Inverse subsemigroups of finite index in finitely generated inverse semigroups

We study some aspects of Schein’s theory of cosets for closed inverse subsemigroups of inverse semigroups. We establish an index formula for chains of subsemigroups, and an analogue of M. Hall’s Theorem on the number of cosets of a fixed finite index. We then investigate the relationships between the following properties of a closed inverse submonoid of an inverse monoid: having finite index; being a recognizable subset; being a rational subset; being finitely generated (as a closed inverse submonoid). A remarkable result of Margolis and Meakin shows that these properties are equivalent for a closed inverse submonoid of a free inverse monoid.     

On Nuclei and Duals of Semifields of Order q 4, q is a Power of 2

The aim of this article is to investigate the nuclei of a semifield A of order q ⁴, q is a power of 2, admitting a four-group E of automorphisms isomorphic to Z ₂ × Z ₂ acting freely on A and having GF(q ²) in its left nucleus. We also study the dual semifield A* of A and show that A and A* are isomorphic.     

Nutraceuticals: Transformation of Conventional Foods into Health Promoters/Disease Preventers and Safety Considerations

Nutraceuticals are essential food constituents that provide nutritional benefits as well as medicinal effects. The benefits of these foods are due to the presence of active compounds such as carotenoids, collagen hydrolysate, and dietary fibers. Nutraceuticals have been found to positively affect cardiovascular and immune system health and have a role in infection and cancer prevention. Nutraceuticals can be categorized into different classes based on their nature and mode of action. In this review, different classifications of nutraceuticals and their potential therapeutic activity, such as anti-cancer, antioxidant, anti-inflammatory and anti-lipid activity in disease will be reviewed. Moreover, the different mechanisms of action of these products, applications, and safety upon consumers including current trends and future prospect of nutraceuticals will be included.