Atomic Sequential Effect Algebras

Various conditions ensuring that a sequential effect algebra or the set of sharp elements of a sequential effect algebra is a Boolean algebra are presented

Conditions that Force an Orthomodular Poset to Be a Boolean Algebra

We introduce two new classes of orthomodular posets—the class of weakly Boolean orthomodular posets and the class of orthomodular posets with the property of maximality. The main result of this paper is that the intersection of these classes is the class of Boolean algebras. Since the first class introduced here contains various classes of orthomodular posets with a given property of its state space and the second class contains, e.g., lattice (orthocomplete, resp.) orthomodular posets, the main theorem can be viewed as a generalization of various results concerning the question when an orthomodular poset has to be a Boolean algebra. Moreover, it gives alternative proofs to previous results and new results of this type

Note on generalizations of orthocomplete and lattice effect algebras

It is proved that two di erent common generalizations of orthocomplete and lattice e ect algebras coincide within the class of separable Archimedean e ect algebras

Effect algebras with the maximality property

The maximality property was introduced in [9] in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of e ect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch{Piron e ect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch{Piron states on an e ect algebra with the maximality property is strongly order determining

Concrete Quantum Logics with Generalised Compatibiliy

We present three results stating when a concrete (= set-representable) quantum logic with covering properties (generalization of compatibility) has to be a Boolean algebra. These results complete and generalize some previous results [3, 5] and answer partially a question posed in [2]

Robust Draws in Balanced Knockout Tournaments

Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decision-making and elections. The traditional computational question, that asks to compute a draw (optimal draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a)~there exist deterministic tournaments (where the pairwise winning probabilities are~0 or~1) where one optimal draw is much more robust than the other; and (b)~in general, there exist tournaments with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time algorithms to compute robust optimal draws

Atomic effect algebras with compression bases

Compression base effect algebras were recently introduced by Gudder [6]. They generalize sequential effect algebras [7] and compressible effect algebras [5]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [11]. The notion of projection-atomicity is introduced and studied and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean

Commutative Bounded Integral Residuated Orthomodular Lattices are Boolean Algebras

We show that a commutative bounded integral orthomodular lattice is residuated iff it is a Boolean algebra. This result is a consequence of [7, Theorem 7.31]; however, our proof is independent and uses other instruments

Strong Amplifiers of Natural Selection: Proofs

We consider the modified Moran process on graphs to study the spread of genetic and cultural mutations on structured populations. An initial mutant arises either spontaneously (aka \emph{uniform initialization}), or during reproduction (aka \emph{temperature initialization}) in a population of $n$ individuals, and has a fixed fitness advantage $r>1$ over the residents of the population. The fixation probability is the probability that the mutant takes over the entire population. Graphs that ensure fixation probability of~1 in the limit of infinite populations are called \emph{strong amplifiers}. Previously, only a few examples of strong amplifiers were known for uniform initialization, whereas no strong amplifiers were known for temperature initialization. In this work, we study necessary and sufficient conditions for strong amplification, and prove negative and positive results. We show that for temperature initialization, graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/f(r)$, where $f(r)$ is a function linear in $r$. Similarly, we show that for uniform initialization, bounded-degree graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/g(r,c)$, where $c$ is the degree bound and $g(r,c)$ a function linear in $r$. Our main positive result complements these negative results, and is as follows: every family of undirected graphs with (i)~self loops and (ii)~diameter bounded by $n^{1-\epsilon}$, for some fixed $\epsilon>0$, can be assigned weights that makes it a strong amplifier, both for uniform and temperature initialization