Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

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.

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.

Explaining the 1914 War in Europe

2019 ◽  
pp. 99-124
Author(s):  
Frank C. Zagare
Keyword(s):  
World War I ◽  
Incomplete Information ◽  
The Other ◽  
Great War ◽  
World War ◽  
International History ◽  
Other Hand ◽  
Game Theoretic ◽  
The One ◽  
The Great War

This chapter focuses on the outbreak of World War I, which remains one of the most perplexing events of international history. It should be no surprise that rationalist interpretations of the July Crisis are a diverse lot, ranging from the sinister to the benign. This chapter constructs a theoretically rigorous rationalist explanation of World War I, the 1914 European war that involved Austria–Hungary, Germany, Russia, and France. On the one hand, this chapter confirms the view that one does not have to take a particularly dark view of German intentions to explain the onset of war in 1914; on the other hand, it also calls into question the “accidental war” thesis. A number of related questions about the Great War are addressed in the context of a generic game-theoretic escalation model with incomplete information.

scholarly journals WEAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY

2016 ◽  
Vol 81 (2) ◽  
pp. 711-717
Author(s):  
DAN HATHAWAY
Keyword(s):  
Boolean Algebra ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Similar Argument ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Image Position ◽  
Weakly Compact Cardinal

AbstractLet $B$ be a complete Boolean algebra. We show that if λ is an infinite cardinal and $B$ is weakly (λω, ω)-distributive, then $B$ is (λ, 2)-distributive. Using a similar argument, we show that if κ is a weakly compact cardinal such that $B$ is weakly (2κ, κ)-distributive and $B$ is (α, 2)-distributive for each α < κ, then $B$ is (κ, 2)-distributive.

The k-Extremally Disconnected Spaces as Projectives

1964 ◽  
Vol 16 ◽  
pp. 253-260 ◽  
Author(s):  
Henry B. Cohen
Keyword(s):  
Continuous Function ◽  
Boolean Algebra ◽  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Proper Subset ◽  
Stone Space ◽  
Infinite Cardinal ◽  
Extremally Disconnected

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.

A Finiteness Criterion for Orthomodular Lattices

1978 ◽  
Vol 30 (02) ◽  
pp. 315-320 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Infinite Chain ◽  
The Other ◽  
Finitely Generated ◽  
Orthomodular Lattices ◽  
Other Hand ◽  
Finiteness Criterion ◽  
Element Set

The main result of this paper is the following: THEOREM. Every finitely generated orthomodular lattice L with finitely many maximal Boolean subalgebras (blocks) is finite. If L has one block only, our theorem reduces to the well-known fact that every finitely generated Boolean algebra is finite. On the other hand, it is known that a finitely generated orthomodular lattice without any further restrictions can be infinite. In fact, in [2] we constructed an orthomodular lattice which is generated by a three-element set with two comparable elements, has infinitely many blocks and contains an infinite chain.

Minimal primal ideal spaces and norms of inner derivations of tensor products of C*-algebras

1996 ◽  
Vol 119 (2) ◽  
pp. 297-308 ◽  
Author(s):  
Eberhard Kaniuth
Keyword(s):  
Cross Sections ◽  
Dense Subset ◽  
Tensor Products ◽  
The Other ◽  
Primitive Ideal ◽  
Ideal Space ◽  
C Algebra ◽  
C Algebras ◽  
Other Hand ◽  
Full Algebra

An ideal I in a C*-algebra A is called primal if whenever n ≥ 2 and J1,…, Jn are ideals in A with zero product then Jk ⊆ I for at least one k. The topologized space of minimal primal ideals of A, Min-Primal (A), has been extensively studied by Archbold[3]. Very much in the spirit of Fell's work [14] it was shown in [3, theorem 5·3] (see also [5, theorem 3·4]) that if A is quasi-standard, then A is *-isomorphic to a maximal full algebra of cross-sections of Min-Primal (A). Moreover, if A is separable the fibre algebras are primitive throughout a dense subset. On the other hand, the complete regularization of the primitive ideal space of A gives rise to the space of so-called Glimm ideals of A, Glimm (A). It turned out that A is quasi-standard exactly when Min-Primal (A) and Glimm (A) coincide as sets and topologically [5, theorem 3·3].

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

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