A Note on Rough Set Algebra and Core Regular Double Stone Algebras

• URL: http://arxiv.org/abs/2101.02313v2
• Date: Mon, 18 Jan 2021 18:57:26 GMT
• Title: A Note on Rough Set Algebra and Core Regular Double Stone Algebras
• Authors: Daniel J. Clouse
• Abstract summary: In our Main Theorem we show $R_theta$ with $|theta_u| > 1 forall u in U$ to be isomorphic to $TP_E$ and $C_3E$, and that the three CRDSA's are complete and atomic. In our Main Corollary we show explicitly how we can embed such $R_theta$ in $TP_U$, $C_3U$, respectively, $phicirc alpha_r:R_thetahookright
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Given an approximation space $\langle U,\theta \rangle$, assume that $E$ is the indexing set for the equivalence classes of $\theta$ and let $R_\theta$ denote the collection of rough sets of the form $\langle\underline{X},\overline{X}\rangle$ as a regular double Stone algebra and what I. Dunstch referred to as a Katrinak algebra.[7],[8] We give an alternate proof from the one given in [1] of the fact that if $|\theta_u| > 1\ \forall\ u \in U$ then $R_\theta$ is a core regular double Stone algebra. Further let $C_3$ denote the 3 element chain as a core regular double Stone algebra and $TP_U$ denote the collection of ternary partitions over the set $U$. In our Main Theorem we show $R_\theta$ with $|\theta_u| > 1\ \forall\ u \in U$ to be isomorphic to $TP_E$ and $C_3^E$, with $E$ is an indexing set for $\theta$, and that the three CRDSA's are complete and atomic. We feel this could be very useful when dealing with a specific $R_\theta$ in an application. In our Main Corollary we show explicitly how we can embed such $R_\theta$ in $TP_U$, $C_3^U$, respectively, $\phi\circ \alpha_r:R_\theta\hookrightarrow TP_U\hookrightarrow C_3^U$, and hence identify it with its specific images. Following in the footsteps of Theorem 3. and Corollary 2.4 of [7], we show $C_3^J \cong R_\theta$ for $\langle U,\theta \rangle$ the approximation space given by $U = J \times \{0,1\}$, $\theta = \{(j0),(j1)\} : j \in J\}$ and every CRDSA is isomorphic to a subalgebra of a principal rough set algebra, $R_\theta$, for some approximation space $\langle U,\theta \rangle$. Finally, we demonstrate this and our Main Theorem by expanding an example from [1]. Further, we know a little more about the subalgebras of $TP_U$ and $C_3^U$ in general as they must exist for every $E$ that is an indexing set for the equivalence classes of any equivalence relation $\theta$ on $U$ satisfying $|\theta_u| > 1\ \forall\ u \in U$.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• LevAttention: Time, Space, and Streaming Efficient Algorithm for Heavy Attentions [54.54897832889028]
  We show that for any $K$, there is a universal set" $U subset [n]$ of size independent of $n$, such that for any $Q$ and any row $i$, the large attention scores $A_i,j$ in row $i$ of $A$ all have $jin U$. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well.
  arXiv  Detail & Related papers  (2024-10-07T19:47:13Z)
• A class of ternary codes with few weights [0.0]
  In this paper, we investigate a ternary code $mathcalC$ of length $n$, defined by $mathcalC$ := (textTr) := (textTr(dx), dots, dots, d_n$. Using recent results on explicit evaluations of exponential sums, we determine the Weil bound, and techniques, we show that the dual code of $mathcalC$ is optimal with respect to the Hamming bound.
  arXiv  Detail & Related papers  (2024-10-05T16:15:50Z)
• Synthesis and Arithmetic of Single Qutrit Circuits [0.9208007322096532]
  We study single qutrit quantum circuits consisting of words over the Clifford+ $mathcalD$ gate set. We characterize classes of qutrit unit vectors $z$ with entries in $mathbbZ[xi, frac1chi]$.
  arXiv  Detail & Related papers  (2023-11-15T04:50:41Z)
• Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits [0.8411424745913132]
  We show that when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $mathcalG_n$. We moreover show that $log(n)-2$ ancillas are always sufficient to construct a circuit for $U$.
  arXiv  Detail & Related papers  (2023-11-13T20:46:51Z)
• Subspace Controllability and Clebsch-Gordan Decomposition of Symmetric Quantum Networks [0.0]
  We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, it qudits Because of the symmetry, the underlying Hilbert space, $cal H=(mathbbCd)otimes n$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(dn)$, denoted here by $uS_n(dn)$.
  arXiv  Detail & Related papers  (2023-07-24T16:06:01Z)
• Realization of an arbitrary structure of perfect distinguishability of states in general probability theory [0.0]
  All subsets with a single element are of course in $mathcal A$, and since smaller collections are easier to distinguish, if $Hin mathcal A$ and $L subset H$ then $Lin mathcal A$; in other words, $mathcal A$ is a so-called $textitindependence system$ on the set of indices $[n]$.
  arXiv  Detail & Related papers  (2023-01-16T18:33:39Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Simplest non-additive measures of quantum resources [77.34726150561087]
  We study measures that can be described by $cal E(rhootimes N) =E(e;N) ne Ne$.
  arXiv  Detail & Related papers  (2021-06-23T20:27:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.