The depth of ultraproducts of Boolean algebras

We show that if μ is a compact cardinal then the depth of ultraproducts of less than μ many Boolean algebras is at most μ plus the ultraproduct of the depths of those Boolean algebras. Received May 18, 2004; accepted in final form December 9, 2004.     

On ultraproducts of operator algebras

Some basic questions on ultraproducts of C~*-algebras and von Neumann al- gebras,including the relation to K-theory of C~*-algebras are considered.More specifically, we prove that under certain conditions,the K-groups of ultraproduct of C~*-algebras are iso- morphic to the ultraproduct of respective K-groups of C~*-algebras.We also show that the ultraproducts of factors of type Ⅱ_1 are prime,i.e.not isomorphic to any non-trivial tensor product.     

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 independent subsets of Boolean algebras

We give here a partial solution to a problem of Monk ([2]). Let κ be a regular cardinal and let {L_i∶i<κ} be a family of linear orderings with first element such that noL _i contains a strictly decreasing sequence of length κ⁺. Then $$Ind\left( {\prod\limits_{i< \kappa } {Intalg(L_i )} } \right) \leqslant 2^\kappa .$$      

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 .

     

On tightly κ-filtered Boolean algebras

In this article we study the notion of tight ?-filteredness of a Boolean algebra for infinite regular cardinals ?. Tight à₀ \aleph_0 -filteredness is projectivity. We give characterizations of tightly ?-filtered Boolean algebras which generalize the internal characterizations of projectivity given by Haydon, Šcepin, and Koppelberg (see [15] or [17]). We show that for each ? there is an rc-filtered Boolean algebra which is not tightly ?-filtered. This generalizes a result of Šcepin (see [15]). We prove that no complete Boolean algebra of size larger than à₂ \aleph_2 is tightly à₁ \aleph_1 -filtered. We give a new example of a model of set theory where \frak P(w) \frak P(\omega) is tightly s-filtered. We study the effect of the tight s-filteredness of \frak P(w) \frak P(\omega) on the automorphism group of \frak P(w)/fin \frak P(\omega)/fin .     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

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.     

On tightness and depth in superatomic Boolean algebras

We introduce a large cardinal property which is consistent with and show that for every superatomic Boolean algebra and every cardinal with the large cardinal property, if tightness, then depth. This improves a theorem of Dow and Monk.

     

On the automorphism groups of countable Boolean algebras

We prove categoricity results for the class {<B, Aut(B)>|B is a countable BA}. This paper is part of the author’s doctoral dissertation prepared at the Hebrew University under the supervision of Professor Saharon Shelah.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

On multiplicity theory for Boolean algebras of projections

The aim of this paper is to present a study of the connections between the commutants of a Boolean algebra of projections of finite multiplicity and the uniformly closed algebra generated by these projections. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.