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     

Finitely generated varieties of distributive double <Emphasis Type="Italic">p</Emphasis>-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.     

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.     

Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the announced proposition. We dually characterize the associated submodel-injection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL.     

Ockham algebras with double pseudocomplementation

The variety dpO consists of those algebras (L; ∧, ∨, f, *, ⁺, 0, 1) with ∧, ∨ binary, f, *, ⁺ unary and 0, 1 nullary, and where (L; ∧, ∨, f, 0, 1) is an Ockham algebra and the unary operations f and * commute, f and⁺ commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass dpK_1,1, characterised by the property f³ = f. This paper is dedicated to Walter Taylor. Received September 29, 2004; accepted in final form September 8, 2005.     

Freely adjoining a relative complement to a lattice

Let K be a lattice, and let a < b < c be elements of K. We adjoin freely a relative complement u of b in [a, c] to K to form the lattice L. For two polynomials A and B over K ∪ {u}, we find a very simple set of conditions under which A and B represent the same element in L, so that in L all pairs of relative complements in [a, c] can be described. Our major result easily follows: Let [a, c] be an interval of a lattice K; let us assume that every element in [a, c] has at most one relative complement. Then K has an extension L such that [a, c] in L, as a lattice, is uniquely complemented.As an immediate consequence, we get the classical result of R. P. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. We also get the stronger form due to C. C. Chen and G. Grätzer: Every at most uniquely complemented bounded lattice has a {0, 1}-embedding into a uniquely complemented lattice. Some stronger forms of these results are also presented.A polynomial A over K ∪ {u} naturally represents an element 〈A 〉 of L. Let us call a polynomial A minimal, if it is of minimal length representing x. We characterize minimal polynomials.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Canonical extensions for congruential logics with the deduction theorem

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.     

Weakly linear quantum MV-algebras

Quantum MV-algebras (QMV-algebras) are a non lattice-theoretic generalization of MV-algebras (multi-valued algebras) and a non-idempotent generalization of orthomodular lattices. In this paper we construct a finite basis for the variety generated by the class of all weakly linear quantum MV-algebras.Dedicated to the memory of Wim BlokReceived October 12, 2000; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

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.     

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.     

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.     

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).     

Ockham algebras with pseudocomplementation

The variety pO consists of those algebras (L; ?, ?, f,*, 0,1) of type (2,2,1,1,0,0) where (L; ?, ?, f, 0,1) is an Ockham algebra, (Z; ?, ?,*, 0,1) is a p-algebra, and the unary operations f and ^* commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass pK_1,1, characterised by the property f³ = f.     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

Finite lattices and congruences. A survey

In the early forties, R.P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L.We have been publishing papers on this topic for 45 years. In this survey paper, we are going to review some of our results and a host of related results by others: Making L nice.If being nice is an algebraic property such as being semimodular or sectionally complemented, then we have tried in many instances to prove a stronger form of these results by verifying that every finite lattice has a congruence-preserving extension that is nice. We shall discuss some of the techniques we use to construct nice lattices and congruence-preserving extensions.We shall describe some results on the spectrum of a congruence of a finite sectionally complemented lattice, measuring the sizes of the congruence classes. It turns out that with very few restrictions, these can be as bad as we wish.We shall also review some results on simultaneous representation of two distributive lattices. We conclude with the magic wand construction, which holds out the promise of obtaining results that go beyond what can be achieved with the older techniques.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 26, 2002; accepted in final form June 18, 2004.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

Complementing lattice-ordered groups: The projectable case

A complemented l-group G is one in which to each a G there corresponds a b G so that |a||b|=0, while |a||b| is a unit of G. For projectable l-groups this is so precisely when the group possesses a unit.The article introduces the notion of complementation, and the situation for projectable l-groups is analyzed in some detail; in particular, it is shown that any projectable l-group having a projectable complementation in which it is convex has a unique maximal one of this kind.A portion of this research was carried out while this author was a Stauffer Visiting Professor at the University of Kansas during the year 1986–87. He thanks his colleagues in mathematics at that institution for their hospitality.