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This article is cited in 2 scientific papers (total in 2 papers)
Multiplicatively idempotent semirings
E. M. Vechtomov, A. A. Petrov
Vyatka State University of Humanities, Vyatka, Russia
Abstract:
The article is devoted to the investigation of semirings with idempotent multiplication. General structure theorems for such semirings are proved. We focus on the study of the class $\mathfrak M$ of all commutative multiplicatively idempotent semirings. We obtain necessary conditions when semirings from $\mathfrak M$ are subdirectly irreducible. We consider some properties of the variety $\mathfrak M$. In particular, we show that $\mathfrak M$ is generated by two of its subvarieties, defined by the identities $3x=x$ and $3x=2x$. We explore the variety $\mathfrak N$ generated by two-element commutative multiplicatively idempotent semirings. It is proved that the lattice of all subvarieties of $\mathfrak N$ is a $16$-element Boolean lattice.
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Journal of Mathematical Sciences (New York), 2015, 206:6, 634–653
Bibliographic databases:
 
UDC:
512.558
Citation:
E. M. Vechtomov, A. A. Petrov, “Multiplicatively idempotent semirings”, Fundam. Prikl. Mat., 18:4 (2013), 41–70; J. Math. Sci., 206:6 (2015), 634–653
Citation in format AMSBIB
\Bibitem{VecPet13}
\by E.~M.~Vechtomov, A.~A.~Petrov
\paper Multiplicatively idempotent semirings
\jour Fundam. Prikl. Mat.
\yr 2013
\vol 18
\issue 4
\pages 41--70
\mathnet{http://mi.mathnet.ru/fpm1528}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431831}
\transl
\jour J. Math. Sci.
\yr 2015
\vol 206
\issue 6
\pages 634--653
\crossref{https://doi.org/10.1007/s10958-015-2340-6}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84956828248}
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http://mi.mathnet.ru/eng/fpm1528 http://mi.mathnet.ru/eng/fpm/v18/i4/p41
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This publication is cited in the following articles:
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E. M. Vechtomov, N. V. Shalaginova, “Semirings of continuous $(0,\infty]$-valued functions”, J. Math. Sci., 233:1 (2018), 28–41
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E. M. Vechtomov, A. A. Petrov, “Pseudocomplements in the lattice of subvarieties of a variety of multiplicatively idempotent semirings”, J. Math. Sci., 237:3 (2019), 410–419
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