A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property.
We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Willard's work ("Varieties Having Boolean Factor Congruences," J. Algebra, 132 (1990)).
We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP…
In previous work, we have introduced and studied a lifting property in congruence--distributive universal algebras which we have defined based on the Boolean congruences of such algebras, and which we have called the Congruence Boolean Lifting Property. In a similar way, a lifting property based on factor congruences can be defined in congruence--distributive algebras; in this paper we introduce this property, which we have called the Factor Congruence Lifting Property, and study it, partly in relation to the Congruence Boolean Lifting Property, and to other lifting properties in particular cl…
The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of philosophical nature as he was looking for an interpretation of the concepts of possibility and necessity. Since then, plenty of research has been developed in this area. In 1968, when Gr.C. Moisil came across Zadeh's fuzzy set theory he found the motivation he had been looking for in order to legitimate the introduction and study of infinitely-valued Lukasiewicz algebras, so he defined θ-valued Lukasiewicz algebras (or LMθ-algebras, for short) (without negation), where θis the order type of a chain. …
The Lifting Idempotent Property ($LIP$) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures ($MV$-algebras, commutative l-groups, $BL$-algebras, bounded distributive lattices, residuated lattices,etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property ($CBLP$) for congruence modular algebras). A lifting ideal of a ring $R$ is an ideal of $R$ fulfilling $LIP$. In a recent paper, Tarizadeh and Sharma obtained new results on liftin…
We give a unified treatment of the model theory of various enrichments of infinite atomic Boolean algebras, with special attention to quantifier-eliminations, complete axiomatizations and decidability. A classical example is the enrichment by a predicate for the ideal of finite sets, and a novel one involves predicates giving congruence conditions on the cardinality of finite sets. We focus on three examples, and classify them by expressive power.
In this paper we define the Boolean Lifting Property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP. BLP behaves interestingly in direct products and involutive residuated lattices, and it is closely related to arithmetic properties involving Boolean elements, nilpotent elements and elements of the radical. When BLP is present, strong representation theorems for semilocal and maximal residuated lattices hold.
In this article, we introduce a lattice congruence with respect to a nonempty ideal $I$ of a distributive lattice $L$ and a derivation $d$ on $L$ denoted by $θ_I^d$. We investigate some necessary and sufficient conditions for the quotient algebra $L/θ_I^d$ to become a Boolean algebra.
We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are "definable" in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0&1 and Boolean Factor Congruences. We also obtain an explicit first order definition of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DF…
A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have natural structure of Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.
The following results are proved: (a) In a model obtained by adding aleph_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 algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…
An algebra $\mathbb A$ is called a perfect extension of its subalgebra $\mathbb B$ if every congruence of $\mathbb B$ has a unique extension to $\mathbb A$. This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$. Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mat…
The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for $K_2$-algebras \cite{6}, is introduced. It is proved that the congruences of the principal MS-algebras $L$ correspond to the MS-congruence pairs on simpler substructures $L^{\circ\circ}$ and $D(L)$ of $L$ that were associated to~$L$ in \cite{4}. An analogy of a well-known Grätzer's problem \cite[Problem 57]{11} formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Pro…
I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this p…
We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -i…
Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to (B union C). While refining the Banach-Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences. The purpose of the present paper is to characterize t…
A Boolean algebra $\mathcal A$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal A$ is uniformly bounded. Assuming the Diamond Principle $\Diamond$, we will construct an example of a minimally generated Boolean algebra $\mathcal A$ with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true - again under $\Diamond$ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The r…
Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations
Czech Academy of Sciences（捷克科学院）；Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机研究所）；University of Silesia in Katowice（卡托维兹西里西亚大学）；Institute of Mathematics Cardinal Stefan Wyszyński University in Warsaw（华沙斯特凡·维辛斯基红衣主教大学数学研究所）
In this paper we prove that in the context of varieties with Right Existentially Definable Factor Congruences, definability of the property "e and f are complementary central elements", stability by complements and coextensivity of its respective algebraic category, are equivalent.
We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal lambda with the large cardinal property, if tightness^+(B) >= lambda^+, then depth (B) >= lambda. This improves a theorem of Dow and Monk.
It is known that by using the commutator operation, for each congruence modular algebra $A$ one can define a notion of prime congruence. The set $Spec(A)$ of prime congruences of $A$ is endowed with a Zariski style topology. The reticulation of the algebra $A$ is a bounded distributive lattice $L(A)$ whose prime spectrum $Spec(L(A))$ (with the Stone topology) is homemorphic to $Spec(A)$. In a recent paper, C. Mureşan and the author have proven the existence of reticulation for a semidegenerate congruence modular algebra $A$. The present paper aims to give an answer to two types of problems: $(…
We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(ω_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(ω)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.
We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ^+,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ,μ)-FN for λ>μform a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (κ,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in t…
We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n-independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, $n$Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n-in…
We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection…
We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Ban…
This paper gives a novel approach to analyze SAT problem more deeply. First, I define new elements of Boolean formula such as dominant variable, decision chain, and chain coupler. Through the analysis of the SAT problem using the elements, I prove that we can construct a k-SAT (k>2) instance where the coefficients of cutting planes take exponentially large values in the input size. This exponential property is caused by the number system formed from the calculation of coefficients. In addition, I show that 2-SAT does not form the number system and Horn-SAT partially forms the number system acc…
