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Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring. Equivalently, a commutative parasemifield $P$ finitely generated as a semiring is additively divisible if and only if $P$ is additively idempotent. We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way. Let $S$ be a semiring that is a factor of a monoid semiring $\mathbb{N}[\mathcal{C}]$ where $\mathcal{C}$ is a submonoid of a free commutative monoid of finite rank. Then the semiring $S$ is additively almost-divisible if and only if $S$ is torsion. In particular, we show that if $S$ is a ring then $S$ cannot contain any non-finitely generated subring of $\mathbb{Q}$.

The paper is devoted to projective Clifford groups of quantum $N$-dimensional systems. Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann-Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that -- in $N$-dimensional quantum mechanics -- the Clifford group is a natural semidirect product provided the dimension $N$ is an odd number. For even $N$ special results on the Clifford groups are scattered in the mathematical literature, but they don't concern the semidirect structure. Using appropriate group presentation of $SL(2,Z_N)$ it is proved that for even $N$ projective Clifford groups are not natural semidirect products if and only if $N$ is divisible by four., Comment: 28 pages

Let $S$ be an additively idempotent semiring and $\mathbf{M}_n(S)$ be the semiring of all $n\times n$ matrices over $S$. We characterize the conditions of when the semiring $\mathbf{M}_n(S)$ is congruence-simple provided that the semiring $S$ is either commutative or finite. We also give a characterization of when the semiring $\mathbf{M}_n(S)$ is subdirectly irreducible for $S$ beeing almost integral (i.e., $xy+yx+x=x$ for all $x,y\in S$). In particular, we provide this characterization for the semirings $S$ derived from the pseudo MV-algebras.

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring $\mathbf{M}_n(S)$, of all $n\times n$ matrices over a semiring $S$, is congruence-simple, provided that either $S$ has a multiplicatively absorbing element or $S$ is commutative and additively cancellative., Comment: 11 pages

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.

We provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without non-trivial nilpotent elements.

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality., Comment: 16 pages

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ 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 $\omega(I(X))$, the chromatic number $\chi(I(X))$ and the convexity number $\gamma(X)$, which is the minimum number of convex subsets of $X$ that cover $X$. We settle a conjecture of Matou\v{s}ek and Valtr claiming that for every planar set $X$, $\gamma(X)$ can be bounded in terms of $\chi(I(X))$. As a part of the proof we show that a disc with $n$ one-point holes near its boundary has $\chi(I(X)) \ge \log\log(n)$ but $\omega(I(X))=3$. We also find sets $X$ in $\mathbb{R}^5$ with $\chi(X)=2$, but $\gamma(X)$ arbitrarily large., Comment: 28 pages, 9 figures; minor changes, added a few definitions and two references regarding metabelian groups

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively divisible semiring posseses no unit, it must contain an ideal of idempotent elements. We also present a series of open questions about finitely generated commutative semirings and their equivalent versions., Comment: 8 pages

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known for quadratic polynomials. We provide it for cubic polynomials and improve known bounds of this value for a general polynomial., Comment: 10 pages