Semicontinuous lattices

In this paper we introduce and study a new type of lattices, semicontinuous lattices, by using semiprime ideals. Such lattices have many properties similar to that of continuous lattices, and are closely related to the theory of continuous lattices. Received November 3, 1995; accepted in final form March 13, 1997.     

半连续格的刻画和映射

本文讨论了半连续格的一些性质,在半连续格中引入半Scott开集族,用半Scott开集族来刻画半连续格,同时定义了半连续格之间的半连续映射,得到闭包算子的像仍是半连续格的条件.最后,研究了半连续格上的半连续映射的全体不动点之集的性质。     

A couple of triples

The standard contravariant adjunction between TOP (the category of topological spaces) and LAT (the category of distributive lattices) induces a triple Λ on LAT and a triple Σ on TOP. We show that the category LAT^Λ of Λ-algebras is just the category of frames, and describe the category TOP^Σ of Σ-algebras as a subcategory of TOP.     

Topological representation of posets

There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.     

Tensor products for bounded posets revisited

The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.     

Ordered groups with a conucleus

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras.     

Remarks on Priestley duality for distributive lattices

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix.     

Localic real functions: A general setting

In pointfree topology the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame L(R) of reals into S(L), the dual of the co-frame of all sublocales of L. The lattice-ordered ring is a pointfree counterpart of the ring R^X with X a topological space. We thus have a pointfree analogue of the concept of an arbitrarynot necessarily (semi) continuous real function on L. One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.     

On the nonexistence of free complete distributive lattices

We prove that there is no free object over a countable set in the category of complete distributive lattices with homomorphisms preserving binary meets and arbitrary joins.     

Coalgebraic representations of distributive lattices with operators

We present a framework for extending Stone's representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor on the category of distributive lattices such that its algebras are exactly the distributive lattices with operators in the original theory. We characterize the topological counterpart of these algebras in terms of suitable coalgebras on spectral spaces. We work out some of these coalgebraic representations, including a new representation theorem for distributive lattices with monotone operators.     

Remarks on the Stone-?ech and Alexandroff compactifications of locales

Relations of strong inclusion are considered on pseudocomplemented distributive lattices to refine existing constructions of (Stone-?ech and Alexandroff) compactifications of frames.     

A Gentzen system for involutive residuated lattices

We establish a cut-free Gentzen system for involutive residuated lattices and provide an algebraic proof of completeness. As a result we conclude that the equational theory of involutive residuated lattices is decidable. The connection to noncommutative linear logic is outlined. Received July 22, 2004; accepted in final form July 19, 2005.     

Reducibility number

Let P be a poset in a class of posets P. A smallest positive integer r is called reducibility number of P with respect to P if there exists a non-empty subset S of P with |S|=r and P-S∈P. The reducibility numbers for the power set 2ⁿ of an n-set (n?2) with respect to the classes of distributive lattices, modular lattices and Boolean lattices are calculated. Also, it is shown that the reducibility number r of the lattice of all subgroups of a finite group G with respect to the class of all distributive lattices is 1 if and only if the order of G has at most two distinct prime divisors; further if r is a prime number then order of G is divisible by exactly three distinct primes. The class of pseudo-complemented u-posets is shown to be reducible. Deletable elements in semidistributive posets are characterized.     

Bigeneration in complete lattices and principal separation in ordered sets

By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.     

Z-半连续格

作为连续格和半连续格的公共推广,引入了广义理想子系统Z、Z-半连续格及强Z-连续格的概念,讨论了它们的基本性质和Z-半连续格的函数空间的结构,给出了强Z-连续格到方体的嵌入,证明了当子系统Z满足一定条件时,Z-半连续格范畴SCLZ是笛卡儿闭的。     

Insertion of lattice-valued and hedgehog-valued functions

Problems of inserting lattice-valued functions are investigated. We provide an analogue of the classical insertion theorem of Lane [Proc. Amer. Math. Soc. 49 (1975) 90-94] for L-valued functions where L is a ?-separable completely distributive lattice (i.e. L admits a countable join-dense subset which is free of completely join-irreducible elements). As a corollary we get an L-version of the Katětov-Tong insertion theorem due to Liu and Luo [Topology Appl. 45 (1992) 173-188] (our proof is different and much simpler). We show that ?-separable completely distributive lattices are closed under the formation of countable products. In particular, the Hilbert cube is a ?-separable completely distributive lattice and some join-dense subset is shown to be both order and topologically isomorphic to the hedgehog J(ω) with appropriately defined topology. This done, we deduce an insertion theorem for J(ω)-valued functions which is independent of that of Blair and Swardson [Indian J. Math. 29 (1987) 229-250]. Also, we provide an iff criterion for inserting a pair of semicontinuous function which yields, among others, a characterization of hereditarily normal spaces.