Distributive lattices with annihilator properties are studied. Abstract characterizations of such lattices are obtained. In particular, new characterization theorems are proved for generalized Boolean lattices and Boolean lattices, as well as for generalized Stone lattices and Stone lattices. It is proved that point distributive lattices are isomorphic to point lattices of sets. Examples are given. Generalizations of the obtained results to one class of commutative multiplicatively idempotent semirings are presented.
Keywords: distributive lattice, ideal, annihilator, annihilator properties, Boolean lattice, Stone lattice, point lattice, semiring
Received April 30, 2024
Revised June 14, 2024
Accepted June 17, 2024
Published online: November 1, 2024
Funding Agency: This work was supported by the Russian Science Foundation within project no. 24-21-00117 “Semirings and Semimodules with Idempotency Conditions.”
Evgenii Mikhailovich Vechtomov, Dr. Phys.-Math. Sci., Prof., Vyatka State University, Kirov, 610000 Russia, e-mail: vecht@mail.ru
REFERENCES
1. Spead T.P. Some remark on a class of distributive lattices. J. Austral. Math. Soc., 1969, vol. 9, no. 3–4, pp. 289–296. https://doi.org/10.1017/S1446788700007205
2. Vechtomov E.M. Annihilator characterizations of Boolean rings and Boolean lattices. Math. Notes, 1993, vol. 53, no. 2, pp. 124–129. https://doi.org/10.1007/BF01208314
3. Birkhoff G. Lattice theory. Providence, R.I., Amer. Math. Soc., 1967, vol. 25, 418 p. Translated to Russian under the title Teoriya reshetok, Moscow, Nauka Publ., 1984, 568 p.
4. Gratzer G. General lattice theory. Berlin, Birkhäuser, 1978, 381 p. Translated to Russian under the title Obshchaya teoriya reshetok, Moscow, Mir Publ., 1982, 456 p.
5. Saliy V.N., Skornyakov L.A. Obshchaya algebra [General Algebra], Moscow, Nauka Publ., 1991, vol. 2, 480 p.
6. Sikorski R. Boolean algebras. New York, Academic Press, 1964, 237 p. Translated to Russian under the title Bulevy algebry, Moscow, Nauka Publ., 1969, 376 p.
7. Skornyakov L.A. Elementy teorii struktur. 2-ye izd. [Elements of the theory of structures. 2nd ed.] Moscow, Nauka Publ, 1982, 160 p.
8. Grätzer G. Lattice theory: Foundation. Basel, Birkhäuser, 2011, 614 p. https://doi.org/10.1007/978-3-0348-0018-1
9. Stone M.H. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 1936, vol. 40, no. 1, pp. 37–111. https://doi.org/10.1090/S0002-9947-1936-1501865-8
10. Vechtomov E.M., Petrov A.A. Multiplicatively idempotent semirings with annihilator condition. Iz. VUZ. Matematika, 2023, vol. 3, pp. 29–40 (in Russian). https://doi.org/10.26907/0021-3446-2023-3-29-40
11. Chermnykh V.V. Functional representations of semirings. J. Math. Sci., 2012, vol. 187, no. 2, pp. 187–267. https://doi.org/10.1007/s10958-012-1062-2
12. Stanley R.P. Enumerative combinatorics. Mathematical Gazette Publ., 1986, vol. 1, p. 306. Translated to Russian under the title Perechislitel’naya kombinatorika, Moscow, Mir Publ., 1990, 440 p. ISBN: 5-03-001348-2 .
13. Ghosh S. A characterization of semirings which are subdirect products of a distributive lattice and a ring. Semigroup Forum, 1999, vol. 59, no. 1, pp. 106–120. doi: 10.1007/PL00005999
14. Vechtomov E.M., Petrov A.A. Funktsional’naya algebra i polukol’tsa. Polukol’tsa s idempotentnym umnozheniyem [Functional algebra and semirings. Semirings with idempotent multiplication]. St. Petersburg, Lan’ Publ., 2023, 180 p. ISBN: 978-5-507-46239-1 .
15. Vechtomov E.M., Petrov A.A. Completely prime ideals in multiplicatively idempotent semirings. Math. Notes, 2022, vol. 111, no. 4, pp. 515–524. https://doi.org/10.1134/S0001434622030191
16. Gleason A. Projective topological spaces. Illinois J. Math., 1958, vol. 2, no. 4A, pp. 482–489. https://doi.org/10.1215/ijm/1255454110
Cite this article as: E.M. Vechtomov. Distributive lattices with different annihilator properties. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 53–65.
