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
Received April 7, 2020
Revised April 23, 2020
Accepted May 11, 2020
Vasiliy Vladimirovich Chermnykh, Dr. Phys.-Math. Sci., Pitirim Sorokin Syktyvkar State University, Syktyvkar, 167001 Russia, e-mail: vv146@mail.ru
Oksana Vladimirovna Chermnykh, Cand. Sci. (Phis.-Math.), Vyatka State University, Kirov, 610000 Russia, e-mail: usr00458@vyatsu.ru
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Cite this article as: V.V. Chermnykh, O.V. Chermnykh. Functional representations of lattice-ordered semirings. III, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 235-248.
