Rothberger's property in all finite powers

A space X has the Rothberger property in all finite powers if, and only if, its collection of ω-covers has Ramseyan properties.     

Rothberger's property in all finite powers

A space X has the Rothberger property in all finite powers if, and only if, its collection of ω-covers has Ramseyan properties.     

Further refinements of the primal algebra characterization theorem

The compatible relations and the automorphisms of finite algebras are investigated. Moreover, stronger versions of Rosenberg's Primal Algebra Characterization Theorem and of Á. Szendrei's Primal Algebra Characterization Theorem are given.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

Categorical equivalence of varieties and invariant relations

Two varieties generated by finite algebras and , respectively are categorically equivalent under an equivalence functor which takes to iff the algebras of invariant relations of the clones of all term operations of and are isomorphic. In this paper we will prove this theorem and will give several applications. Received September 10, 1999; accepted in final form October 16, 2000.     

Model theoretic properties in the variety generated by a primal algebra

We show that the model theoretic and syntactic properties of the structures in the variety generated by a primal algebra are essentially the same as those in the variety generated by the two element Boolean algebra. The particular properties that we study here are the finite axiomatizability of their theories, the quantifier complexity of axioms for their theories, and various kinds of model completeness for these theories. It turns out that the proofs of these results can be extracted from the corresponding results for Boolean algebras if one takes the proper viewpoint.Presented by S. Burris.The author acknowledges the useful comments of the editor which led to an improved version of Proposition 1.17 as well as the formulation of some questions at the end of this paper.     

A Ramsey property for graph invariants

We consider the problem of which graph invariants have a certain property relating to Ramsey's theorem. Invariants which have this property are called Ramsey functions. We examine properties of chains of graphs associated with Ramsey functions. Methods are developed which enable one to prove that a given invariant is not a Ramsey function. Results for several familiar invariants are presented.     

Ramsey property, ultrametric spaces, finite posets, and universal minimal flows

We introduce the class COU _S of finite ultrametric spaces with distances in the set S and with two additional linear orderings. We also introduce the class EOP of finite posets with two additional linear orderings. In this paper, we prove that COU _S and EOP are Ramsey classes. In addition, we give an application of our results to calculus of universal minimal flows.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

A note on the Ramsey property

An elementary setting of the classical Ramsey property is given, which leads to simple proofs of the relevant theorems of Galvin-Prikry and Silver.

     

Congruence lattices of powers of an algebra

In Memory of Evelyn Nelson     

Categorical abstract algebraic logic: The criterion for deductive equivalence

Equivalent deductive systems were introduced in [4] with the goal of treating 1‐deductive systems and algebraic 2‐deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π‐institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence of deductive systems was generalized to a criterion for the deductive equivalence of term π‐institutions, forming a subclass of all π‐institutions that contains those π‐institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π‐institutions.     

Multiplicity function for tensor powers of modules of the A n algebra

We consider the decomposition of the pth tensor power of the module $L^{\omega _1 }$ over the algebra A_n into irreducible modules, $(L^{\omega _1 } )^{ \otimes p} = \sum\nolimits_v {m(v,p)L^v }$ . This problem occurs, for example, in finding the spectrum of an invariant Hamiltonian of a spin chain with p nodes. To solve the problem, we propose using the Weyl symmetry properties. For constructing the coefficients m(??, p) as functions of p, we develop an algorithm applicable to powers of an arbitrary module. We explicitly write an expression for the multiplicities m(??, p) in the decomposition of powers of the first fundamental module of sl(n+1). Based on the obtained results, we find new properties of systems of orthogonal polynomials (multivariate Chebyshev polynomials). Our algorithm can also be applied to tensor powers of modules of other simple Lie algebras.     

Weighted arcs,the finite radon transform and a Ramsey problem

We establish a link between the theory of (k, v)-arcs in affine planes and a graph theoretic Ramsey problem: A (n, k)-coloring of the complete graphK _u is a coloring of the edges ofK _u withk colours such that monochromatic connected subgraphs have at mostn vertices. The Ramsey numberf(n, k) is the smallestu such thatK _u does not admit a (n, k)-coloring. Let be an affine plane of orderq. Aweighted v-arc is a function w: such that line integrals are at mostv. Themass ofw is the sum of all the weights.In certain cases the maximum massM _q (v) can be determined (maximum over all weightedv-arcs and all affine planes of orderq). This yields the exact value off(n, q + 1) for all large enoughn.