Endoprimal implication algebras

An algebra A is endoprimal if, for all , the only maps which preserve the endomorphisms of A are the n-ary term functions of A. The theory of natural dualities has been a very effective tool for finding finite endoprimal algebras. We study endoprimality within the variety of implication algebras, which does not contain any non-trivial dualisable algebras. We show that there are no non-trivial finite endoprimal implication algebras. We also give some examples of infinite implication algebras which are endoprimal. Received July 28, 1998; accepted in final form January 18, 1999.     

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.     

Grothendieck-like duality for Heyting algebras

In this paper we shall use the multipliers in order to define a sheaf on the prime spectrum of a Heyting algebra. This construction allows us to obtain a Grothendieck-style duality for Heyting algebras. Received March 3, 2000; accepted in final form February 16, 2001.     

A criterion for a finite endoprimal algebra to be endodualisable

In this note the strategy for finding endoprimal algebras due to B. A. Davey and J. G. Pitkethly is further explored in the finite case. The Retraction Test Algebra Lemma is used as a tool to show that, in many quasivarieties, endoprimality is equivalent to endodualisability for finite algebras which are suitably related to finitely generated free algebras. Received September 2, 1998; accepted in final form June 3, 1999.     

Full duality among graph algebras and flat graph algebras

We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non--dualizable for every infinite cardinal . A new, general method for proving strong duality is presented. Received August 30, 1999; accepted in final form September 22, 1999.     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

Functional monadic Heyting algebras

We show every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. This solves a 1957 problem of Monteiro and Varsavsky [9]. Received May 18, 2001; accepted in final form October 18, 2001.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

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.     

Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals of length less than , with each cardinal in the sequence being either or , we construct a -poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence . The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under Under Martin's Axiom, the construction allows any cardinals in the sequence, provided it has length Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build -posets for any given cardinal sequence of length with each cardinal in the sequence being either or . Received September 2, 1998; accepted in final form September 13, 2001.     

伪补分配格上的主同余关系

本文研究了伪补分配格Ｌ的＊－同余关系的基本性质。主要给出了：⑴主同余θ＊（ａ,ｂ）的一个具体刻划;⑵理想Ⅰ成为同余理想的一个充要条件的以Ⅰ为核心的最小＊－同余θ＊（Ｉ）;⑶Ｌ上的格同余θＬ（ａ,ｂ）成为＊－同余的干充要条件。其中的大部分结果必进或包含了已有的结论。     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Equational bases for varieties of Ockham algebras

We consider the variety O of Ockham algebras and its subvarieties of the form P _m,n (m > n ≥0), sometimes with an additional condition. We use Priestley duality and a remarkable theorem of Urquhart to develop a simple method for determining the equational bases of the subvarieties. The axioms that we obtain have the same canonical form and involve few variables. We illustrate our method by the detailed study of the variety MS ₂ and some considerations about P _3,2. Received July 28, 1998; accepted in final form May 25, 2000.     

Lattice orders on matrix algebras

We construct all the lattice orders (up to isomorphism) on a full matrix algebra over a subfield of the field of real numbers so that it becomes a lattice-ordered algebra. Received June 26, 2001; accepted in final form February 9, 2002.     

Notes on sums of algebras, subdirect products and subdirectly irreducible algebras

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras. Received September 10, 1999; accepted in final form January 3, 2001.     

Mailbox¶A Note on minimal varieties generated by order-primal algebras

We prove that the order-primal algebra of a non-trivial finite connected poset P generates a minimal variety if and only if P is dismantlable. Received November 26, 2002; accepted in final form January 24, 2003.     

Strong semimodular lattices and Frankl's conjecture

In this paper, we show that Frankl's conjecture holds for strong semimodular lattices. The result is the first step to deal with the case of upper semimodular lattices. Received May 30, 2000; accepted in final form August 14, 2000.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the cofinalities of Boolean algebras and the ideal of null sets

We will show that if the cofinality of the ideal of Lebesgue measure zero sets is equal to then there exists a Boolean algebra B of cardinality which is not a union of strictly increasing -sequence of its subalgebras. This generalizes a result of Just and Koszmider who showed that it is consistent with that such an algebra exists. Received November 21; accepted in final form April 12, 2001.