Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. Received February 24, 1999; accepted in final form May 10, 1999.     

Endoprimal algebras

An algebra A is endoprimal if, for all the only maps from A ^k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that , regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable. Received May 16, 1997; accepted in final form November 6, 1997.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

The word problem for involutive residuated lattices and related structures

It will be shown that the word problem is undecidable for involutive residuated lattices, for finite involutive residuated lattices and certain related structures like residuated lattices. The proof relies on the fact that the monoid reduct of a group can be embedded as a monoid into a distributive involutive residuated lattice. Thus, results about groups by P. S. Novikov and W. W. Boone and about finite groups by A. M. Slobodskoi can be used. Furthermore, for any non-trivial lattice variety , the word problem is undecidable for those involutive residuated lattices and finite involutive residuated lattices whose lattice reducts belong to . In particular, the word problem is undecidable for modular and distributive involutive residuated lattices. The author would like to thank the Deutsche Telekom Stiftung for financial support. Received: 10 November 2005     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Affine complete Stone algebras

In [1] R. Beazer characterized affine complete Stone algebras having a smallest dense element. We remove this latter assumption and describe affine complete algebras in the class of all Stone algebras.Dedicated to the memory of M. KolibiarPresented by Á. Szendrei.     

Small congruence distributive varieties: Retracts, injectives, equational compactness and amalgamation

The relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence distributive varieties with 1- element subalgebras is considered and several characterization theorems are proven. Amongst others, we prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class and that every equationally compact reduced power of a finite absolute retract is an absolute retract. We also show that any elementary amalgamation class is Horn if and only if it is closed under finite direct products. The second author's work was supported by grants from the South African Council for Scientific and Industrial Research and the University of Cape Town Research Committee.     

Some of our primal algebras are missing: the canonical primal theory

It has been proven elsewhere that every variety has associated with it a unique canonical theory, where idempotent morphisms split. This article exhibits models of the canonical theory associated with any primal variety, for example, Boolean algebras. One such variety of models is generated by the several-sorted algebra with carriers of all prime cardinalities and with a clone of all finitary operations ω on and between carriers. This primal algebra was unknown. There are more. Presented by R. McKenzie. Received December 20, 2005; accepted in final form May 2, 2006.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

Amorphe Potenzen kompakter Räume

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT ₂ space is compact. TA2: An amorphous power of a compactT ₂ space which as a set is wellorderable is compact. In ZF⁰TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF⁰RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF⁰. But TA2 cannot be proved in ZF⁰ alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD ₃ (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet Mathematische Analysis an der Technischen Universität Wien.     

Residual type a posteriori error estimates for elliptic obstacle problems

Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Received June 19, 1998 / Published online December 6, 1999     

Ultrapowers and semigroups of operators

We study two semigroups of operators between Banach spaces, related with the finite representability ofc ₀ and ℓ₁. We show that these semigroups are open, have nice duality properties and can be characterized in terms of compact perturbations, and in terms of the properties of their ultrapowers. We obtain analogous results for their associated dual semigroups. Supported in part by DGICYT Grant PB 94-1052 (Spain). Supported by a postdoctoral Grant of the Ministry of Spain for Education and Science     

Equations on real intervals

A (finite or infinite) set ∑ of equations, in operation symbols F_t (t ∈T) and variables x_i, is said to be compatible with iff there exist continuous operations F_t^A on such that the algebra satisfies the equations ∑ (with the variables x_i understood as universally quantified). It is proved that there is no algorithm to decide -compatibility for all finite ∑. If the definition is restricted to C¹ idempotent operations F_t^A , then there does exist an algorithm for compatibility. Received August 9, 2005; accepted in final form February 14, 2006.     

On the (un)decidability of a near-unanimity term

We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem. While working on this paper, the author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant nos. T 37877 and K 60148.     

Relations compatible with near unanimity operations

Given a system of k-ary relations on a finite set A which are compatible with a (k + 1)-ary near unanimity operation on A, we provide a characterization of when is the system of all k-ary subuniverses of an algebra A on A.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 26, 2003; accepted in final form July 10, 2004.     

Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates. Presented by R. Willard.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

The structure of max-min hyperplanes

In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form

a₁⊗x₁⊕…⊕a_n⊗x_n⊕a_n+1=b₁⊗x₁⊕…⊕b_n⊗x_n⊕b_n+1,     

Structural reducts and the full implies strong problem

Let be a finite algebra. We show that if there exists a particular dualising alter ego that satisfies a weak form of injectivity, then the notions of full duality and strong duality for are equivalent. Presented by R. Willard. Received August 30, 2005; accepted in final form August 1, 2006.