In this paper some properties of the uniformity topology on a BCI-algebras are discussed. Keywords—(Fuzzy) ideal, (Fuzzy) subalgebra, Uniformity, clopen sets.

We construct a fundamental group for subshifts of nite type and prove that its abelianization gives a homology dual to ordered cohomology. We characterize cohomology classes representable as characteristic functions of clopen sets and give examples to illustrate what can happen for inclusions among clopen sets. We show that existence of unital ordered cohomology equivalence arising from isomorp...

1.1. Types. Most of the time we fix a complete countable first-order theory T in a language L, and we let M be a very saturated model of it, a “monster” model. All our sets and models are subsets and elementary submodels of M. A set is called small if it is cardinality is smaller than the saturation of M. For a set A, we let Def (A) be the boolean algebra of A-definable subsets of M, i.e. sets ...

We consider the ω1-Borel subsets of the reals in models of ZFC. This is the smallest family of sets containing the open subsets of the 2 and closed under ω1 intersections and ω1 unions. We show that Martin’s Axiom implies that the hierarchy of ω1-Borel sets has length ω2. We prove that in the Cohen real model the length of this hierarchy is at least ω1 but no more than ω1 + 1. Some authors have...

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. Rough approximations on a complete completely distributive lattice(CCD lattice for short) and brings generalizations of rough sets into a unified framework are discussed in [3]. This paper is devoted to the discussion of the relationship between approximations and topologies on a CCD lattice. It ...

A partial ordering P has the property SEP (or SEP(P )), if, for a sufficiently large regular cardinal χ, the family of elementary submodels of H(χ) of cardinality א1 with the property that [M ]א0 ∩M is cofinal subset (with respect to ⊆) of [M ]א0 and P ∩M ≤σ P , is cofinal (also with respect to ⊆) in [H(χ)]א1 (see Proposition 2 below). We describe, in section 1, against which historical backgro...

It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all sep...

Both spaces were introduced and shown to be one-dimensional but totally disconnected by Paul Erdős [9] in 1940. This result together with the obvious fact that E and Ec are homeomorphic to their squares make these spaces important examples in Dimension Theory. Both E and Ec are universal spaces for the class of almost zero-dimensional spaces; see [7, Theorem 4.15]. A subset of a space is called...