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Article

Pawar, Y. S.
Reticulation of a 0-distributive Lattice. (English). Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, vol. 54 (2015), issue 1, pp. 121-128
MSC: 06D99 | MR 3468605 | Zbl 1347.06015
Keywords:
0-distributive lattice; ideal; prime ideal; congruence relation; prime spectrum; minimal prime spectrum; maximal spectrum
Summary:
A congruence relation $\theta $ on a 0-distributive lattice is defined such that the quotient lattice $L/\theta $ is a distributive lattice and the prime spectrum of $L$ and of $L/\theta $ are homeomorphic. Also it is proved that the minimal prime spectrum (maximal spectrum) of $L$ is homeomorphic with the minimal prime spectrum (maximal spectrum) of $L/\theta $.
References:
[1] Belluce, L. P.: Semisimple Algebras of Infinite Valued Logic and Bold Fuzzy Set Theory. Can. J. Math. 38, 6 (1986), 1356–1379. DOI 10.4153/CJM-1986-069-0 | MR 0873417 | Zbl 0625.03009
[2] Belluce, L. P.: Spectral Spaces and Non-commutative Rings. Comm. Algebra 19 (1991), 1855–1865. DOI 10.1080/00927879108824234 | MR 1121110 | Zbl 0728.16002
[3] Balasubramani, P.: Stone Topology of The Set of Prime Ideals of a 0-distributive Lattice. Indian J. Pure Appl. Math. 35 (2004), 149–158. MR 2040729
[4] Dan, C. T.: Reticulation in Heyting Algebra. Annals of University of Craiova, Math. Comp. Sci. Ser. 30, 2 (2003), 66–70. MR 2064622
[5] Muresan, C.: The Reticulation of a Residuated Lattice. Bull. Math. Soc. Sci. Math. Roumanie 51 (99), 1 (2008), 47–65. MR 2396283 | Zbl 1164.06011
[6] Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices. W. H. Freeman, San Francisco, 1971. MR 0321817
[7] Kelley, J. L.: General Topology. Van Nostrand, New York, 1969. MR 0070144
[8] Leustean, L.: The Prime and Maximal Spectra and The Reticulation of BL-algebras. Central European Journal of Mathematics 1, 3 (2003), 382–397. DOI 10.2478/BF02475217 | MR 1992899 | Zbl 1039.03052
[9] Pawar, Y. S.: 0-1 distributive lattices. Indian J. Pure Appl. Math. 24 (1993), 173–179. MR 1210389 | Zbl 0765.06015
[10] Simmons, H.: Reticulated Rings. J. Algebra 66 (1980), 169–192. DOI 10.1016/0021-8693(80)90118-0 | MR 0591251 | Zbl 0462.13002
[11] Varlet, J.: A generalization of the notion of pseudo-complementedness. Bull. Soc. Liege 37 (1968), 149–158. MR 0228390 | Zbl 0162.03501
[12] Varlet, J.: On The Characterizations of Stone Lattices. Acta Sci. Math. (Szeged) 27 (1966), 81–84. MR 0194370
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