The problem of determining the maximum size La(n, P ) that a P -free subposet of the Boolean lattice Bn can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this paper we determine the asymptotic behavior of La∗(n, P ), the maximum size that an induced P -free subposet of the Boolean lattice Bn can have for the case when P is...

We apply the graph container method to prove a number of counting results for the Boolean lattice P(n). In particular, we: (i) Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in P(n), and we also give an upper bound on the number of transportation codes; (ii) Provide an alternative proof of Kleitman’s theorem on the number of antichains in P(n...

D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on {0, 1} that contain all total functions preserving the relation ρ0,2 = {(0, 0), (0, 1), (1, 0...

Managing large numbers of complex objects or descriptions requires sophisticated storage and retrieval mechanisms. Hierarchical classiication techniques have proven to be useful for managing complex objects in practice. Previous classiication methods have not used all the information recorded in hierarchies. We show how lattice operations can be used to navigate around a hierarchy. This is achi...

If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Zn of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a la...

A lattice-based model for information retrieval has been suggested in the 1960’s but has been seen as a theoretical possibility hard to practically apply ever since. This paper attempts to revive the lattice model and demonstrate its applicability in an information retrieval system, FaIR, that incorporates a graphical representation of a faceted thesaurus. It shows how Boolean queries can be la...

We study the compatibility of a construction proposed by Aït-Kaci to extend a poset of sorts into a lattice with the specification on type constraints on the value of attributes, through Gloess assorted signatures. We first show that it is natural to extend the poset into a boolean algebra, when one wishes to obtain a complete and complementary lattice endowed with a smooth semantics. We then s...

Professor Rudeanu published numerous research papers and authored or coauthored some important books. A simple enumeration of some titles of his books, from the rst one, Axioms of Lattices and Boolean Algebras published in Romanian by the Romanian Academy Publishing House in 1963 to the pioneering Pseudo-Boolean Methods for Bivalent Programming, Springer-Verlag, 1966 and Boolean Methods in Oper...

The probability p(s) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [0, 1]. The function p is called a numerical event or multidimensional probability. When appropriately structured, sets P of numerical events form so-called algebras of Sprobabilities. Their main feature is t...

In this tutorial, I will discuss the concepts behind generalizing ordering to measuring and apply these ideas to the derivation of probability theory. The fundamental concept is that anything that can be ordered can be measured. Since we are in the business of making statements about the world around us, we focus on ordering logical statements according to implication. This results in a Boolean...