Geodesic
Cyclic semirings with nonidempotent noncommutative addition
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 6, pp. 17-41.

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The article discusses the structure of cyclic semirings with nonidempotent noncommutative addition. Also here are some general results about cyclic semirings with noncommutative addition. 
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E. M. Vechtomov; I. V. Orlova. Cyclic semirings with nonidempotent noncommutative addition. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 6, pp. 17-41. http://geodesic.mathdoc.fr/item/FPM_2015_20_6_a1/

[1] Bestuzhev A. S., “O stroenii konechnykh multiplikativno-tsiklicheskikh polukolets”, Yaroslav. ped. vestn., 3:2 (2013), 14–18

[2] Vechtomov E. M., Vvedenie v polukoltsa, Vyat. gos. ped. un-t, Kirov, 2000

[3] Vechtomov E. M., Lubyagina I. V., “Tsiklicheskie polukoltsa s idempotentnym nekommutativnym slozheniem”, Fundament. i prikl. matem., 17:1 (2011/2012), 33–52 | MR

[4] Brown T., Lazerson E., “On finitely generated idempotent semigroups”, Semigroup Forum, 78:1 (2009), 183–186 | DOI | MR | Zbl