Full ideals of polynomial rings

An idealI of the ringK[x ₁, ...,x _n ] of polynomials over a fieldK inn indeterminates is a full ideal ifI is closed under substitution,f I,g ₁...g_n K[x ₁, ...,x _n ] implyf(g ₁, ...,g _n ) I. In this paper we continue the investigation of full ideals ofK[x ₁, ...,x _n ]. In particular we determine several classes of full ideals ofK[x, y] (K a finite field) and investigate properties of these classes.The first author gratefully acknowledges support from theDeutsche Forschungsgemeinschaft     

Solving multi-metric network problems: An interplay between idempotent semiring rules

We motivate computations in a multifunctional networked system as instances of algebraic path problems on labeled graphs. We illustrate, using examples, that composition operators used in many function computations in a networked system follow semiring axioms. We present an abstract framework, using a special idempotent semiring algebraic path problem, to handle multiple metrics for composition. We show that using different vector order relations in this abstract framework, we can obtain different rules of compositions such as Pareto, lexicographic and max-order efficiency. Under this framework, we identify a class of tractable composition rules that can be solved in different multi-criteria settings at affordable computational cost. We demonstrate using an example of trusted routing in which logical security rules of admission control can be combined with delay performance metrics in the multi-criteria optimization framework.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

Different orthomodular orthocomplementations on a lattice

A new method for constructing separable Hilbert lattices is described. Examples of lattices are given that admit infinitely many different orthomodular orthocomplementations.     

A lattice isomorphism between sets of ideals of the near-rings in a near-ring morita context

We establish a one-to-one correspondence between certain sets of ideals of two near-rings L and R respectively if they form a near-ring morita context (L,G,H,R) for some groups G and H.     

Prime elements from prime ideals

The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive lattice with a compact unit has a prime element, which is then used to deduce from PIT that (1) every nontrivial ring with unit has a prime ideal, and (2) every Wallman locale is spatial.     

Loewy series,V-modules and trace ideals

If _R Vis a V-module and (R V W S) is a Morita context in which (S/WV) _s is flat, then the trace ideal WVis left V-module over S. If, in addition_S:(S/WV) is flat and S/WVis a fully left idempotent ring, then Sis also fully left idempotent. The lower (upper) Loewy length of _R Vprovides an upper bound for the corresponding Loewy length of _s(WV).     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

The geometry of the chamber system of a semimodular lattice

In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric d is the gallery distance. We show that X has many properties in common with buildings. More specifically, Tits [17] has recently described buildings in terms of Weyl-group valued distance functions. We consider the Jordan-Hölder permutation (C, D) corresponding to a pair C, D of chambers and show that it has most properties of such a distance with values in the symmetric group.     

Summand absorbing submodules of a module over a semiring

An R-module V over a semiring R lacks zero sums (LZS) if $x+y=0$ implies $x=y=0$. More generally, we call a submodule W of V “summand absorbing” (SA) in V if $?x,y\in V:x+y\in W?x\in W,y\in W$. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.     

Chip firing and the tutte polynomial

It is shown that the generating function of critical configurations of a version of a chip firing game on a graphG is an evaluation of the Tutte polynomial ofG, thus proving a conjecture of Biggs [3]. Supported by a grant from D.G.A.P.A.     

Six problems of Gian-Carlo Rota in lattice theory and universal algebra

No Abstract. .The algebraic structure sooner or later comes to dominate, whether or not it is recognized when a subject is born. Algebra dictates the analysis. Gian-Carlo Rota [33]     

The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra S_n in the title which is an algebra obtained from a polynomial algebra P_n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P_n. The algebra S_n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S_n is 2n; but the weak and the global dimensions of S_n are n. The prime and maximal spectra of S_n are found, and the simple S_n-modules are classified. It is proved that the algebra S_n is central, prime, and catenary. The set I_n of idempotent ideals of S_n is found explicitly. The set I_n is a finite distributive lattice and the number of elements in the set I_n is equal to the Dedekind number d_n.     

On annihilator ideals of a polynomial ring over a noncommutative ring

Let R be a ring and let R[x] denote the polynomial ring over R. We study relations between the set of annihilators in R and the set of annihilators in R[x].     

All hereditary torsion theories are differential

Let α and β be automorphisms on a ring R and δ:R→R an (α,β)-derivation. It is shown that if F is a right Gabriel filter on R then F is δ-invariant if it is both α and β-invariant. A consequence of this result is that every hereditary torsion theory on the category of right R-modules is differential in the sense of Bland (2006). This answers in the affirmative a question posed by Vaš (2007) and strengthens a result due to Golan (1981) on the extendability of a derivation map from a module to its module of quotients at a hereditary torsion theory.     

The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1

Let S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex Δ. The purpose of this paper is investigating the Buchsbaum property of S/I^(r) when Δ is pure dimension 1. We shall characterize the Buchsbaumness of S/I^(r) in terms of the graphical property of Δ. That is closely related to the characterization of the Cohen-Macaulayness of S/I^(r) due to the first author and N.V. Trung.     

The Shifting Lemma and shifting lattice identities

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity and to any variable y occurring in we introduce a Horn sentence S(, y). When S(, y) happens to be equivalent to , we call it a shift of . When has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift.     

On strongly Π-regular ideals

In this paper, we introduce the concept of strongly π-regular ideal of a ring. We prove that every square regular matrix over a strongly π-regular ideal of a ring admits a diagonal reduction.     

Weakening partial orderings to lattice and semilattice orderings

A partial ordering on a set P can be weakened to an upper or lower semilattice ordering, respectively a lattice ordering, if and only if P is filtered in the appropriate direction(s).This work was done while the author was partly supported by NSF contract DMS 85-02330.