Endomorphism Classes of Ordered Sets,Graphs and Lattices

Order - Given a structure S, what other structures S' on the same underlying set satisfy End S $ \subseteq$ End S'? We answer this question for the cases of ordered sets, reflexive...     

Decidability of Some Classes of Modal Logic

We describe a modal sequential calculus some modal rules of which do not contain duplications of the main formula. We also prove its equivalence to the modal calculus S4 and the decidability of some classes of formulas containing only oneplace predicate variables.     

A Theorem of the Alternative in Banach Lattices

We consider a linear sytem in a Banach lattice and provide a simple theorem of the alternative (or Farkas lemma) without the usual closure condition.

     

Corrigendum: Gluings of Modular Lattices

In Day and Hermann (Order, 5:85–101, 1988) it has been stated that for every S-cover of a lattice L there is an extension L′ of L in the variety of L and a bounded S-cover of L′ which restricts to the given S-cover and has each block L′(x) in the variety of L(x). A correct proof of this statement is given, here.     

Totally Order-disconnected Compact Topologies

Posets with property DINT which are Priestely spaces with respect interval topologies are characterized. Also, posets which are Priestley spaces with respect to bi-Scott topologies are characterized.     

Lattices with Complemented Tolerance Lattice

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.     

The Essentially Equational Theory of Horn Classes

It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic (see [2]). In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known (see [1]). Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes are in fact (equivalent to) quasivarieties.     

Free Distributive Completions of Partial Complete Lattices

The free distributive completion of a partial complete lattice is the complete lattice that it is freely generated by the partial complete lattice in the most distributive way. This can be described as being a universal solution in the sense of universal algebra. Free distributive completions generalize the constructions of tensor products and of free completely distributive complete lattices over partially ordered sets.     

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.     

K-Theory of Noncommutative Lattices

Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their K-theory. We shall do it algebraically, by studying the algebraic K-theory of the associated algebras of continuous functions which turn out to be noncommutative approximately finite dimensional (AF)C^*. We also work out several examples.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

关于具有$Q$-逆断面的正则半群的同余格

In this paper, a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed. The classes of this complete congruence which are intervals are discussed and their least and greatest elements are presented clearly.     

Ordered Compactifications of Products of Two Totally Ordered Spaces

We describe the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y where X and Y are certain totally ordered topological spaces, and where _o Z denotes the Stone–ech ordered- or Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y for arbitrary totally ordered topological spaces X and Y. Such products X × Y provide many counterexamples in the theory of ordered compactifications.     

A Note of Regularity on Completely Distributive Lattices

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Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Lattices on Nonuniform Trees

Let X be a locally finite tree, and let G=Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup such that the quotient graph of groups \\X is infinite but has finite covolume, then G contains a nonuniform lattice, that is, a discrete subgroup such that \G is not compact, yet has a finite G-invariant measure.     

对称拓扑分子格的直和

文章给出了对称拓扑分子格的直和概念,给出了拓扑分子格的直和的特征,证明了对称拓扑分子格的分离性Ti(i=-1,0,1,2)及可数性CⅠ,CⅡ是可和性质.     

Join-Independent and Meet-Independent Sets in Complete Lattices

In this paper join-independent and meet-independent sets in a complete lattice are defined. The relations between them are presented and substructures of a complete lattice induced by such sets are investigated.