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Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.

In this paper, we provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without nontrivial nilpotent elements.

The invisibility graph I(X) of a set X ⊆ Rd is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X . We consider the following three parameters of a set X : the clique number ω(I(X)), the chromatic number χ(I(X)) and the inimum number γ(X) of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3. We also find sets X in R5 with χ(I(X)) = 2, but γ(X) arbitrarily large. Czech Science Foundation Ministry of Education, Youth and Sports of the Czech Republic European Science Foundation Országos Tudományos Kutatási Alapprogramok (OTKA) Centre Interfacultaire Bernoulli Swiss National Science Foundation