On poset Boolean algebras of scattered posets with finite width

We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.Mathematics Subject Classification (2000):Primary 03G05, 06A06, 06A11; Secondary 08A05, 54G12     

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.     

On the cohomology of incidence algebras of partially ordered sets

We show how to construct a representation of the Lyons sporadic simple group in 651 dimensions over G F(3), which was apparently not previously known. This is the smallest faithful representation in charac-teristic 3.     

Fibres of width 3 ordered sets

We show that in an ordered setP of width 3 there always exists a fibre (i.e., a subset intersecting each maximal nontrivial antichain) of size at most 11/18|P|. This improves previously known results.Research supported by Polish KBN Grant 2 2037 92 03.     

On some Menger algebras of multiplace transformations of ordered sets

Presented by B. M. Schein.     

On maximal and minimal elements of partially ordered sets of Boolean degrees

The paper addresses the “weakest” algorithmic reducibility—Boolean reducibility. Under study are the partially ordered sets of Boolean degrees L _Q corresponding to the various closed classes of Boolean functions Q. The set L _Q is shown to have no maximal elements for many closed classes Q. Some examples are given of a sufficiently large classes Q for which L _Q contains continuum many maximal elements. It is found that the sets of degrees corresponding to the closed classes T ₀₁ and SM contain continuum many minimal elements.     

Boolean and distributive ordered sets: Characterization and representation by sets

Boolean ordered sets generalize Boolean lattices, and distributive ordered sets generalize distributive lattices. Ideals, prime ideals, and maximal ideals in ordered sets are defined, and some well-known theorems on Boolean lattices, such as the Glivenko-Stone theorem and the Stone representation theorem, are generalized to Boolean ordered sets. A prime ideal theorem for distributive ordered sets is formulated, and the Birkhoff representation theorem is generalized to distributive ordered sets. Fundamental are the embedding theorems for Boolean ordered sets and for distributive ordered sets.Financial support of the Grant Agency of the Czech Republic under the grant No. 201/93/0950 is gratefully acknowledged.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.     

On axiomatizations of Boolean algebras

We construct some new axiomatic systems for the Boolean algebra. In particular, an axiomatic system for disjunction and logical negation consists of three axioms. We prove the independence of the axiomatic systems proposed. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 937–942, July, 1997.     

On the strong amalgamation of Boolean algebras

This paper presents a proof that Boolean algebras have the strong amalgamation property which does not use any choice principle, based on a feature of their binary coproducts which is derived by means of a construction from pointfree topology.     

The length,the width and the cutset-number of finite ordered sets

The partially ordered set P is an (, , ) ordered set if the width of P, the length of any chain of P and the cut-set number . We will prove that if P is an (, , ) ordered set then P contains a simple (, , ) ordered set and use this result to prove that if P has the 3 cutset property, then width of P length of P+3.     

On ultraproducts of Boolean algebras and irr

1. Consistent inequality [We prove the consistency of irrirr(B_i)/D where D is an ultrafilter on and each B_i is a Boolean algebra and irr(B) is the maximal size of irredundant subsets of a Boolean algebra B, see full definition in the text. This solves the last problem, 35, of this form from Monk's list of problems in [M2]. The solution applies to many other properties, e.g. Souslinity.] 2. Consistency for small cardinals [We get similar results with =₁ (easily we cannot have it for =₀) and Boolean algebras B_i (i<) of cardinality .] This article continues Magidor Shelah [MgSh:433] and Shelah Spinas [ShSi:677], but does not rely on them: see [M2] for the background. I would like to thank Alice Leonhardt for the beautiful typing. This research was partially supported by the Israel Science Foundation. Publication 703     

On subalgebras of Boolean interval algebras

We prove that the following three conditions are necessary and sufficient for a Boolean algebra to be embeddable into an interval algebra.

(i)

is generated by a subset such that for all .

(ii)

has a complemented subalgebra lattice, where complements can be chosen in a monotone way.

(iii)

is isomorphic to ClopX for a compact zero-dimensional topological semilattice such that for all .