Lattice Of Ideals Research Articles

Congruence modularity at 0

Given a variety \({\mathcal{V}}\) with a constant 0 in its type and a lattice identity p ≤ q, we say that p ≤ q holds for congruences in \({\mathcal{V}}\) at 0 if the p-block of 0 is included in the q-block of 0 for all substitutions of congruences of \({\mathcal{V}}\) -algebras for the variables of p and q. Varieties that are congruence modular at 0 are characterized by a Mal’tsev condition. This result generalizes the classical characterization of congruence modularity by Day terms.

Rings with finite decomposition of identity

A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also present a brief survey of some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given.

The strong no loop conjecture for mild algebras

Let Λ be a finite dimensional associative algebra over an algebraically closed field with a simple module S of finite projective dimension. The strong no loop conjecture says that this implies Ext Λ 1 ( S , S ) = 0 , i.e. that the quiver of Λ has no loop at the point corresponding to S. In this paper we prove the conjecture in case Λ is mild, which means that Λ has a distributive lattice of two-sided ideals and each proper factor algebra Λ / J is representation-finite. In fact, it is sufficient that a “small neighborhood” of the support of the projective cover of S is mild.

CIRCUIT INTEGRATION THROUGH LATTICE HYPERTERMS

Reducing the size of a logic circuit through lattice identities is an important and well-studied discrete optimization problem. In this paper, we consider a related problem of integrating several circuits into a single hypercircuit using the recently developed concept of lattice hyperterms. We give a combinatorial algorithm for integrating k-out-of-n symmetrical diagrams which play an important role in reliability theory. Our results show that the integration can reduce the number of circuit gates by more than twice.

Ideals of generalized matrix rings

Let and be rings, and and bimodules. In the paper, in terms of isomorphisms of lattices, relationships between the lattices of one-sided and two-sided ideals of the generalized matrix ring and the corresponding lattices of ideals of the rings and are described. Necessary and sufficient conditions for a pair of ideals , of rings and , respectively, to be the main diagonal of some ideal of the ring are also obtained.Bibliography: 8 titles.

A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or −1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. We prove product formulas counting these order ideals and give the rank generating function of some of the corresponding lattices of order ideals. We also prove an expansion of the tournament generating function as a sum over TSSCPPs. This result is analogous to a result of Robbins and Rumsey expanding the tournament generating function as a sum over alternating sign matrices.

Lattice Polly Cracker cryptosystems

Using Gröbner bases for the construction of public key cryptosystems has been often attempted, but has always failed. We review the reason for these failures, and show that only ideals generated by binomials may give a successful cryptosystem. As a consequence, we concentrate on binomial ideals that correspond to Euclidean lattices. We show how to build a cryptosystem based on lattice ideals and their Gröbner bases, and, after breaking a simple variant, we construct a more elaborate one. In this variant the trapdoor information consists in a “small” change of coordinates that allows one to recover a “fat” Gröbner basis. While finding a change of coordinates giving a fat Gröbner basis is a relatively easy problem, finding a small one seems to be a hard optimization problem. This paper develops the details and proofs related to computer algebra, the cryptographic details related to security, the comparison with other lattice cryptosystems and discusses the implementation.

Weak dimension and right distributivity of skew generalized power series rings

Let R be a ring, S a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of skew polynomial rings, skew power series rings, skew Laurent polynomial rings, skew group rings, and Mal'cev-Neumann Laurent series rings. In the case where S is positively ordered we give sufficient and necessary conditions for the skew generalized power series ring R[[S, ω]] to have weak dimension less than or equal to one. In particular, for such an S we show that the ring R[[S, ω]] is right duo of weak dimension at most one precisely when the lattice of right ideals of the ring R[[S, ω]] is distributive and ω(s) is injective for every s ∈ S.

On identities in orthocomplemented difference lattices

Abstract In this note we continue the investigation of algebraic properties of orthocomplemented (symmetric) difference lattices (ODLs) as initiated and previously studied by the authors. We take up a few identities that we came across in the previous considerations. We first see that some of them characterize, in a somewhat non-trivial manner, the ODLs that are Boolean. In the second part we select an identity peculiar for set-representable ODLs. This identity allows us to present another construction of an ODL that is not set-representable. We then give the construction a more general form and consider algebraic properties of the ‘orthomodular support’.

