We investigate the numbers d k of all (isomorphism classes of) distributive lattices with k eleme... more We investigate the numbers d k of all (isomorphism classes of) distributive lattices with k elements, or, equivalently, of (unlabeled) posets with k antichains. Closely related and useful for combinatorial identities and inequalities are the numbers v k of vertically indecomposable distributive lattices of size k. We present the explicit values of the numbers d k and v k for k < 50 and prove the following exponential bounds: 1.67 k < v k < 2.33 k and 1.84 k < d k < 2.39 k (k k 0 ). Important tools are (i) an algorithm coding all unlabeled distributive lattices of height n and size k by certain integer sequences 0 = z 1 • • • z n k -2, and (ii) a "canonical 2-decomposition" of ordinally indecomposable posets into "2indecomposable" canonical summands. the electronic journal of combinatorics 9 (2002), #R24
We investigate various weak conditions ensuring that a lattice be complemented. Using these gener... more We investigate various weak conditions ensuring that a lattice be complemented. Using these general results in connection with a famous result due to Lampe, we show that the lattice of all equational theories containing a fixed theory must be complemented if it is lower semicomplemented, thereby answering in the affirmative a question raised by Volkov and Vernikov. Moreover, such a lattice must be a finite Boolean algebra if it has one of the following properties: upper or lower sectionally complemented; incomparably complemented; lower semicomplemented and lower semimodular; or atomisfic and upper semimodular. of all equational theories containing a fixed theory S, or dually, the lattices of subvarieties of a variety. For convenience, we say L is a lattice of equational theories iff L is isomorphic to some L(S). Such lattices are certainly algebraic and coatomic, possessing a compact top element 1; but stronger properties were not known before Lampe's surprising discovery [8] that any lattice of equational theories obeys the following rule: (Lo) For alla, cEL, W{b:a^b---c}=l impliesa=c. Moreover, (L0) is valid in the congruence lattice of any O-l-algebra, that is, any algebra possessing a binary term with a right unit and a right zero; and any lattice of equational theories is isomorphic to such a lattice -in fact, to the congruence lattice of a monoid with a right zero and one additional unary operation, as was recently shown by Newrly [10]. Denoting the least element by 0, we see that (L0) is stronger than (L~) For alla~L, W{b:a/xb=0}=l impliesa=0, which in turn is stronger than the "binary version" (L2) a/xb=a/xc=0and bvc=l implya=0, Presented by R. Freese.
A classical tensor product A ⊗ B of complete lattices A and B, consisting of all down-sets in A ×... more A classical tensor product A ⊗ B of complete lattices A and B, consisting of all down-sets in A × B that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A, B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.
A standard completion γ assigns a closure system to each partially ordered set in such a way that... more A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called “subset systems”. Related facts are derived for conditional completions.
We establish formulas for the number of all downsets (or equivalently, of all antichains) of a fi... more We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset P. Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points and inducing the poset P on the non- minimal points. It turns out that these counting functions are closely related to a collection of downset numbers of certain subposets. Since any function of that kind is an exponential sum (with the number of minimal points as exponent), we call it the exponential function of the poset. Some linear equations, divisibility relations, upper and lower bounds, and asymptotical equalities for the counting functions are deduced. A list of all such exponential functions for posets with up to five points concludes the paper.
We present old and new characterizations of core spaces, alias worldwide web spaces, originally d... more We present old and new characterizations of core spaces, alias worldwide web spaces, originally defined by the existence of supercompact neighborhood bases. The patch spaces of core spaces, obtained by joining the original topology with a second topology having the dual specialization order, are the so-called sector spaces, which have good convexity and separation properties and determine the original space. The category of core spaces is shown to be concretely isomorphic to the category of fan spaces; these are certain quasi-ordered spaces having neighborhood bases of so-called fans, obtained by deleting a finite number of principal filters from a principal filter. This approach has useful consequences for domain theory. In fact, endowed with the Scott topology, the continuous domains are nothing but the sober core spaces, and endowed with the Lawson topology, they are the corresponding fan spaces. We generalize the characterization of continuous lattices as meet-continuous lattice...
Abstract. A basic tool in domain theory and point-free topology is given by the system of (Scott)... more Abstract. A basic tool in domain theory and point-free topology is given by the system of (Scott) open,lters in a poset. A systematic investigation of that concept shows that central notions and facts like Lawson’s famous self-duality of the category of domains,and their representation by suitable topological spaces may be established without invoking any choice principles. Many of the conclusions remain valid for the rather exible notion of -domains, comprising important variants such as algebraic or hypercontinuous domains. Mathematics Subject Classication: Primary: 06B35. Secondary: 03E25,
A distributor in an m-semilattice (a join-semilattice with an isotone multiplication) is a nonemp... more A distributor in an m-semilattice (a join-semilattice with an isotone multiplication) is a nonempty upper set containing both the join of a and c and the join of b and c iff it contains the join of ab and c. Distributors provide a far-reaching extension of the filter theory for distributive lattices, quantales and similar objects to structures where no distributive laws are assumed a priori. The closure system of all distributors is a universal locale over the given m-semilattice, and the principal distributors form its universal distributive lattice quotient. Moreover, distributors are a helpful tool for the spectral theory of m-semilattices and various related kinds of (ordered) algebras. We present diverse alternative characterizations of Scott-open distributors in complete m-semilattices, for example as kernels of join-preserving homomorphisms onto compact locales, and we establish a one-to-one correspondence between Scott-open distributors and those nuclei whose range is a Wallman locale, the pointfree analogue of a compact T1-topology
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Commentationes Mathematicae Universitatis Carolinae, 1995
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
For instance, we show that order convergence in L x is pretopological if it is pretopological in ... more For instance, we show that order convergence in L x is pretopological if it is pretopological in L, while topological order convergence is in general not preserved under complete images. We conclude with some applications and examples.
Proceedings of the American Mathematical Society, 1980
In Theorem 1 of this note, results of Kogan [2], Kolibiar [3], Matsushima [4] and Wölk [7] concer... more In Theorem 1 of this note, results of Kogan [2], Kolibiar [3], Matsushima [4] and Wölk [7] concerning interval topologies are presented under a common point of view, and further characterizations of the T2 axiom are obtained. A sufficient order-theoretical condition for regularity of interval topologies is established in Theorem 2. In lattices, this condition turns out to be equivalent both to the T2 and to the T3 axiom. Hence, a Hausdorf f interval topology of a lattice is already regular. However, an example of a poset is given where the interval topology is T2 but not T3. Lemma 1. The interval topology of any poset is T,.
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