Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebra in superextensions of groups, I: zeros and commutativity   Given a group $X$ we study the algebraic structure of its superextension
  $\lambda(X)$. This is a right-topological semigroup consisting of all maximal
linked systems on $X$ endowed with the operation $$\mathcal A\circ\mathcal
B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}$$ that extends
the group operation of $X$. We characterize right zeros of $\lambda(X)$ as
invariant maximal linked systems on $X$ and prove that $\lambda(X)$ has a right
zero if and only if each element of $X$ has odd order. On the other hand, the
semigroup $\lambda(X)$ contains a left zero if and only if it contains a zero
if and only if $X$ has odd order $|X|\le5$. The semigroup $\lambda(X)$ is
commutative if and only if $|X|\le4$. We finish the paper with a complete
description of the algebraic structure of the semigroups $\lambda(X)$ for all
groups $X$ of cardinality $|X|\le5$.

9
The categories of this paper are: nlin
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess