The concept of soft sets based on complete atomic Boolean lattice, which can be seen as a generalization of soft sets, is introduced. Some operations on these soft sets are discussed, and new types of soft sets such as full, keeping infimum, and keeping supremum are defined and supported by some illustrative examples. Two pairs of new soft rough approximation operators are proposed and the relationship among soft set is investigated, and their related properties are given. We show that Järvinen's approximations can be viewed as a special case of our approximation. If = ℘(), then our soft approximations coincide with crisp soft rough approximations (Feng et al. 2011).

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Abstract:A soft ideal on a non empty set X is a non empty collection of soft subsets of X with heredity property which is also closed under finite unions. The concept of soft generalized closed sets in soft topological spaces was introduced by Kannan [1]. In this paper, we introduce and study the concept of soft generalized closed sets with respect to a soft ideal, which is the extension of the concept of soft generalized closed sets.

Background: Atherosclerosis has been extensively studied in thalassemia major (TM) and sickle cell disease but not yet in β thalassemia intermedia (TI). Previous studies concerned with TM were performed in children. TI patients usually live longer and, thus, are more prone to complications of atherosclerosis. Aim: In our study, we applied color Doppler for the determination of arterial conduit and flow velocities in β TI patients. Methods: For central circulation, we measured right and left middle cerebral arteries (MCAs) and basilar artery (BA) mean flow velocity (MFV), pulsatility index (PI), and peak systolic velocity (PSV) as well as carotid intimal media thickness, and to assess peripheral circulation, we studied ankle/brachial index and posterior and anterior tibial arteries' (ATA, PTA) pressure and PSV. This was applied for 30 adult TI patients and 20 age-, sex-, and ethnic group-matched controls. Results: Transcranial Doppler findings among cases and controls showed that the MFV, PSV of MCAs, and PSV, PI, and MFV of the BA were statistically higher in cases than controls. A comparison between splenectomized and non-splenectomized patients showed that total leukocyte count, platelet count, lactate dehydrogenase, ferritin, PSV and MFV of the left MCA were all statistically higher in splenectomized cases. Differences between males and females with TI with respect to laboratory and Doppler findings were all statistically insignificant except for intima media thickness, PTA pressure, ATA pressure, and PSV. Conclusion: More than one parameter should be applied to assess atherosclerosis in TI. There is evidence of an increased risk of central ischemia rather than peripheral ischemia in these patients.

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