Lattice Of Ideals Research Articles

Associated prime ideals of a complete lattice

Associated prime ideals of a complete lattice

Support varieties: an ideal approach

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer.

An Example of Set-theoretic Complete Intersection Lattice Ideal

We prove that the monomial curve $(t^{17}, t^{19}, t^{25}, t^{27})$ is set-theoretic complete intersection.

Constructions of Representations of Rank Two Semisimple Lie Algebras with Distributive Lattices

We associate one or two posets (which we call "semistandard posets") to any given irreducible representation of a rank two semisimple Lie algebra over ${\Bbb C}$. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these "semistandard lattices") can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the "solitary" and "edge-minimal" properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.

The generic Gröbner walk

The Gröbner walk is an algorithm for conversion between Gröbner bases for different term orders. It is based on the polyhedral geometry of the Gröbner fan and involves tracking a line between cones representing the initial and target term order. An important parameter is the explicit numerical perturbation of this line. This usually involves both time and space, demanding arithmetic of integers much larger than the input numbers. In this paper we show how the explicit line may be replaced by a formal line using Robbiano’s characterization of group orders on Q n . This gives rise to the generic Gröbner walk involving only Gröbner basis conversion over facets and computations with marked polynomials. The infinite precision integer arithmetic is replaced by term order comparisons between (small) integral vectors. This makes it possible to compute with infinitesimal numbers and perturbations in a consistent way without introducing unnecessary long integers. The proposed technique is closely related to the lexicographic (symbolic) perturbation method used in optimization and computational geometry. We report on an implementation of our algorithm specifically tailored to computations with lattice ideals.

Polynomial approximations of the relational semantics of imperative programs

We present a static analysis that approximates the relational semantics of imperative programs by systems of low-degree polynomial equalities. Our method is based on Abstract Interpretation in a lattice of polynomial pseudo ideals — finite-dimensional vector spaces of degree-bounded polynomials that are closed under degree-bounded products. For a fixed degree bound, the sizes of bases of pseudo ideals and the lengths of chains in the lattice of pseudo ideals are bounded by polynomials in the number of program variables. Despite the approximate nature of our analysis, for several programs taken from the literature on non-linear polynomial invariant generation our method produces results that are as precise as those produced by methods based on polynomial ideals and Gröbner bases.

Hereditary and Semiperfect Distributive Rings

A ring R is called right distributive if its lattice of right ideals is distributive. In this paper, we investigate distributive rings. We prove that if a ring R is right hereditary, then R is right distributive if and only if R is weakly right duo. We also prove that right semiperfect right distributive rings are right quasi-continuous. Finally, it is proved that right perfect distributive rings are quasi-Frobenius. In addition, we add examples to the situations that occur naturally in the process of this paper.

Matchings in simplicial complexes, circuits and toric varieties

Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties.

Positive definite functions of finitary isometry groups over fields of odd characteristic

This paper is part of a programme to describe the lattice of all two-sided ideals in complex group algebras of simple locally finite groups. Here we determine the extremal normalized positive definite functions for finitary groups of isometries, defined over fields of odd characteristic.

CONGRUENCE LIFTING OF DIAGRAMS OF FINITE BOOLEAN SEMILATTICES REQUIRES LARGE CONGRUENCE VARIETIES

We construct a diagram [Formula: see text], indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 24, such that for any variety V of algebras, if [Formula: see text] has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M3. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.

The monomial ideal of a finite meet-semilattice

Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice, the Alexander dual is computed.

Primal decomposition in rings, modules and lattice modules

In a work by L. Fuchs, W. Heinzer and B. Olberding, a decomposition of ideals in a commutative ring as intersections of primal isolated component ideals is investigated. In subsequent work by L. Fuchs and R. Reis, these ideas are developed in multiplicative lattices. The object of this note is to point out that, when specialized to the lattice of ideals of a commutative ring, the decomposition of L. Fuchs and R. Reis does not give the decomposition obtained in the paper by Fuchs, Heinzer and Olberding, and to give two variations of the decomposition of Fuchs and Reis. One of these variations, when specialized to the lattice of ideals of a ring, does give the decomposition obtained by Fuchs, Heinzer and Olberding, and the other one gives a decomposition which is superior in some ways.

The near-ring of congruence-preserving functions on an expanded group

Let V be a finite expanded group, e.g., a ring or a group. We investigate the near-ring 〈 C 0 ( V ) ;+ , ∘ 〉 of zero-preserving congruence-preserving functions on V. We obtain some information on the structure of 〈 C 0 ( V ) ;+ , ∘ 〉 from the lattice of ideals of V: for example, the number of maximal ideals of 〈 C 0 ( V ) ;+ , ∘ 〉 is completely determined by the isomorphism class of the ideal lattice of V.

Set-theoretic complete intersection lattice ideals in monoid rings

Assume that the characteristic of the base field is zero. We give a necessary and sufficient condition for the defining ideal of a monomial curve to be generated by one polynomial on another lattice ideal, up to radical. By applying it, we provide two examples of monomial curves in affine 4-space; in the first one, a monomial curve is set-theoretic complete intersection, in the second, it is never generated by two binomials and one polynomial up to radical.

Optimal Mal’tsev conditions for congruence modular varieties

For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jonsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.

Stanley–Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I L , ρ ⊂ K [ x 1 , … , x m ] a cone σ and a simplicial complex Δ σ with vertices the minimal generators of the Stanley–Reisner ideal of σ . We assign a simplicial subcomplex Δ σ ( F ) of Δ σ to every polynomial F. If F 1 , … , F s generate I L , ρ or they generate rad ( I L , ρ ) up to radical, then ⋃ i = 1 s Δ σ ( F i ) is a spanning subcomplex of Δ σ . This result provides a lower bound for the minimal number of generators of I L , ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.

Matrices Defining Gorenstein Lattice Ideals

We study a class of integer matrices that define Gorenstein lattice ideals. We call them Gorenstein matrices. We give a combinatorial characterization of those which are of size (n + 1) × n and we relate them to the Frobenius problem in integer programming theory. We also give a necessary and sufficient condition for Gorensteinness of generic matrices which are defined in integer programming theory.

Divisibility and Factorization of Kernel Functors#

In this article we explore the semigroup of kernel functors associated to a generalized discrete valuation ring. We will show that, among other interesting features, this semigroup is linearly ordered under divisibility, and we will present a factorization of kernel functors in terms of irreducible elements. The asymmetry of the divisibility condition will also be considered. In a way, these results are closely related to (and in the spirit of) the corresponding results obtained by Brungs for the lattice of ideals of this type of rings.

Lattices of two-sided ideals of locally matricial algebras and the Γ-invariant problem

We develop a method of representation of distributive (V,0,1)-semilattices as semilattices of finitely generated ideals of locally matricial algebras. We use the method to reprove two representation results by G. M. Bergman and prove a new one that every distributive (0,1)-lattice is, as a semilattice, isomorphic to the semilattice of all finitely generated ideals of a locally matricial algebra. We apply this fact to solve the Γ-invariant problem.

Complete intersection lattice ideals

In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of Z n ⊕ T with no invertible elements, where T is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.