Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $\sigma$-effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $\sigma$-MV algebra, and every observable is defined by a smearing of a sharp...

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.

In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,0, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties...

If element $z$ of a lattice effect algebra $(E,\oplus ,\mathbf{0},\mathbf{1})$ is central, then the interval $[\mathbf{0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C\left(E\right)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C\left(E\right)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C\left(E\right)$ is a bifull sublattice...

In the present paper we show that free $MV$-algebras can be constructed by applying free abelian lattice ordered groups.

Fuzzy ideals of pseudo MV-algebras are investigated. The homomorphic properties of fuzzy prime ideals are given. A one-to-one correspondence between the set of maximal ideals and the set of fuzzy maximal ideals μ satisfying μ(0) = 1 and μ(1) = 0 is obtained.

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 $𝒜$ we denote by ${𝒜}^{\text{'}},A$ and $\ell \left(𝒜\right)$ the idempotent modification, the underlying set or the underlying lattice of $𝒜$, respectively. In the present paper we prove that if $𝒜$ is semisimple and $\ell \left(𝒜\right)$ is a chain, then ${𝒜}^{\text{'}}$ is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.

Let Int $𝒜$ be the lattice of all intervals of an $MV$-algebra $𝒜$. In the present paper we investigate the relations between direct product decompositions of $𝒜$ and (i) the lattice Int $𝒜$, or (ii) 2-periodic isometries on $𝒜$, respectively.

In this paper we apply the notion of the product $MV$-algebra in accordance with the definition given by B. Riečan. We investigate the convex embeddability of an $MV$-algebra into a product $MV$-algebra. We found sufficient conditions under which any two direct product decompositions of a product $MV$-algebra have isomorphic refinements.

It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations – multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.

Let $𝔪$ be an infinite cardinal. We denote by $C_{𝔪}$ the collection of all $𝔪$-representable Boolean algebras. Further, let $C_{𝔪}^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_{𝔪}$. In this paper we prove that $C_{𝔪}^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.

A variant of Alexandrov theorem is proved stating that a compact, subadditive $D$-poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.