Lattice Of Ideals Research Articles

On Linear Codes over Local Rings of Order p4

Suppose R is a local ring with invariants p,n,r,m,k and mr=4, that is R of order p4. Then, R=R0+uR0+vR0+wR0 has maximal ideal J=uR0+vR0+wR0 of order p(m−1)r and a residue field F of order pr, where R0=GR(pn,r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p,n,r,m,k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.

On the interaction of the Coxeter transformation and the rowmotion bijection

Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.

MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

Suppose that R=Zp4[u] with u2=p3β and pu=0, where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p≠2, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.

The prime spectrum of an 𝐿-algebra

We prove that the lattice of ideals of an arbitrary L L -algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of L L -algebras and characterize prime ideals in topological terms.

Topological representation of some lattices

Abstract In this paper, we first give a topological representation of some algebraic lattices of ideals of C(X). Next, we apply these results and prove that a space X is normal if and only if the lattice of closed fixed ideals of C(X) is a sublattice of the lattice of ideals of C(X). It is proved that if two rings C(X) and C(Y) are isomorphic, then two lattices Z ∘[X] and Z ∘[Y] are isomorphic. We conclude that two rings C *(X) and C *(Y) are isomorphic if and only if two lattices Z[βX] and Z[βY] are isomorphic.

Lattice Ordered G􀀀Semirings

Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this study is that the concept of gamma is new for the study of lattices. Novelty: We consider a condition (c.f. Theorem 4.1.5) for an additively idempotent semiring due to which it becomes a distributive lattice ordered semiring. Again, in general, the sum of ideals of a semiring need not be ideal. Indeed, and are ideals of is a set of non-negative integers. Clearly, (say) is not a ideal, because , but . However, this condition does not hold in the case of a lattice ordered semiring. AMS Mathematics subject classification (2020): 16Y60. Keywords: Lattices, additive idempotent, multiplicative Γ-idempotent, k-ideal, lattice ideal, Γ-morphism

Wiener Indices of Minuscule Lattices

The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.

Computing the binomial part of a polynomial ideal

Given an ideal I in a polynomial ring K[x1,…,xn] over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal Bin(I) of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate Bin(I) when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of Bin(I) for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called (s,t)-binomial parts. All algorithms have been implemented in SageMath.

M− Prime Ideals in Lattices and Their M− Prime Spectrum

M− Prime Ideals in Lattices and Their M− Prime Spectrum

The image of the pop operator on various lattices

Extending the classical pop-stack sorting map on the lattice given by the right weak order on Sn, Defant defined, for any lattice M, a map PopM:M→M that sends an element x∈M to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study PopM(M) when M is the weak order of type Bn, the Tamari lattice of type Bn, the lattice of order ideals of the root poset of type An, and the lattice of order ideals of the root poset of type Bn. In particular, we settle four conjectures proposed by Defant and Williams on the generating functionPop(M;q)=∑b∈PopM(M)q|UM(b)|, where UM(b) is the set of elements of M that cover b.

O-Fuzzy ideals in distributive lattices

O-Fuzzy ideals in distributive lattices

LATTICE OF A FUZZY PRIME IDEALS IN AN ORDERED SEMIGROUP

This paper introduces the concept of lattices of prime fuzzy ideals, which extends the classical theory of lattices of prime ideals in ring theory to the fuzzy setting. In this paper, we propose some examples and properties related to lattices of prime fuzzy ideals, contributing to the knowledge in fuzzy algebra. Think about an ordered semigroup ℛ. Plotting the function f from ℛ to the closed interval [0,1] defines a fuzzy subset of ℛ, where [0,1] ϵ ℛ. In order to explore prime ideals in rings, semigroups, and ordered semigroups in terms of fuzzy subsets, we build an ordered semigroup ℛ with ordered fuzzy points. To study, we presented the thoughts of ℒ-fuzzy prime ideals of an ordered semigroups ℛ, which forms a completelattice obeying the law relating the operations of multiplication and addition.

Semi-Extending Ideals and St-Closed Ideals in Lattices

Semi-Extending Ideals and St-Closed Ideals in Lattices

Rational points of lattice ideals on a toric variety and toric codes

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature.

Lie Structures and Chain Ideal Lattices

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed.

On the Generating Function for Intervals in Young's Lattice

In this paper, we study a family of generating functions whose coefficients are polynomials that enumerate partitions in lower order ideals of Young's lattice. Our main result is that this family satisfies a rational recursion and are therefore rational functions. As an application, we calculate the asymptotic behavior of the cardinality of a lower order ideals for the "average" partition of fixed length and give a homological interpretation of this result in relation to Grassmannians and their Schubert varieties.

Congruence Lattices of Ideals in Categories and (Partial) Semigroups

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

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.

Some Examples of BL-Algebras Using Commutative Rings

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2r+1 elements, r≥2, and BL-chains with k elements, k≥4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size (2≤n≤5).

Rank polynomials of fence posets are unimodal

We prove a conjecture of Morier-Genoud and Ovsienko that says that rank polynomials of the distributive lattices of lower ideals of fence posets are unimodal. We do this by proving a stronger version of the conjecture due to McConville, Sagan, and Smyth. Our proof involves introducing a related class of posets, which we call circular fence posets and showing that their rank polynomials are symmetric. We also apply the recent work of Elizalde, Plante, Roby, and Sagan on rowmotion on fences and show many of their homomesy results hold for the circular case as well.