The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notions of closed intersectional soft BCI-ideals and intersectional soft commutative BCI-ideals are introduced, and related properties are investigated. Conditions for an intersectional soft BCI-ideal t...

The concept of soft ideal was first introduced by Kandil et al. [13]. In 1999, Molodtsov [22] introduced the concept of soft sets as a general mathematical tool for dealing with uncertain objects. The concept of generalized closed soft sets in soft topological spaces was introduced by Kannan [15] in 2012. The notions of supra soft topological space were first introduced by Kandil et al. [14]. I...

in this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. we study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. for strongly z-ideals, we analyze prime ideals using the concept of zero sets. moreover, it is proven that the intersection of all zero sets of a prime ideal of c(l),...

We study Baire category for downward-closed subsets of 2ω , showing that it behaves better in this context than for general subsets of 2ω . We show that, in the downward-closed context, the ideal of meager sets is prime and b-complete, while the complementary filter is g-complete. We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results fo...

We solve two open problems concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a left ideal or a suffix-closed language with n left quotients (that is, with state complexity n) is at most n n−1 + n − 1, and that of a two-sided ideal or a factor-closed language is at most n n−2 + (n − 2)2 n−2 + 1. Since these bounds are known to be reachable, this settles...

We compare open loop versus closed loop identiication when the identiied model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. For three diierent control design criteria (minimum variance, LQG and model reference control) we show that, under those conditions, a better performance is achieved by closing the loop during...

The classical Nullstellensatz asserts that a reduced affine variety is known by its closed points; algebraically, a prime ideal in an affine ring is the intersection of the maximal ideals containing it. A leading special case of our theorem says that any affine scheme can be distinguished from its subschemes by its closed points with a bounded index of nilpotency; algebraically, an ideal I in a...

An Ideal on a set X is a non empty collection of subsets of X with heredity property which is also closed under finite unions. In this paper, (i,j)g*s closed and open sets are introduced with respect to an ideal in a bitopological space and their properties are investigated. Additionally, we compare them with other sets to show their relationships and characterize many other results.

We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.