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Start Over Creator Jakubík, Ján Language English Subject $MV$-algebra

2. Higher degrees of distributivity in $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, archimedean $MV$-algebra, completeness, singular $MV$-algebra, and higher degrees of distributivity
Language:
English
Description:
In this paper we deal with the of an $MV$-algebra $\mathcal A$, where $\alpha $ and $\beta $ are nonzero cardinals. It is proved that if $\mathcal A$ is singular and $(\alpha,2)$-distributive, then it is . We show that if $\mathcal A$ is complete then it can be represented as a direct product of $MV$-algebras which are homogeneous with respect to higher degrees of distributivity.
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3. On a homogeneity condition for $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, generalized cardinal property, projectability, orthogonal completeness, and direct product
Language:
English
Description:
In this paper we deal with a homogeneity condition for an $MV$-algebra concerning a generalized cardinal property. As an application, we consider the homogeneity with respect to $\alpha $-completeness, where $\alpha $ runs over the class of all infinite cardinals.
Rights:
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5. On idempotent modifications of $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, idempotent modification, and subdirect reducibility
Language:
English
Description:
The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an $MV$-algebra $\mathcal A$ we denote by $\mathcal A^{\prime }, A$ and $\ell (\mathcal A)$ the idempotent modification, the underlying set or the underlying lattice of $\mathcal A$, respectively. In the present paper we prove that if $\mathcal A$ is semisimple and $\ell (\mathcal A)$ is a chain, then $\mathcal A^{\prime }$ is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.
Rights:
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6. On intervals and isometries of $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, duality, interval, autometrization, and 2-periodic isometry
Language:
English
Description:
Let Int $\mathcal A$ be the lattice of all intervals of an $MV$-algebra $\mathcal A$. In the present paper we investigate the relations between direct product decompositions of $\mathcal A$ and (i) the lattice Int $\mathcal A$, or (ii) 2-periodic isometries on $\mathcal A$, respectively.
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7. State-homomorphisms on $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, state homomorphism, and $\sigma $-closed maximal ideal
Language:
English
Description:
Riečan [12] and Chovanec [1] investigated states in $MV$-algebras. Earlier, Riečan [11] had dealt with analogous ideas in $D$-posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on $MV$-algebras. We remark that a different definition of a state in an $MV$-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term “state-homomorphism” (instead of “state”). The author is indebted to the referee for his suggestion concerning terminology. Let $\mathcal A$ be an $MV$-algebra which is defined on a set $A$ with $\mathop {\mathrm card}A>1$. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $\mathcal A$ and the system of all $\sigma $-closed maximal ideals of $\mathcal A$. For $MV$-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between $MV$-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.
Rights:
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8. Weak homogeneity and Pierce’s theorem for $MV$-algebras

Creator:
Jakubík, Ján
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$MV$-algebra, weak homogeneity, and internal direct product decomposition
Language:
English
Description:
In this paper we prove a theorem on weak homogeneity of $MV$-algebras which generalizes a known result on weak homogeneity of Boolean algebras. Further, we consider a homogeneity condition for $MV$-algebras which is defined by means of an increasing cardinal property.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public