Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties

We introduce new sufficient conditions for a finite algebraU to possess a finite basis of identities. The conditions are that the variety generated byU possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatizable. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.     

A note on residually small varieties of semigroups

It is proved for any varietyG of groups that if the subdirectly irreducible groups inG form a set, and if the subdirectly irreducible representation algebras of groups inG form a set, then every finite group inG is Abelian. The result is essential for the characterization of residually finite varieties of semigroups.     

Two results for monocompact measures

For every finite measure space (Ω,A, P) whereA is K₁-generated we prove the equivalence of compactness and monocompactness for P . Moreover, we prove the existence of a perfect, not monocompaot probability, thus answering an open question in [6]. Let P be a charge on the algebraA andK ?A be a monocompact class. We show that P is o-additive ifK ^S P-approximatesK ^S, the family of finite unions inK , needs not to be monocompact.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

When is the radical associated with a module a torsion?

For an arbitrary R-module M we consider the radical (in the sense of Maranda)G _M, namely, the largest radical among all radicalsG, such thatG(M). We determine necessary and sufficient on M in order for the radicalG(M) to be a torsion. In particular,G(M) is a torsion if and only if M is a pseudo-injective module.     

On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

Semiconic idempotent residuated structures

An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains.     

Spaces of compact operators and their dual spaces

LetE andF be reflexive Banach spaces andC the space of all compact linear operators fromE toF. A representation of the dual space ofC is given and it is proved thatC is either reflexive or nonconjugate. Applications of these results are also given.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

A note on compact Kähler manifolds with nef tangent bundles

In this paper, we study the compact Kähler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kähler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(?K _X )=1 if and only if there exists a finite étale coverY→X such thatY??¹×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].     

A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra ${\mathcal{A}}$ and a finite n ?? 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra ${\mathcal{P}}$ such that for any subsignature L of the signature of ${\mathcal{P}}$ that contains the boolean operations and cylindrifications, the L-reduct of ${\mathcal{P}}$ is completely representable if and only if ${\mathcal{A}}$ is completely representable. If ${\mathcal{A}}$ is finite then so is ${\mathcal{P}}$ . It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.     

Finite union theorem with restrictions

The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties:

1. ifS ₁,S ₂, ...,S _{k + 1} are distinct pariwise disjoint elements ofL then there exists nonemptyI ? {1, 2, ...,k + 1} with ∪ _i∈I S _i ?L
2. ifL =L ₁ ?...?L _r is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS ₁,S ₂, ...,S _k ∈L _j , such thatL _i∈I S _i ∈L _j for every nonemptyI ? {1, 2, ...,k}.

     

Expansion of open sets and decomposition of continuous mappings

The notion of expansionA of open sets is introduced. ThenA-expansion continuous mappingf:X→Y is defined. The main result of this note is that a mappingf is continuous if and only if it is bothA-expansion continuous andB-expansion continuous, whereA-expansion,B-expansion are two mutually dual expansions.     

Eine explizite Einbettung für Möbius-Geometrien endlicher Dimension

A new and explicit embedding is given for geometries ?(R,L) of Möbius type (i.e.L any extension field ofR) and finite dimension n = (L:R). The image of a chain is an algebraic curve. Till now explicit embeddings of Möbius geometries have been known only in the case n = 2.