Narrow Results

In this paper, we study the concept of nearness between $L$-fuzzy rough sets and the concepts of $L$-fuzzy rough grills and $L$-fuzzy rough ${\delta }_{{ℱ}}$-clusters on a Čech $L$-fuzzy rough proximity space $(X,{\delta }_{{ℱ}})$. Also, we investigate the relationship among $L$-fuzzy rough grills, $L$-fuzzy rough ${\delta }_{{ℱ}}$-clan and $L$-fuzzy rough ${\delta }_{{ℱ}}$-clusters. Moreover, we introduce the $L$-fuzzy rough LO-proximity spaces and study some of its basic properties. An adequate number of examples support the study.

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let be the statement that a non-principal ultrafilter on ℕ exists and let be the higher-order extension of ACA₀. We show that is -conservative over and thus that is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly statement ∀ f ∃ g A_qf(f, g) in a realizing term in Gödel's system T can be extracted. This means that one can extract a term t ∈ T, such that ∀ f A_qf(f, t(f)).

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers $B$ on $\omega$ as the prototype structures, we construct a class of continuum many topological Ramsey spaces ${{ℰ}}_B$ which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces ${{ℰ}}_B$, extending the Pudlák–Rödl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings ${𝒫}({[\omega ]}^k)/{\mathrm{\text{Fin}}}^{\otimes k}$ to the countable transfinite. The $\sigma$-closed partial order $({{ℰ}}_B,{\subseteq }^{{\mathrm{\text{Fin}}}^B})$ is forcing equivalent to ${𝒫}(B)/{\mathrm{\text{Fin}}}^B$, which forces a non-p-point ultrafilter ${{𝒢}}_B$. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters ${{𝒢}}_B$.

The notion of a $\delta$-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each $\delta <{\omega }_2$, any $\delta$-generic sequence of P-points can be extended to an ${\omega }_2$-generic sequence. This shows that the CH implies that there is a chain of P-points of length ${𝔠}^{+}$ with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question of Andreas Blass.

The Ultrapower Axiom is an abstract combinatorial principle inspired by the fine structure of canonical inner models of large cardinal axioms. In this paper, it is established that the Ultrapower Axiom implies that the Generalized Continuum Hypothesis holds above the least supercompact cardinal.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

This paper characterizes ideal structure of the uniform Roe algebra B*(X) over simple cores X. A necessary and sufficient condition for a principal ideal of B*(X) to be spatial is given and an example of non-spatial ideal of B*(X) is constructed. By establishing an one-one correspondence between the ideals of B*(X) and the ω-filters on X, the maximal ideals of B*(X) are completely described by the corona of the Stone-Čech compactification of X.

In order to define the concept of nearness between $L$-fuzzy rough sets, we introduce the concept of Čech $L$-fuzzy rough proximity spaces. On this new nearness structure, we prove some basic topological results. To support the proposed approach, examples are given. Further, we introduce $L$-fuzzy rough grills, $L$-fuzzy rough filters and $L$-fuzzy rough clusters and on Čech $L$-fuzzy rough proximity spaces, and find the relationship between them.