Does there exist an atomic lattice effect algebra with non-atomic subalgebra of sharp elements? An affirmative answer to this question (and slightly more) is given: An example of an atomic MV-effect algebra with a non-atomic Boolean subalgebra of sharp or central elements is presented.
Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given., Wenchang Li, Jingshi Xu., and Seznam literatury
Following the study of sharp domination in effect algebras, in particular, in atomic Archimedean MV-effect algebras it is proved that if an atomic MV-effect algebra is {\it uniformly Archimedean} then it is sharply dominating.
Let L := −Δ + V be a Schrödinger operator on\mathbb{R}^{n} with n\geqslant 3 and V\geqslant 0 satisfying \Delta ^{-1}V\in L^{\infty }(\mathbb{R}^{n}). Assume that φ: {R}^{n} × [0,∞) → [0,∞) is a function such that φ(x,,) is an Orlicz function, φ(•, t) \in A_{\infty }({R}^{n}) (the class of uniformly Muckenhoupt weights). Let w be an L-harmonic function on {R}^{n} with 0< C_{1}\leq w\leq C_{2}, where C_{1} and C_{2} are positive constants. In this article, the author proves that the mapping H_{\phi ,L} (\mathbb{R}^n ) \mathrel\backepsilon f \mapsto wf \in H_\phi (\mathbb{R}^n ) is an isomorphism from the Musielak-Orlicz-Hardy space associated with L,H_{\phi ,L} (\mathbb{R}^n ), to the Musielak-Orlicz-Hardy space H_\phi (\mathbb{R}^n ) under some assumptions on φ. As applications, the author further obtains the atomic and molecular characterizations of the space H_{\phi ,L} (\mathbb{R}^n ) associated with w, and proves that the operator {( - \Delta )^{ - 1/2}}{L^{1/2}} is an isomorphism of the spaces H_{\phi ,L} (\mathbb{R}^n ) and H_\phi (\mathbb{R}^n ). All these results are new even when φ(x, t) ≔ t^{p}, for all x \in \mathbb{R}^{n} and t \in [0,∞), with p ∞ (n/(n + μ_{0}), 1) and some μ_{0} \in (0, 1]., Sibei Yang., and Obsahuje seznam literatury
If element z of a lattice effect algebra (E,⊕,0,1) is central, then the interval [0,z] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕. It is a known fact that if the set C(E) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C(E) in E equals to the top element of E, then E is isomorphic to a subdirect product of irreducible effect algebras (\cite{R2}). This means that if there exists a MacNeille completion E^ of E which is its extension (i.e. E is densely embeddable into E^) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of E^. For example, the existence of a state in E^ implies the existence of a state in E. In this context, a natural question arises if the MacNeille completion of the center of E (denoted as MC(C(E))) is necessarily the same as the center of E^, i.e., if MC(C(E))=C(E^) is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of C(E) and its bifullness in E is not sufficient to guarantee the mentioned equality.
If element z of a lattice effect algebra (E,⊕,0,1) is central, then the interval [0,z] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕. It is a known fact that if the set C(E) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C(E) in E equals to the top element of E, then E is isomorphic to a direct product of irreducible effect algebras (\cite{R2}). In \cite{PR} Paseka and Riečanová published as open problem whether C(E) is a bifull sublattice of an Archimedean atomic lattice effect algebra E. We show that there exists a lattice effect algebra (E,⊕,0,1) with atomic C(E) which is not a bifull sublattice of E. Moreover, we show that also B(E), the center of compatibility, may not be a bifull sublattice of E.
It is proved that a radical class $\sigma $ of lattice-ordered groups has exactly one cover if and only if it is an intersection of some $\sigma $-complement radical class and the big atom over $\sigma $.