Initial segments of the lattice of Π10 classes

2001 ◽  
Vol 66 (4) ◽  
pp. 1749-1765 ◽  
Author(s):  
Douglas Cenzer ◽  
Andre Nies
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Limit Point ◽  
Finite Differences ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Distributive Lattice ◽  
Property A ◽  
Decidable Theory ◽  
Reduction Property

Abstract.We show that in the lattice of classes there are initial segments [∅, P] = (P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a class P such that L is isomorphic to the lattice (P)*, which is (P). modulo finite differences. For the 2-element lattice, we obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new class P constructed, P has a single, non-computable limit point and (P)* has three elements, corresponding to ∅, P and a minimal class P0 ⊂ P, The element corresponding to P0 has no complement in the lattice. On the other hand, the theory of (P) is shown to be decidable.A class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then (P)* is always a Boolean algebra. We show that if P is a decidable class and (P) is not a Boolean algebra, then the theory of (P) interprets the theory of arithmetic and is therefore undecidable.

scholarly journals Imbedding Infinitely Distributive Lattices Completely Isomorphically Into Boolean Algebras

1959 ◽  
Vol 15 ◽  
pp. 71-81 ◽  
Author(s):  
Nenosuke Funayama
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Algebras ◽  
The Other ◽  
Distributive Lattices ◽  
Algebra 1 ◽  
Infinite Distributivity ◽  
Distributive Law ◽  
Finite Sums ◽  
Finite Products

It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).

scholarly journals Four games on Boolean algebras

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3389-3395
Author(s):  
Milos Kurilic ◽  
Boris Sobot
Keyword(s):  
Boolean Algebra ◽  
Winning Strategy ◽  
Zero Element ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Other Hand

The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

On Boolean algebras and integrally closed commutative regular rings

1992 ◽  
Vol 57 (4) ◽  
pp. 1305-1318
Author(s):  
Misao Nagayama
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
The Other ◽  
Model Completeness ◽  
Integrally Closed ◽  
Regular Rings ◽  
Definitional Extension

AbstractIn this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai, partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S′(ai, l) for l < n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ϵ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.

scholarly journals The left, the right and the sequential topology on Boolean algebras

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4451-4459
Author(s):  
Milos Kurilic ◽  
Aleksandar Pavlovic
Keyword(s):  
Boolean Algebra ◽  
A Priori ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
A Posteriori ◽  
Other Hand ◽  
Convergence Of Sequences ◽  
Algebraic Convergence ◽  
The Right

For the algebraic convergence ?s, which generates the well known sequential topology ?s on a complete Boolean algebra B, we have ?s = ?ls ? ?li, where the convergences ?ls and ?li are defined by ?ls(x) = {lim sup x}? and ?li(x) = {lim inf x+}? (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology Olsi extending the (unique) sequential topologies O?s (left) and O?li (right) generated by the convergences ?ls and ?li and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences. In addition, we observe some special classes of algebras and, in particular, show that in (?,2)-distributive algebras we have limOlsi = lim?s = ?s, while the equality Olsi = ?s holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.

Chains and antichains in

1980 ◽  
Vol 45 (1) ◽  
pp. 85-92 ◽  
Author(s):  
James E. Baumgartner
Keyword(s):  
Boolean Algebra ◽  
Continuum Hypothesis ◽  
Boolean Algebras ◽  
The Other ◽  
Martin's Axiom ◽  
Counter Example ◽  
Martin’S Axiom ◽  
Infinite Sequences ◽  
The Continuum

Consider the following propositions:(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α < ω1〉 such that α < β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α < ω1〉 exists.Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.

A Stochastic Comparison for Arrangement Increasing Functions

1994 ◽  
Vol 3 (3) ◽  
pp. 345-348 ◽  
Author(s):  
Abba M. Krieger ◽  
Paul R. Rosenbaum
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Random Permutation ◽  
The Other ◽  
Stochastic Comparison ◽  
Finite Distributive Lattice ◽  
Set Of Permutations ◽  
Arrangement Increasing ◽  
Increasing Functions ◽  
Increasing Function

Let h(·) be an arrangement increasing function, let X have an arrangement increasing density, and let XE be a random permutation of the coordinates of X. We prove E{h(XE)} ≤ E{h(X)}. This comparison is delicate in that similar results are sometimes true and sometimes false. In a finite distributive lattice, a similar comparison follows from Holley's inequality, but the set of permutations with the arrangement order is not a lattice. On the other hand, the set of permutations is a lattice, though not a distributive lattice, if it is endowed with a different partial order, but in this case the comparison does not hold.

Profinite Heyting Algebras and Profinite Completions of Heyting Algebras

2009 ◽  
Vol 16 (1) ◽  
pp. 29-47
Author(s):  
Guram Bezhanishvili ◽  
Patrick J. Morandi
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Heyting Algebra ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Profinite Completion ◽  
Bounded Distributive Lattice ◽  
Recent Developments ◽  
Profinite Completions

Abstract This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.

scholarly journals Characterising scoring rules by their solution in iteratively undominated strategies

2021 ◽  
Author(s):  
Christian Basteck
Keyword(s):  
Monotonicity Property ◽  
Scoring Rules ◽  
The Other ◽  
Approval Voting ◽  
Property A ◽  
Borda Rule ◽  
Scoring Rule ◽  
Direct Mechanism ◽  
Social Choice Correspondence ◽  
Choice Correspondence

AbstractWe characterize voting procedures according to the social choice correspondence they implement when voters cast ballots strategically, applying iteratively undominated strategies. In elections with three candidates, the Borda Rule is the unique positional scoring rule that satisfies unanimity (U) (i.e., elects a candidate whenever it is unanimously preferred) and is majoritarian after eliminating a worst candidate (MEW)(i.e., if there is a unanimously disliked candidate, the majority-preferred among the other two is elected). In a larger class of rules, Approval Voting is characterized by a single axiom that implies both U and MEW but is weaker than Condorcet-consistency (CON)—it is the only direct mechanism scoring rule that is majoritarian after eliminating a Pareto-dominated candidate (MEPD)(i.e., if there is a Pareto-dominated candidate, the majority-preferred among the other two is elected); among all finite scoring rules that satisfy MEPD, Approval Voting is the most decisive. However, it fails a desirable monotonicity property: a candidate that is elected for some preference profile, may lose the election once she gains further in popularity. In contrast, the Borda Rule is the unique direct mechanism scoring rule that satisfies U, MEW and monotonicity (MON). There exists no direct mechanism scoring rule that satisfies both MEPD and MON and no finite scoring rule satisfying CON.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

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