covering spaces $(\tilde{X},\tilde{x},p)$ of $(X,x)$ where $(X,x)$ are path connected,locally path connected and semilocally simply connected pointed topological spaces.\\In this paper we show that:\\(i)$(C(X),\geq)$ is a lattice and $(C^r(X),\geq)$ is a subllatice of $(C(X),\geq)$ without assuming $\pi(X,x)$ is abelian,where $C^r(X)$ is the set of all regular covering spaces of $(X,x)$.\\(ii)$(C(X),\geq)$ is a modular,bounded and complete lattice when $\pi(X,x)$ is abelian.