Ring geometries and their lattices

We construct a notion of a generalized geometrical lattice, allowing us to obtain an analog of matroid theory for the case of arbitrary principal ideal rings. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 127–137, 2005.     

Two families of Newman lattices

Dedicated to Bjarni Jonsson on his seventieth birthday     

Power contexts and their concept lattices

We introduce a framework for the study of formal contexts and their lattices induced by the additional structure of self-relations on top of the traditional incidence relation. The induced contexts use subsets as objects and attributes, hence the name power context and power concept. Six types of new incidence relations are introduced by taking into account all possible combinations of universal and existential quantifiers as well as the order of the quantifications in constructing the lifted power contexts. The structure of the power concept lattice is investigated through projection mappings from the baseline objects and attributes to those of the power context, respectively. We introduce the notions of extensional consistency and intensional consistency, corresponding to the topological notions of continuity in the analogous setting when concepts are viewed as closed sets. We establish Galois connections for these notions of consistency. We further introduce the notion of faithfulness for the first type of lifted incidence relation based on the fact that it can be equivalently characterized by a concept-faithful morphism. We also present conditions under which the power concept lattice serves as a factor lattice of the base concept lattice.     

Reflexivity in Banach lattices and their subspaces

The present paper studies in detail necessary and sufficient conditions for a subspace of a Banach Lattice to be weakly complete, weakly compact or reflexive.     

Skew-symmetric vanishing lattices and their monodromy groups

Mathematische Annalen -     

Two congruence lattices of completely simple semigroups

For a Rees matrix semigroupS with normalized sandwich matrix and C(S), the congruence lattice ofS, we consider the lattice generated by {itpT_l, pK, pT_r, pt_l, pk, pt_r}. HerepT ₁ andpt _l are the upper and lower ends of the interval which makes up the _i -class of , _i being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {T _l, K, T_r, t_l, _k, t_r} where and are the equality and the universal relations onS, respectively. In both cases, we find lattices freest relative to these lattices and represent them as distributive lattices with generators and relations.With 3 Figures     

Two embedding theorems for lower bounded lattices

The purpose of this note is to present the following two results that were announced in [2].Presented by B. Jónsson.     

Conformal nets associated with lattices and their orbifolds

In this paper, we study representations of conformal nets associated with positive definite even lattices and their orbifolds with respect to isometries of the lattices. Using previous general results on orbifolds, we give a list of all irreducible representations of the orbifolds, which generate a unitary modular tensor category.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.     

Weak distributive laws and their role in free lattices

In this note we use concepts from Marcel Erné's work on weak distributive laws to show that free lattices have a property called *-distributivity. While this property, which is preserved by bounded epimorphisms, is equivalent to a -semidistributivity for finite lattices, it is much stronger in the infinity case, entailing not only -continuity but staircase distributivity, a property that is inherited by sublattices.Presented by R. Freese.