Free products of unital ℓ-groups and free products of generalized MV-algebras

We generalize Komori’s characterization of the proper subvarieties of MV-algebras. Namely, within the variety of generalized MV-algebras (GMV-algebras) such that every maximal ideal is normal, we characterize the proper top varieties. In addition, we present equational bases for these top varieties. We show that there are only countably many different proper top varieties and each of them has uncountably many subvarieties. Finally, we study coproducts and we show that the amalgamation property fails for the class of n-perfect GMV-algebras, i.e., GMV-algebras that can be split into n + 1 comparable slices. This paper has been supported by the Center of Excellence SAS -Physics of Information-I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contracts No. APVT-51-032002, APVV-0071-06, Bratislava.     

Diagonal sequence property in Banach spaces with weaker topologies

We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet–Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.     

Semi-Fredholm problems on periodic solutions of functional-differential equations of neutral type

We consider the problem on periodic solutions for linear systems of functionaldifferential equations of neutral type with periodic coefficients and periodic deviations of the argument. By reduction to associated functional equations, we derive necessary and sufficient conditions under which the problem on periodic solutions for such a system is Fredholm or semi-Fredholm.     

α‐keto‐β‐diimine nickel‐catalyzed olefin polymerization: Effect of ortho‐aryl substituents and preparation of stereoblock copolymers

A series of α‐keto‐β‐diimine nickel complexes (Ar‐N = C(CH₃)‐C(O)‐C(CH₃)=N‐Ar)NiBr₂; Ar = 2,6‐R‐C₆H₃‐, R = Me, Et, iPr, and Ar = 2,4,6‐Me₃‐C₆H₃‐) was prepared. All corresponding ligands are unstable even under an inert atmosphere and in a freezer. Stable copper complex intermediates of ligand synthesis and ethyl substituted nickel complex were isolated and characterized by X‐ray. All nickel complexes were used for the polymerization of ethene, propylene, and hex‐1‐ene to investigate their livingness and the extent of chain‐walking. Low‐temperature propene polymerization with less bulky ortho‐substituents was less isospecific than the one with isopropyl derivative. Propene stereoblock copolymers were prepared by iPr derivative combining the polymerization at low temperature to prepare isotactic polypropylene (PP) block and at a higher temperature, supporting chain‐walking, to obtain amorphous regioirregular PP block. Alternatively, a copolymerization of propene with ethene was used for the preparation of amorphous block. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2017 , 55, 2440–2449     

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.     

On States on Orthogonally Closed Subspaces of an Inner Product Space

We show that every finitely additive state on the system F(S) of all orthogonally closed subspaces of an infinite-dimensional inner product space S attains all values from the real interval [0,1]. In particular, we show that there is no finitely additive countably-valued state on F(S) whenever dim S=. The main technique we use is an embedding of L(H _n ) into F(S).     

Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.In this paper, we discuss the property of archimedeanness for PE-algebras on the one hand and for po-groups on the other hand. We prove that under the assumption of suphomogeneity, archimedeanness holds for a PE-algebra with the Riesz property if and only if it holds for its representing group. The algebra is in that case commutative. This result is established by using the technique of MacNeille completion. We give the exact condition for this completion to exist, and we clearly exhibit the role played by archimedeanness and by sup-homogeneity.     

Converse of the Eilers-Horst theorem

We generalize the theorem of Eilers and Horst, showing that any finite as well as any-finite measure on a quantum logic of all closed subspaces of a Hilbert spaceH of dimension 2 is a Gleason one iff the dimension ofH is a nonmeasurable cardinal.     

State on splitting subspaces and completeness of inner product spaces

We show that an inner product spaceV is complete iff the system of all splitting subspaces, i.e., of all subspacesM for whichM + M =V, possesses at least one completely additive state.