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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{TIMM_2020_26_3_a19,
     author = {V. V. Chermnykh and O. V. Chermnykh},
     title = {Functional representations of lattice-ordered semirings. {III}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {235--248},
     year = {2020},
     volume = {26},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a19/}
}
TY  - JOUR
AU  - V. V. Chermnykh
AU  - O. V. Chermnykh
TI  - Functional representations of lattice-ordered semirings. III
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2020
SP  - 235
EP  - 248
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a19/
LA  - ru
ID  - TIMM_2020_26_3_a19
ER  - 
%0 Journal Article
%A V. V. Chermnykh
%A O. V. Chermnykh
%T Functional representations of lattice-ordered semirings. III
%J Trudy Instituta matematiki i mehaniki
%D 2020
%P 235-248
%V 26
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a19/
%G ru
%F TIMM_2020_26_3_a19
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/

[1] Rao P.R., “Lattice ordered semirings”, Math. Sem. Notes, Kobe Univ, 9 (1981), 119–149 | MR | Zbl

[2] Swamy K.L.N., “Dually residuated lattice ordered semigroups”, Math. Ann., 159 (1965), 105–114 | DOI | MR | Zbl

[3] Chermnykh O.V., “O $drl$-polugruppakh i $drl$-polukoltsakh”, Chebyshevskii sb., 14:4 (2016), 167–179 | DOI | MR

[4] Chermnykh V.V., Chermnykh O.V., “Funktsionalnye predstavleniya reshetochno uporyadochennykh polukolets”, Sib. elektron. mat. izvestiya, 14 (2017), 946–971 | DOI | MR | Zbl

[5] Chermnykh O.V., “Funktsionalnye predstavleniya reshetochno uporyadochennykh polukolets. II”, Sib. elektron. mat. izvestiya, 15 (2018), 677–684 | DOI | MR | Zbl

[6] Pierce R.S., “Modules over commutative regular rings”, Mem. Amer. Math. Soc., 70 (1967), 1–112 | DOI | MR | Zbl

[7] Koh K., “On a representation of strongly harmonic ring by sheaves”, Pacif. J. Math., 41:2 (1972), 459–468 | DOI | MR | Zbl

[8] Mulvey C.J., “A generalization of Gelfand duality”, J. Algebra, 56:2 (1979), 499–505 | DOI | MR | Zbl

[9] Filipoiu A., “Compact sheaves of lattices and normal lattices”, Math. Jap., 36:2 (1991), 381–386 | MR | Zbl

[10] Chermnykh V.V., “Predstavlenie polozhitelnykh polukolets secheniyami”, Uspekhi mat. nauk, 47:5 (1992), 180–182 | DOI | MR | Zbl

[11] Chermnykh V.V., “Funktsionalnye predstavleniya polukolets”, Fundam. i prikl. matematika, 17:3 (2012), 111–227 | DOI | MR | Zbl

[12] Mulvey C.J., “Compact ringed spaces”, J. Algebra, 52:2 (1978), 411–436 | DOI | MR | Zbl

[13] Simmons H., “Compact representations - the lattice theory of compact ringed spaces”, J. Algebra, 126 (1989), 493–531 | DOI | MR | Zbl

[14] Dauns J., Hofmann K.H., “The representation of biregular rings by sheaves”, Math. Z., 91 (1966), 103–123 | DOI | MR | Zbl

[15] Davey B.A., “Sheaf spaces and sheaves of universal algebras”, Math. Z., 134:4 (1973), 275–290 | DOI | MR | Zbl

[16] Birkhoff G., Lattice theory, Amer. Math. Soc., Providence, R.I., 1967, 418 pp. | MR | Zbl

[17] Miklin A.V., Chermnykh V.V., “O $drl$-polukoltsakh”, Mat. vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 16 (2014), 87–95