Geodesic
Functional representations of lattice-ordered semirings. III
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 235-248

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Lattice-ordered semirings ($drl$-semirings) are considered. Compact sheaves of $drl$-semirings are defined and their characterization is obtained. The properties of compact sheaves are studied; in particular, the structure of irreducible and maximal $l$-ideals in the $drl$-semiring of sections of a compact sheaf is described. A compact sheaf of functional semirings ($f$-semirings) is described in terms of a continuous mapping of the irreducible (and maximal) spectrum of this sheaf onto a compact Hausdorff space. The paper also contains a proof that an $f$-semiring is Gelfand if and only if it is isomorphic to the semiring of all sections of a compact sheaf of $f$-semirings with a unique maximal ideal.
Keywords: lattice-ordered semiring, functional semiring, compact sheaf, Gelfand $f$-semiring. 
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V. V. Chermnykh; O. V. Chermnykh. Functional representations of lattice-ordered semirings. III. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 235-248. http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a19/