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Extensions of Soft Ideals over Semigroups
Essam Hamouda, A. A. Ramadan and S. Amira
Let (๐น, ๐ด) be a soft ideal over a semigroup ๐. We introduce the concept of the extension of (๐น, ๐ด). By using the mapping ฮฑ๐น corresponding to (๐น, ๐ด), we define the notions ฮฑโprime and ฮฑโsemiprime soft sets over ๐. Relationships between these concepts are investigated and illustrative examples are provided. Also, by means of the extensions of a soft ideal (๐น, ๐ด), a congruence relation ฮท๐น on ๐ is defined. It is shown that ฮท๐น is a semilattice congruence if (๐น, ๐ด) is a ฮฑโsemiprime soft left ideal over ๐. Finally, the soft sets (๐๐น , ๐ด) and (๐ผ๐น , ๐ด) over ๐ are defined in terms of the concept of soft points.
Keywords: Soft semigroups, soft left (right) ideals, soft points, extensions of soft ideals