Rings and Gödel algebras

In this paper, we study those rings whose semiring of ideals can be given the structure of a Godel algebra. Such rings are called Godel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Godel ring R is a subdirect product of prime Godel rings R i , and the Godel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into $${{\prod_{i}}Id(R_{i})}$$ , where each Id(R i ) is the algebraic lattice of ideals of R i that can be equipped with the structure of a Godel algebra. We see that the mapping associating to each Godel ring its Godel algebra of ideals is functorial from the category of Godel rings with epimorphisms into the full subcategory of frames whose objects are Godel algebras and whose morphisms are complete epimorphisms.

A slice algorithm for koszul simplicial complexes on the lcm lattice of monomial ideals

poster A slice algorithm for koszul simplicial complexes on the lcm lattice of monomial ideals Share on Author: Bjarke Hammersholt Roune View Profile Authors Info & Claims ACM Communications in Computer AlgebraVolume 43Issue 3/4September/December 2009 pp 96–98https://doi.org/10.1145/1823931.1823946Online:24 June 2010Publication History 0citation29DownloadsMetricsTotal Citations0Total Downloads29Last 12 Months1Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Ideals, Natural Classes, and Functors

For any ring R, the set 𝒩(R) of all natural classes of R-modules is a complete Boolean lattice, which is a direct sum of two convex and complete Boolean sublattices 𝒩(R) = 𝒩 t (R) ⊕ 𝒩 f (R), where the last summand is the set of all nonsingular natural classes. The ring R contains a unique lattice of ideals 𝒥(R) which is lattice isomorphic to 𝒩 f (R). The present note develops the analogue of all of the above for an arbitrary R-module M, so that in the special case when M R = R R , the known lattice isomorphism 𝒥(R) ≅ 𝒩 f (R) is recovered.

Specializations of Multigradings and the Arithmetical Rank of Lattice Ideals

In this article, we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal I L 𝒢 ⊂ K[x 1,…, x n ]. The arithmetical rank of I L 𝒢 equals the ℱ-homogeneous arithmetical rank of I L 𝒢 , for an appropriate specialization ℱ of 𝒢. To the lattice ideal I L 𝒢 and every specialization ℱ of 𝒢, we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the ℱ-homogeneous arithmetical rank of I L 𝒢 .

Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

Extending work of von Neumann, Jonsson has shown that each complemented modular lattice, L admitting a large partial n-frame with n ≥ 4, or with n ≥ 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink’s embedding of L into to a direct product of subspace lattices of irreducible projective spaces and coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici’s thesis [6] is an important source for this approach.

Quotient of Ideals of an Intuitionistic Fuzzy Lattice

The concept of intuitionistic fuzzy ideal of an intuitionistic fuzzy lattice is introduced, and its certain characterizations are provided. We defined the quotient (or residual) of ideals of an intuitionistic fuzzy sublattice and studied their properties.

On the generalized Scarf complex of lattice ideals

Let k be a field, L ⊂ Z n be a lattice such that L ∩ N n = { 0 } , and I L ⊂ R = k [ x 1 , … , x n ] the corresponding lattice ideal. We present the generalized Scarf complex of I L and show that it is indispensable in the sense that it is contained in every minimal free resolution of R / I L .

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL ( n , C ) is an equivariant fibration X → X S of the complete flag variety X of C n . To each such fibration we associate an algebra J S of operators on L 2 ( X ) , or more generally on L 2 -sections of vector bundles over X . This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X → X S and X → X T , is a compact operator if S ∪ T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU ( n ) , which may be described as ‘essential orthogonality of subrepresentations’.

EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO ITS LATTICE OF FUZZY IDEALS

We show that the lattice of all ideals of a ring R can be embedded in the lattice of all its fuzzy ideals in uncountably many ways. For this purpose, we introduce the concept of the generalized characteristic function r(A) of a subset A of a ring R for fixed r,s 2 (0,1) and show that A is an ideal of R if, and only if, its generalized characteristic function r(A) is a fuzzy ideal of R. We also show that the set of all generalized characteristic functions Cr(I(R)) of the members of I(R) for fixed r,s 2 (0,1) is a complete sublattice of the lattice of all fuzzy ideals of R and establish that this latter lattice is generated by the union of all its complete sublattices Cr s(I(R)).

Commutative rings whose ideals form an MV‐algebra

AbstractIn this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi‐ring of ideals, is isomorphic to an MV‐algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On ideals of triangular matrix rings

We provide a formula for the number of ideals of complete block-triangular matrix rings over any ring R such that the lattice of ideals of R is isomorphic to a finite product of finite chains, as well as for the number of ideals of (not necessarily complete) block-triangular matrix rings over any such ring R with three blocks on the diagonal.