

First name: Singer
Last name: Ivan
E-mail: ivan.singer@imar.ro

Title and abstract of your presentation
(Maximum 5000 characters)
 \section MS7 {Your title here} Max-min convexity
By {\sl names of all authors here} Ivan Singer

\noindent
Insert your abstract here

The max-min semifield is the set $\overline{R}=R\cup \{-\infty ,+\infty \}$
endowed with the operations $\oplus =\max ,\otimes =\min $. We study the
semimodule $\overline{R}^{n}=\overline{R}\times ...\times \overline{R}$ ($n$
times), with the operations $\oplus $ and $\otimes $ defined componentwise.

A subset $G$ of $\overline{R}^{n}$ (respectively, a function $f:\overline{R}%
^{n}\rightarrow \overline{R}$) is said to be max-min convex if the relations
$x,y\in G$ (respectively, $x,y\in \overline{R}^{n}$) and $\alpha ,\beta \in
\overline{R}$, $\alpha \oplus \beta =+\infty $, where $+\infty $ is the
neutral element for $\otimes =\min $, imply $(\alpha \otimes x)\oplus (\beta
\otimes y)\in G$ (respectively, $f((\alpha \otimes x)\oplus (\beta \otimes
y))\leq (\alpha \otimes f(x))\oplus (\beta \otimes f(y)$).

We give some results on max-min convexity of sets and functions in $%
\overline{R}^{n}$ (e.g. on segments, semispaces, separation, multi-order
convexity, ...) that correspond to some results for max-plus convexity,
replacing $\otimes =+$ of the max-plus case by the semi-group operation $%
\otimes =\min $ of the max-min case.

References
K. Zimmermann, Convexity in semimodules. Ekonom.-Mat.
Obzor 17 (1981), 199-213.
V. Nitica and I. Singer, Contributions to max-min convex geometry. I:
Segments. Lin. Alg. Appl. 428 (2008), 1439-1459. II: Semispaces and convex
sets. Ibidem 2085-2115.




Keywords: Max-min convex set; Max-min convex function

AMS primary classification: 08A72
