Characterizations of Upward and Downward Sets in Semimodules by Using Topical Functions

ABSTRACT

In this article, we study topical functions f:X→K defined on a b-complete idempotent semimodule X over a b-complete idempotent semifield K with values in K. We characterize the abstract concavity and support set of this class of functions. Next, we investigate the abstract concavity of extended valued topical functions , where and ⊤: = supK. Finally, as an application, we present characterizations of upward and downward sets by using extended valued elementary topical functions.

KEYWORDS:

• Abstract concavity; b-complete; downward set; elementary topical function; extended valued function; semifield; semimodule; topical function; support set; upward set

MATHEMATICS SUBJECT CLASSIFICATION:

• 06F07
• 26A48
• 26B25
• 52A01
• 06F20

Acknowledgments

The authors are very grateful to the anonymous referee for his/her useful suggestions on an earlier version of this article. The comments of the referee were very fruitful and these comments enabled the authors to improve the article significantly and, moreover, the connection between our results and the Hilbert’s projective metric and further research were mentioned by the referee.