Matrix representation of finite effect algebras

Grzegorz Bińczak; Joanna Kaleta; Andrzej Zembrzuski

Displaying similar documents to “Matrix representation of finite effect algebras”

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

Trivialization of $𝒞 (X)$ -algebras with strongly self-absorbing fibres

Marius Dadarlat, Wilhelm Winter (2008)

Bulletin de la Société Mathématique de France

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Suppose $A$ is a separable unital $𝒞 (X)$ -algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$ -injective $C^{*}$ -algebra $𝒟$ . We show that $A$ and $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

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We investigate derived equivalences between subalgebras of some $Φ$ -Auslander-Yoneda algebras from a class of $n$ -angles in weakly $n$ -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$ -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to $n$ -angle cases. Finally, we give an explicit example to illustrate our result.

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- $n$ commutative algebras equipped with an ordered basis is an affine scheme $𝔅_{n}$ of finite type over $ℤ$ , with geometrically connected fibers. It is smooth if and only if $n \leq 3$ . It is reducible if $n \geq 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$ ). The relative dimension of $𝔅_{n}$ is $\frac{2}{27} n^{3} + O (n^{8 / 3})$ . The subscheme parameterizing étale algebras is isomorphic to ${GL}_{n} / S_{n}$ , which is of dimension only $n^{2}$ . For $n \geq 8$ , there exist algebras that are not limits of étale algebras. The dimension...

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...

A generalization of a formalized theory of fields of sets on non-classical logics

Helena Rasiowa

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Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮 *$ ..................................................................................................................... 5 § 3. $𝒮 *$ -algebras.....................................................................................................................

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

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Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵 (𝒢)$ be the center of $𝒢$ . Suppose that $𝔮 : 𝒢 \times 𝒢 \to 𝒢$ is an $ℛ$ -bilinear mapping and that $𝔗_{𝔮} : 𝒢 \to 𝒢$ is a trace of $𝔮$ . The aim of this article is to describe the form of $𝔗_{𝔮}$ satisfying the centralizing condition $[𝔗_{𝔮} (x), x] \in 𝒵 (𝒢)$ (and commuting condition $[𝔗_{𝔮} (x), x] = 0$ ) for all $x \in 𝒢$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $𝔗_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let $m$ and $n$ be positive integers, and let $R = (r_{1}, ..., r_{m})$ and $S = (s_{1}, ..., s_{n})$ be nonnegative integral vectors. Let $A (R, S)$ be the set of all $m \times n$ $(0, 1)$ -matrices with row sum vector $R$ and column vector $S$ . Let $R$ and $S$ be nonincreasing, and let $F (R)$ be the $m \times n$ $(0, 1)$ -matrix, where for each $i$ , the $i$ th row of $F (R, S)$ consists of $r_{i}$ 1’s followed by $(n - r_{i})$ 0’s. Let $A \in A (R, S)$ . The discrepancy of A, $disc (A)$ , is the number of positions in which $F (R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m \times n$ matrices over...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

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By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a \oplus b = max {a, b}$ and $a \otimes b = a + b$ for $a, b \in ℝ .$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B \in ℝ (m, n)$ if $A \otimes x = λ \otimes B \otimes x$ for some $λ \in ℝ$ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$ ; the $c$ -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$ . Its maximal cones are naturally indexed by the $c$ -sortable elements of $W$ . The main result of this paper is that the known bijection cl $_{c}$ between $c$ -sortable elements and $c$ -clusters induces a combinatorial isomorphism of fans. In particular, the $c$ -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

The $ℒ_{n}^{m}$ -propositional calculus

Carlos Gallardo, Alicia Ziliani (2015)

Mathematica Bohemica

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T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $ℒ_{n}^{m}$ -propositional calculus, denoted by $ℓ_{n}^{m}$ , is introduced in terms of the binary connectives $\to$ (implication), $↠$ (standard implication), $\land$ (conjunction), $\lor$ (disjunction) and the unary ones $f$ (negation) and $D_{i}$ , $1 \leq i \leq n - 1$ (generalized Moisil operators). It is proved that $ℓ_{n}^{m}$ belongs to the class of standard systems...

Bicrossed products of generalized Taft algebra and group algebras

Dingguo Wang, Xiangdong Cheng, Daowei Lu (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m, d} (q) ⋈ k [G]$ between the generalized Taft algebra $H_{m, d} (q)$ and group algebra $k [G]$ is actually the smash product $H_{m, d} (q) ♯ k [G]$ . Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$ . As an application, the classification of $H_{m, d} (q) ⋈ k [C_{n_{1}} \times C_{n_{2}}]$ is completely presented by generators and relations, where $C_{n}$ denotes the $n$ -cyclic group.

On generalized Beurling’s theorem and symmetry of $L_{1}$ -group algebras

Zenobia Anusiak (1971)

Colloquium Mathematicae

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On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let $𝕄_{n, m}$ be the set of all $n \times m$ real or complex matrices. For $A, B \in 𝕄_{n, m}$ , we say that $A$ is row-sum majorized by $B$ (written as $A ≺^{rs} B$ ) if $R (A) ≺ R (B)$ , where $R (A)$ is the row sum vector of $A$ and $≺$ is the classical majorization on $ℝ^{n}$ . In the present paper, the structure of all linear operators $T : 𝕄_{n, m} \to 𝕄_{n, m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on $ℝ^{n}$ and then find the linear preservers of row-sum majorization of these relations on $𝕄_{n, m}$ . ...

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

Maps on upper triangular matrices preserving zero products

Roksana Słowik (2017)

Czechoslovak Mathematical Journal

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Consider $𝒯_{n} (F)$ —the ring of all $n \times n$ upper triangular matrices defined over some field $F$ . A map $φ$ is called a zero product preserver on $𝒯_{n} (F)$ in both directions if for all $x, y \in 𝒯_{n} (F)$ the condition $x y = 0$ is satisfied if and only if $φ (x) φ (y) = 0$ . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $φ$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $φ$ as a...