Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A combinatorial approach to the set-theoretic solutions of the
  Yang-Baxter equation   A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$
is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter
equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$
holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying
$r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There
exist close relations between the square-free set-theoretic solutions of YBE,
the semigroups of I-type, the semigroups of skew polynomial type, and the
Bieberbach groups, as it was first shown in a joint paper with Michel Van den
Bergh.
  In this paper we continue the study of square-free solutions $(X,r)$ and the
associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the
group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by
$X$ and with quadratic defining relations naturally arising and uniquely
determined by $r$. We study the properties of the associated Yang-Baxter
structures and prove a conjecture of the present author that the three notions:
a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a
semigroup of skew-polynomial type, are equivalent. This implies that the
Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra,
with respect to some appropriate ordering of $X$. We conjecture that every
square-free solution of YBE is retractable, in the sense of Etingof-Schedler.

22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess