Tensor products of orthoalgebras

We define a tensor product via a universal mapping property on the class oforthoalgebras, which are both partial algebras and orthocomplemented posets. We show how to construct such a tensor product forunital orthoalgebras, and use the Fano plane to show that tensor products do not always exist.     

Lattice ordered effect algebras

The purpose of this paper is to discuss some structural properties of lattice ordered effect algebras. We will use these structural properties to find certain lattices and classes of lattices that do not admit an effect algebra structure. Finally, using these structural properties, we will show that if L is the face lattice of a convex polytope in $ R^3 $ with more than 3 vertices, then L does not admit an effect algebra structure.Dedicated to the memory of Gian-Carlo Rota     

Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.In this paper, we discuss the property of archimedeanness for PE-algebras on the one hand and for po-groups on the other hand. We prove that under the assumption of suphomogeneity, archimedeanness holds for a PE-algebra with the Riesz property if and only if it holds for its representing group. The algebra is in that case commutative. This result is established by using the technique of MacNeille completion. We give the exact condition for this completion to exist, and we clearly exhibit the role played by archimedeanness and by sup-homogeneity.     

Topological duality for Tarski algebras

In this paper we will generalize the representation theory developed for finite Tarski algebras given in [7]. We will introduce the notion of Tarski space as a generalization of the notion of dense Tarski set, and we will prove that the category of Tarski algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called T-relations. By these results we will obtain that the algebraic category of Tarski algebras is dually equivalent to the category of Tarski spaces with certain partial functions. We will apply these results to give a topological characterization of the subalgebras. Received August 21, 2005; accepted in final form December 5, 2006.     

The overlap algebra of regular opens

Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, overlap algebras offer a negation-free framework suitable, among other things, for the development of point-free topology. A lot of topology can be done “inside” the language of overlap algebra. In particular, we prove that the collection of all regular open subsets of a topological space is an example of overlap algebra which, under natural hypotheses, is atomless. Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful both in constructive mathematics and for applications in computer science.     

On modular lattices generated by chains

We describe the free modular lattice generated by two chains and a single point, under the assumption that there are few meets. Received February 11, 2005; accepted in final form August 11, 2005.     

CoGalois and strongly coGalois actions

For a profinite group acting continuously on a discrete quasicyclic group, certain classes of closed subgroups called coGalois and strongly coGalois having natural field theoretic interpretations are investigated. Criteria for closed subgroups being coGalois and strongly coGalois as well as a complete classification of the associated actions are given.     

σ-additivity in manuals and orthoalgebras

Notions of -additivity are introduced for orthoalgebras and for manuals. It is shown that the logic of a -additive manual is a -additive orthoalgebra, and that, conversely, every -additive orthoalgebra arises as such a logic. Using this theorem, it is shown that a given orthoalgebra admits at most one reasonable extension of its orthogonal sum operation to countable jointly orthogonal sets. It is also shown that every orthoalgebra can be embedded in a sigma-orthoalgebra and every orthomodular poset, in a -OMP.     

An Abstract Cogalois Theory for profinite groups

The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.     

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies. We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved. Received August 21, 2005; accepted in final form October 17, 2006.     

Positive Sugihara monoids

It is proved that in the variety of positive Sugihara monoids, every finite subdirectly irreducible algebra is a retract of a free algebra. It follows that every quasivariety of positive Sugihara monoids is a variety, in contrast with the situation in several neighboring varieties. This result shows that when the logic R-mingle is formulated with the Ackermann constant t, then its full negation-free fragment is hereditarily structurally complete. Presented by R. W. Quackenbush. Received August 28, 2005; accepted in final form July 31, 2006.     

Weak effect algebras

Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is restricted to a partial cancellative operation. We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known structure theory of BL-algebras. Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups. This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).     

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]     

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

POWER STRUCTURES AND LOGIC

ABSTRACT

This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic.     

On amalgamation of reducts of polyadic algebras

Following research initiated by Tarski, Craig and Németi, and further pursued by Sain and others, we show that for certain subsets G of w, G polyadic algebras have the strong amalgamation property. G polyadic algebras are obtained by restricting the (similarity type and) axiomatization of -dimensional polyadic algebras to finite quantifiers and substitutions in G. Using algebraic logic, we infer that some theorems of Beth, Craig and Robinson hold for certain proper extensions of first order logic (without equality).     

Generalized ordinal sums and the decidability of BL-chains

We present a general construction of a family of ordinal sums of a sequence of structures and prove an elimination theorem for the class of ordinal sums in an expanded language. From this we deduce the decidability of the class of -ordinal sums of models of a decidable theory T. As an application of this result we prove that the theory of BL-chains is decidable.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived June 9, 2002; accepted in final form June 19, 2003.