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2. Linear maps that strongly preserve regular matrices over the Boolean algebra

Creator:
Kang, Kyung-Tae and Song, Seok-Zun
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Boolean algebra, regular matrix, and $(U,V)$-operator
Language:
English
Description:
The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb M}_{m,n}$. We call a matrix $A\in {\mathbb M}_{m,n}$ regular if there is a matrix $G\in {\mathbb M}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb M}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $\min \{m,n\}\le 2$, then all operators on ${\mathbb M}_{m,n}$ strongly preserve regular matrices, and if $\min \{m,n\}\ge 3$, then an operator $T$ on ${\mathbb M}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T(X)=UXV$ for all $X\in {\mathbb M}_{m,n}$, or $m=n$ and $T(X)=UX^TV$ for all $X\in {\mathbb M}_{n}$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. On some types of radical classes

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Boolean algebra, generalized Boolean algebra, $\frak m$-representability, lattice ordered group, generalized $MV$-algebra, and radical class
Language:
English
Description:
Let $\frak m$ be an infinite cardinal. We denote by $C_\frak m$ the collection of all $\frak m$-representable Boolean algebras. Further, let $C_\frak m^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\frak m$. In this paper we prove that $C_\frak m^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. Orthocomplemented difference lattices with few generators

Creator:
Matoušek, Milan and Pták, Pavel
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
orthomodular lattice, quantum logic, symmetric difference, Gödel´s coding, Boolean algebra, and free algebra
Language:
English
Description:
The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., \cite{pp:book,wata}). Recently an effort has been exercised to advance with logics that possess a symmetric difference (\cite{matODL,MP1}) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In \cite{matODL} the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is MO3×24.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public