Geodesic
Cyclic semirings with idempotent noncommutative addition
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 1, pp. 33-52

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The article discusses the structure of cyclic semirings with noncommutative addition. In the infinite case, the addition is idempotent and is either left or right. Addition of a finite cyclic semirings can be either idempotent or nonidempotent. In the finite additively idempotent cyclic semiring, addition is reduced to the addition of cyclic subsemiring with commutative addition and absorbing element for multiplication and the addition of a cycle that is a finite semifield. 
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     author = {E. M. Vechtomov and I. V. Lubyagina},
     title = {Cyclic semirings with idempotent noncommutative addition},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_1_a2/}
}
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E. M. Vechtomov; I. V. Lubyagina. Cyclic semirings with idempotent noncommutative addition. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 1, pp. 33-52. http://geodesic.mathdoc.fr/item/FPM_2012_17_1_a2/