Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Reducibility of $n$-ary semigroups: from quasitriviality towards
  idempotency   Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of
associative operations $F\colon X^n\to X$ satisfying the condition
$F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements
$x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are
said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be
idempotent. We show that $\mathcal{F}^n_1=\cdots
=\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ and
we give conditions on the set $X$ for the last inclusions to be strict. The
class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet,
who showed that its elements are reducible to binary associative operations.
However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper,
we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and
show that its elements are reducible. We give a full description of the
corresponding reductions and show how each of them is built from a quasitrivial
semigroup and an Abelian group whose exponent divides $n-1$.

31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
19
The categories of this paper are: eess
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex