Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.
We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such state MV-algebras with the category of unital Abelian ℓ-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.
Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201–206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138–152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic 161:161–173, 2009).
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. (shrink)
Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.
We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.
We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.
We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.
A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G 0), where G is an Abelian ℓ-group, G 0 is a subset of the positive cone generating G + (...) such that if u, v ∈ G 0, then 0 ∨ (u - v) ∈ G 0, and morphisms are ℓ-group homomorphisms h: (G, G 0) → (G′,G′0) with f(G 0) ⫅ G′0. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). (shrink)
The perspectives of Ukraine which are outlined by the notion of the “Ukrainian project”, and determined by potential of development of civil society, as congruent with perspectives of steady international development for the sake of collaboration and peace, are examined in the article. Determination of such basic analytical notions as discursive practices of “prevailing” and “understanding” is offered with this purpose. Discourse is considered as reason for choice and giving the advantage to one meaning over the others that is set (...) in the certain modes of signification. Discourse of understanding – is the process of creation of such knowledge, the nominative function of which stops being the function of power, and becomes the instrument of the renewed perception and understanding at new level of communicative space. As discursive reality is reflected not only on the methods of thinking, but also on the practical behaviour of people, we have warrants to speak about its ethical conditionality and the corresponding discursive‐ethical practices of “freedom and authenticity”, “paternalism and clientism” and “nihilism and anarchy”. Last two discursive-ethical practices are examined as obstacles on the way to realization of the “Ukrainian project” in relation to itsdemocratic development in direction to the civilization values common with the European Union. (shrink)
The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In (...) particular, if dim H=n, then on ℒ(H) there is a unique discrete state, namely m(A)=dim A/dim H, A∈ℒ(H). (shrink)
We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.
For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. (...) Finally, the result will be generalized to general inner product spaces. (shrink)
We study positive bilinear forms on a Hilbert space which are not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete generalized effect algebras.
We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a \-monotone \-complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral (...) resolutions of observables. (shrink)
An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure.
From "word-images' to "chapter-shots: The irnagiuist montage of Anatolij Mariengof. The article discusses the three dominant imaginist principles of Anatolij Mariengofs (1897-1962) poetic technique, as they are translated into prose in his first fictional novel Cynics (1928). These principles include the "catalogue of images", a genre introduced by Vadim Shershenevich, i.e. poetry formed of nouns, which Mariengof makes use of in his longer imaginist poems. Another dominant imaginist principle, to which Mariengof referred in his theoretic articles and poetic (...) texts, is similar to the creating of shocking images typical of Russian futurism. Mariengofs application is the juxtaposition of "pure" (chistyj) and "impure" (nechistyj), either a conflict between the vehicle and the object within a metaphor or a conflict between metaphors. This is an essential poetic feature in both Mariengofs poetry and prose. The third, maybe the most Mariengofian imaginist principle, relevant to the study of Cynics, is the poetics of transition (poetika sdviga), i.e. a certain fragmented structure of the text, which is related to Mariengors use of heteroaccentual rhyme. All these principles can be treated as fundamental elements in Mariengofs use of montage technique in his fictional prose. (shrink)
"The revolutionary desire to realize the Kingdom of God is the beginning of modern history." "Friedrich Schlegel" "The world has to be purified, recreated." "Anatolij Lunačarskij" "Ubi Lenin, ibi Israel." "Ernst Bloch" "Socialism is the religion that will kill off Christianity." "Antonio Gramsci"I. The Millennial Vision of History The millennialism that penetrated the heart of western civilization was one of the most incisive and enduring results of the spiritual victory of Christianity over the Greco-Roman culture. This is a vision (...) of history, formulated by the prophets of Israel from the eighth century…. (shrink)
From "word-images' to "chapter-shots: The irnagiuist montage of Anatolij Mariengof. The article discusses the three dominant imaginist principles of Anatolij Mariengofs poetic technique, as they are translated into prose in his first fictional novel Cynics. These principles include the "catalogue of images", a genre introduced by Vadim Shershenevich, i.e. poetry formed of nouns, which Mariengof makes use of in his longer imaginist poems. Another dominant imaginist principle, to which Mariengof referred in his theoretic articles and poetic texts, is (...) similar to the creating of shocking images typical of Russian futurism. Mariengofs application is the juxtaposition of "pure" and "impure", either a conflict between the vehicle and the object within a metaphor or a conflict between metaphors. This is an essential poetic feature in both Mariengofs poetry and prose. The third, maybe the most Mariengofian imaginist principle, relevant to the study of Cynics, is the poetics of transition, i.e. a certain fragmented structure of the text, which is related to Mariengors use of heteroaccentual rhyme. All these principles can be treated as fundamental elements in Mariengofs use of montage technique in his fictional prose. (shrink)
The article discusses the concept of montage as used by the Russian Imaginist poetic group: the montage principle in their poetry, theoretical writings and theatre articles. The leading Imaginist figures Vadim Shershenevich and Anatolij Mariengof were active both in theorizing and practising montage in their oeuvre at the beginning of the 1920s. Shershenevich’s application of the principle in poetry was called “image catalogue”, a radical poetic experiment in the spirit of both Walt Whitman and Sergei Eisenstein. Mariengof ’s main (...) contribution to the montage poetics was his first fictional novel The Cynics. The article also discusses the Imaginists’ writings on the essence of theatre as an autonomous art form – Shershenevich’s actitivy in the OGT and Mariengof ’s participation in the work of the MKT. (shrink)