Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coproduct Cancellation on \textbf{Act}-$S$   The themes of cancellation, internal cancellation, substitution have led to a
lot of interesting research in the theory of modules over commutative and
noncommutative rings. In this paper, we introduce and study cancellation
problem in the theory of acts over monoids. We show that if $A$ is an $S$-act
and $A={\dot\bigcup_{i\in I}}A_i$ is the unique decomposition of $A$ into
indecomposable subacts $A_i, i\in I$ such that the set $P=\{{\rm Card} [i] \mid
i\in I\}$ is finite, then $A$ is cancellable if and only if the equivalence
class $[i]=\{j\in I \mid A_i\cong A_j\}$ is finite, for every $i\in I$.
Likewise, we prove that every $S$-act is cancellable if and only if it is
internally cancellable. Thus, the concepts cancellation and internal
cancellation coincide here.

31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat