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\title[On distributive lattices of left $k$-Archimedean semirings]{\large On distributive lattices \\ of left $k$-Archimedean semirings}

\author[T. K. Mondal]{TAPAS KUMAR MONDAL}

\author[A. K. Bhuniya]{ANJAN KUMAR BHUNIYA}
\begin{abstract}
Here we introduce the notion of left $k$-Archimedean semirings which
generalize the notion of $k$-Archimedean semirings \cite{BM1}, and
characterize the semirings which are distributive lattices (chains)
of left $k$-Archimedean semirings.
 A semiring $S$ is a left $k$-Archimedean semiring if for all $a, b \in S, \ b \in \sqrt{Sa}$, the $k$-radical of $Sa$.
 A semiring $S$ is a distributive lattice of left $k$-Archimedean semirings if and only if for all $a, b \in S, \  ab \in \sqrt{Sa}$ and $S$ is a
 chain of left $k$-Archimedean semirings if and only if $\sqrt{L}$ is a completely prime $k$-ideal, for every left $k$-ideal $L$ of $S$.
\end{abstract}

\begin{amssubject}
16Y60.
\end{amssubject}

\begin{keyword}
$k$-radical, left $k$-Archimedean semiring, completely prime
$k$-ideal, semiprimary  $k$-ideal, congruence, distributive lattice
congruence.
\end{keyword}

\thanks{The authors thank the editor for communicating the paper and the referee for his careful reading 
of the manuscript and helpful suggestions.}

\maketitle

\let\thefootnote\relax\footnote{ \ }\hspace{-5mm}\let\thefootnote\relax\footnote{\hspace{-13pt}DOI: 10.24193/mathcluj.2020.2.07}




\begin{thebibliography}{99}

\bibitem {BM1} A. K. Bhuniya and T. K. Mondal, {\it Distributive lattice
decompositions of semirings with a semillatice additive reduct},
Semigroup Forum, {\bf 80} (2010), 293--301.

\bibitem {BM2015} A. K. Bhuniya and T. K. Mondal, {\it On the least distributive
lattice congruence on a semiring with a semilattice additive
reduct}, Acta Math. Hungar., {\bf 147} (2015), \textit{1}, 189--204.

\bibitem {CB1996}
M. \'Ciri\'c and S. Bogdanovi\'c, {\it Semilattice decompositions of
semigroups}, Semigroup Forum, {\bf 52} (1996), 119--132.

\bibitem {clifford}
A. H. Clifford, {\it Semigroups admitting relative inverses}, Ann. of Math., {\bf 42} (1941), 1037--1049.

\bibitem {HW1998}
U. Hebisch and H. J. Weinert, {\it Semirings: Algebraic Theory and
Applications in Computer Science}, World Scientific, Singapore,
1998.

\bibitem {howie}
J. M. Howie, {\it Fundamentals in Semigroup Theory}, Clarendon Press, 1995.

\bibitem {kmet}
F. Kmet, {\it Radicals and their left ideal analogues in a
semigroup}, Math. Slovaca, {\bf 38} (1988), 139--145.

\bibitem {putcha}
M. S. Putcha, {\it Semilattice decomposition of semigroups},
Semigroup Forum, {\bf 6} (1973), 12--34.

\end{thebibliography}

\vspace{10pt}

\hspace{-4mm}{\small{Received May 5, 2019}}\\
\hspace{-4mm}{\small{Accepted November 3, 2019}}

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\ \hfill
\begin{tabular}{c}
{\small\em Dr. Bhupendra Nath Dutta Smriti}\\ 
{\small\em Mahavidyalaya}\\
{\small\em Department of Mathematics}\\
{\small\em Hagobindapur - 713407, Purba Bardhaman}\\
{\small\em West Bengal, India}\\ 
{\small\em E-mail: {\tt tapumondal@gmail.com}} \\ \\
{\small\em Visva Bharati University}\\
{\small\em Department of Mathematics}\\
{\small\em Santiniketan - 731235, Birbhum}\\
{\small\em West Bengal, India}\\
{\small\em E-mail: {\tt anjankbhuniya@gmail.com}}
\end{tabular}

\end{document}
