Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note on Semigroup Algebras of Permutable Semigroups   Let $S$ be a semigroup and $\mathbb F$ be a field. For an ideal $J$ of the
semigroup algebra ${\mathbb F}[S]$ of $S$ over $\mathbb F$, let $\varrho _J$
denote the restriction (to $S$) of the congruence on ${\mathbb F}[S]$ defined
by the ideal $J$. A semigroup $S$ is called a permutable semigroup if $\alpha
\circ \beta =\beta \circ \alpha$ is satisfied for all congruences $\alpha$ and
$\beta$ of $S$. In this paper we show that if $S$ is a semilattice or a
rectangular band then $\varphi _{\{S;{\mathbb F}\}}:\ J\mapsto \varrho _J$ is a
homomorphism of the semigroup $(Con ({\mathbb F}[S]);\circ )$ into the
relations semigroup $({\cal B}_S; \circ )$ if and only if $S$ is a permutable
semigroup.

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The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
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The categories of this paper are: eess