On radiant sets, downward sets, topical functions and sub-topical functions in lattice ordered groups

Abstract

We extend some recent results on radiant and normal sets in

, increasing positively homogeneous and increasing co-radiant functions

plus-radiant and downward sets in

, topical and sub-topical functions

to the unifying framework of subsets of Aⁿ and functions f:Aⁿ →

where A=(A,≤, ⊗ ) is a conditionally complete lattice ordered group and

is the minimal enlargement of A. For results involving closedness, continuity and semi-continuity, we assume that the lattice A is continuous and we use the order topology on Aⁿ and A.

Keywords:

• Radiant set
• Downward set
• Topical function
• Sub-topical function
• Lattice ordered group
• Continuous lattice
• Abstract convexity
• Mathematics Subject Classification 2000: Primary: 26B25
• Secondary: 49N15
• 06F20

Acknowledgements

We wish to thank the referee for the careful reading of the manuscript and for valuable remarks which contributed to the improvement of this article.

Notes

We presented this article at the International Workshop on Max-Algebra, held in Birmingham, June 30–July 3, 2003.

E-mail: [email protected]

E-mail: [email protected]

Additional information

Notes on contributors

Ivan Singer †

We presented this article at the International Workshop on Max-Algebra, held in Birmingham, June 30–July 3, 2003. E-mail: [email protected]