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Functional representations of lattice-ordered semirings. (Russian. English summary) Zbl 1377.16042
Summary: The paper is devoted to lattice-ordered semirings ($$drl$$-semirings) and their representations by sections of sheaves. We build two sheaves of $$drl$$-semirings. The first sheaf construction is generalization of Keimel sheaf of $$l$$-rings, the second sheaf is analogy of Lambek sheaf of abstract semirings. The classes of Gelfand, Rickart, biregular and strongly regular $$f$$-semirings are investigated in this paper. The main aim is to study sheaf representations of such algebras.

##### MSC:
 16Y60 Semirings 06B75 Generalizations of lattices
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##### References:
 [1] G. Birkhoﬀ, Lattice Theory, Amer. Math. Soc., Providence, R.I., 1967. MR0227053 · Zbl 0153.02501 [2] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York- Paris, 1963. MR0171864 · Zbl 0137.02001 [3] F. Paoli, C. Tsinakis, On Birkhoﬀ ’s common abstraction problem, Studia Logica, 100 (2012), 1079-1105. MR3001048 · Zbl 1298.08002 [4] K. L. N. Swamy, Dually residuated lattice ordered semigroups, Math. Ann., 159 (1965), 105- 114. MR0183797 · Zbl 0135.04203 [5] K. L. N. Swamy, Dually residuated lattice ordered semigroups, II , Math. Ann., 160 (1965), 64-71. MR0191851 · Zbl 0138.02104 [6] N. A. Kovar, A general theory of dually residuated lattice-ordered monoids, Ph.D Thests, Palacky Univ., Olomouc., 1996. [7] T. Kovar, Two remarks on dually residuated lattice ordered semigroups, Math. Slovaka, 49 (1999), 17-18. MR1804468 · Zbl 0943.06007 [8] J. Kuhr, Representable dually residuated lattice-ordered monoids, Discuss. Math. General Alg. and Appl., 23 (2003), 115-123. MR2070377 · Zbl 1066.06008 [9] O. V. Chermnykh, On drl-semigroups and drl-semirings. Chebyshevskiy sbornik, 14:4 (2016), 167-179. · Zbl 1373.06018 [10] P. R. Rao, Lattice ordered semirings, Math. Sem. Notes, Kobe Univ., 9 (1981), 119-149. Zbl 0476.06018 · Zbl 0476.06018 [11] A. Grothendieck, J. Dieudonne, Elements de Geometrie Algebrique 1, I.H.E.S., Publ. Math. 4, Paris, 1960. MR0163908 [12] V. V. Chermnykh, Representation of positive semirings by sections, Uspehi matemat. nauk, 47:5 (1992), 193-194. MR1211839 · Zbl 0796.16039 [13] V. V. Chermnykh, Sheaf representations of semirings, Uspehi matemat. nauk, 48:5 (1993), 185-185. MR1258775 · Zbl 0823.16033 [14] V. V. Chermnykh, Functional representation of semirings, Fundament. i prikl. matemat., 17:3 (2012), 111-227. MR2954727 [15] E. M. Vechtomov, A. V. Cheraneva, Semiﬁlds and their properties, Fundament. i prikl. matemat., 14:5 (2008), 3-54. [16] K. Keimel, The representation of lattice ordered groups and rings by sections in sheaves, Lect. Notes Math., 248, Springer-Verlag, 1971. MR0422107 [17] J. Dauns, Representation of l-groups and f-rings, Paciﬁc J. Math., 31 (1969), 629-654. MR0255468 ФУНКЦИОНАЛЬНЫЕ ПРЕДСТАВЛЕНИЯ971 · Zbl 0192.36402 [18] K. Keimel, Representations of lattice-ordered rings, Proc Univ. of Houston. Lattice Theory Conf., (1973), 277-293. MR0401594 [19] J. Lambek, On the representation of modules by sheaves of factor modules, Can. Math. Bull., 41:3 (1971), 359-368. MR0313324 · Zbl 0217.34005 [20] J. S. Golan, Semirings and their applications, Kluwer Acad. Publ., Dordrecht, 1999. MR1746739 · Zbl 0947.16034 [21] B. A. Davey, Sheaf spaces and sheaves of universal algebras, Math. Z., 134:4 (1973), 275-290. MR0330006 · Zbl 0259.08002 [22] G. Bredon Sheaf theory, McGray-Hill, New York, 1967. MR0221500 [23] G. Birkhoﬀ, R. S. Pierce, Lattice – ordered rings, An. Acad. Brasil. Ci., 28 (1956), 41-69. MR0080099 · Zbl 0070.26602 [24] A. V. Miklin, V. V. Chermnykh, On drl-semirings, Matematicheskiy vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 16 (2014), 87-95. [25] General algebra 1
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