Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. In this paper, we propose two family of sets and study the conditions that these two sets become some lattice structures. These two sets are consisted by ...

Laser has proven to be an effective and safe method of treating ureteral stones. When the electromagnetic energy of a laser pulse is selectively absorbed by stone, a plasma fonns at the surface. This plasma, which is composed of ions and electrons, continues to absorb laser energy reaching very high pressure and generating a shock wave that fragment the stone. The degree of stone fragmentation ...

The idea of the mini-course is to review the formulation of quantum theories from an advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions due to von Neumann, discussing some fundamental ideas, mathematical tools and theorems in particular related to the representation of physical symmetries (Gleason, Wigner-Kadison, Stone Stone-von Neumann, Garding, Nelso...

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone ...

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice, whenever the partially ordered set of the equivalence classes of R∩R−1 does not contain infinite ascending chains. We show that on this lattice can be defined three different kinds of complementation operations, and we...

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice, whenever the partially ordered set of the equivalence classes of R∩R does not contain infinite ascending chains. We show that on this lattice can be defined three different kinds of complementation operations, and we d...

In this note we prove several duality theorems in lattice theory. We also discuss the connection between spectral spaces and Priestley spaces, and interpret Priestley duality in terms of spectral spaces. The organization of this note is as follows. In the first section we collect appropriate definitions and basic results common to many of the various topics. The next four sections consider Birk...

BACKGROUND JJ stents are often encountered in patients with pelvic renal stones referred for shock wave lithotripsy, most of them being placed either for obstructive renal pelvic stones or for ureteric stones mobilized retrograde during the JJ stent insertion. The aim of the study was to determine whether the relative stone position in the upper loop of the JJ stent during extracorporeal shock ...

It is well known that the lattice $${{\,\mathrm{Id_c}\,}}{G}$$ of all principal $$\ell $$ -ideals any Abelian -group G a completely normal distributive 0-lattice; yet not every 0-lattice homomorphic image some , via counterexample cardinality $$\aleph _2$$ . We prove with at most _1$$ elements By Stone duality, this means generalized spectral space compact open sets homeomorphic to subspace -sp...