Countably categorical and autostable Boolean algebras with distinguished ideals

We study countable Boolean algebras with finitely many distinguished ideals (countable I-algebras) whose elementary theory is countably categorical, and autostable I-algebras which form their subclass. We propose a new characterization for the former class that allows to answer a series of questions about the structure of countably categorical and autostable I-algebras.     

Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6.     

The Lindenbaum-Tarski algebra for Boolean algebras with distinguished ideals

A complete solution is given to the problem of describing algebras with distinguished ideals, formulated by Peretyatkin. It is proven that such an algebra is isomorphic to _× , an interval algebra of the linear ordering × . I-algebras the elementary theory of each of which is axiomatizable by a single atom in some finite quotient with respect to the Frechet ideal of the Lindenbaum-Tarski algebra for the class of Boolean algebras with distinguished ideals are fully described in terms of direct summands.Translated fromAlgebra i Logika, Vol. 34, No. 1, pp. 88–116, January–February, 1995.     

Unsupported Boolean algebras and forcing

If κ is an infinite cardinal, a complete Boolean algebra B is called κ‐supported if for each sequence 〈b_β : β < κ〉 of elements of B the equality _{α<κ β>α} b_β = equation/tex2gif-inf-5.gif _β∈A b_β holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras (distributivity, caliber, etc.). The set of regular cardinals κ for which B is not κ‐supported is investigated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterations of Boolean algebras with measure

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Some Boolean Algebras with Finitely Many Distinguished Ideals I

We consider the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th_prin and Th_sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th_prin) ? intalg(ω⁴), Sentalg(Th_max) ? intalg(ω³) ? Sentalg(Th_ac), and Sentalg(Th_sa) ? intalg(ω² + ω²). For Th_max and Th_ac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic.     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean Algebras with Finite Families of Computable Automorphisms

We prove that each computable Boolean algebra has a computable presentation in which for every computable family of automorphisms the set of atoms moved by at least one of its members is finite. This implies that each computable atomic Boolean algebra has a computable presentation in which its every computable family of automorphisms is finite. The priority argument is not used in the proof.     

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 weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

The finiteness of compact Boolean algebras

We show that it consistent with Zermelo‐Fraenkel set theory that there is an infinite, compact Boolean algebra (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

Modules and ideals of algebras of associative type

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.     

Finiteness conditions and distributive laws for Boolean algebras

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)