Riesz Decomposition Properties and the Lexicographic Product of po-groups

We establish conditions when a certain type of the Riesz Decomposition Property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $\ell$-group if and only if the first one is linearly ordered and the second one is an $\ell$-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDP's. RDP's of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas

When Lexicographic Product of Two po-Groups has the Riesz Decomposition Property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even RDP$_1$, a stronger type of RDP. We recall that a very strong type of RDP, RDP$_2$, entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas

The lexicographic product of po-groups and nn-perfect pseudo effect algebras

We will study the existence of different types of the Riesz Decomposition Property for the lexicographic product of two partially ordered groups. A special attention will be paid to the lexicographic product of the group of the integers with an arbitrary po-group. Then we apply these results to the study of $n$-perfect pseudo effect algebras. We show that the category of strong $n$-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups with RDP$_1.

Lexicographic Product vs Q\mathbb Q-perfect and H\mathbb H-perfect Pseudo Effect Algebras

We study the Riesz Decomposition Property types of the lexicographic product of two po-groups. Then we apply them to the study of pseudo effect algebras which can be decomposed to a comparable system of non-void slices indexed by some subgroup of real numbers. Finally, we present their representation by the lexicographic product

Kite Pseudo Effect Algebras

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected with partially ordered groups not necessarily with strong unit. In such a case, starting even with an Abelian po-group, we can obtain a noncommutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. Kites are so-called perfect pseudo effect algebras, and we define conditions when kite pseudo effect algebras have the least non-trivial normal ideal

Discrete (n+1)(n+1)-valued states and nn-perfect pseudo-effect algebras

We give sufficient and necessary conditions to guarantee that a pseudo-effect algebra admits an $(n+1)$-valued discrete state. We introduce $n$-perfect pseudo-effect algebras as algebras which can be split into $n+1$ comparable slices. We prove that the category of strong $n$-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups of a special type

Some Remarks on Kite Pseudo Effect Algebras

Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group $G$, a set $I$ and with two bijections $\lambda,\rho:I \to I.$ Using a clever construction on the ordinal sum of $(G^+)^I$ and $(G^-)^I,$ we can define a pseudo effect algebra which can be non-commutative even if $G$ is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.Comment: arXiv admin note: text overlap with arXiv:1306.030

Riesz space-valued states on pseudo MV-algebras

We introduce Riesz space-valued states, called $(R,1_R)$-states, on a pseudo MV-algebra, where $R$ is a Riesz space with a fixed strong unit $1_R$. Pseudo MV-algebras are a non-commutative generalization of MV-algebras. Such a Riesz space-valued state is a generalization of usual states on MV-algebras. Any $(R,1_R)$-state is an additive mapping preserving a partial addition in pseudo MV-algebras. Besides we introduce $(R,1_R)$-state-morphisms and extremal $(R,1_R)$-states, and we study relations between them. We study metrical completion of unital $\ell$-groups with respect to an $(R,1_R)$-state. If the unital Riesz space is Dedekind complete, we study when the space of $(R,1_R)$-states is a Choquet simplex or even a Bauer simplex

On EMV-algebras

The paper deals with an algebraic extension of $MV$-algebras based on the definition of generalized Boolean algebras. We introduce a new algebraic structure, not necessarily with a top element, which is called an $EMV$-algebra and every $EMV$-algebra contains an $MV$-algebra. First, we present basic properties of $EMV$-algebras, give some examples, introduce and investigate congruence relations, ideals and filters on this algebra. We show that each $EMV$-algebra can be embedded into an $MV$-algebra and we characterize $EMV$-algebras either as $MV$-algebras or maximal ideals of $MV$-algebras. We study the lattice of ideals of an $EMV$-algebra and prove that any $EMV$-algebra has at least one maximal ideal. We define an $EMV$-clan of fuzzy sets as a special $EMV$-algebra. We show any semisimple $EMV$-algebra is isomorphic to an $EMV$-clan of fuzzy functions on a set. We consider the variety of $EMV$-algebra and we present an equational base for each proper subvariety of the variety of $EMV$-algebras. We establish a categorical equivalencies of the category of proper $EMV$-algebras, the category of $MV$-algebras with a fixed special maximal ideal, and a special category of Abelian unital $\ell$-groups

Lifting, nn-Dimensional Spectral Resolutions, and nn-Dimensional Observables

We show that under some natural conditions, we are able to lift an $n$-dimensional spectral resolution from one monotone $\sigma$-complete unital po-group into another one, when the first one is a $\sigma$-homomorphic image of the second one. We note that an $n$-dimensional spectral resolution is a mapping from $\mathbb R^n$ into a quantum structure which is monotone, left-continuous with non-negative increments and which is going to $0$ if one variable goes to $-\infty$ and it goes to $1$ if all variables go to $+\infty$. Applying this result to some important classes of effect algebras including also MV-algebras, we show that there is a one-to-one correspondence between $n$-dimensional spectral resolutions and $n$-dimensional observables on these effect algebras which are a kind of $\sigma$-homomorphisms from the Borel $\sigma$-algebra of $\mathbb R^n$ into the quantum structure. An important used tool are two forms of the Loomis--Sikorski theorem which use two kinds of tribes of fuzzy sets. In addition, we show that we can define three different kinds of $n$-dimensional joint observables of $n$ one-dimensional observables