Geodesic
On ideals in De Morgan residuated lattices
Kybernetika, Tome 54 (2018) no. 3, pp. 443-475
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In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, $\odot$-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.
In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, $\odot$-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.
DOI : 10.14736/kyb-2018-3-0443
Classification : 03B22, 03G05, 03G25, 06A06, 08A72
Keywords: residuated lattice; De Morgan laws; filter; deductive system; ideal; $\cap $-prime; $\cap $-irreducible; annihilator 
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Holdon, Liviu-Constantin. On ideals in De Morgan residuated lattices. Kybernetika, Tome 54 (2018) no. 3, pp. 443-475. doi: 10.14736/kyb-2018-3-0443

[1] Balbes, R., Dwinger, Ph.: Distributive Lattices. University of Missouri Press 1974. | MR

[2] Blyth, T. S.: Lattices and Ordered Algebraic Structures. Springer, London 2005. | DOI | MR

[3] Buşneag, C., Piciu, D.: The stable topologies for residuated lattices. Soft Computing 16 (2012), 1639-1655. | DOI

[4] Buşneag, D., Piciu, D., Paralescu, J.: Divisible and semi-divisible residuated lattices. Ann. Alexandru Ioan Cuza University-Mathematics (2013), 14-45. | MR

[5] Buşneag, D., Piciu, D., Holdon, L. C.: Some Properties of Ideals in Stonean residuated lattice. J. Multiple-Valued Logic Soft Computing 24 (2015), 5-6, 529-546. | MR

[6] Cignoli, R.: Free algebras in varieties of Stonean residuated lattices. Soft Computing 12 (2008), 315-320. | DOI

[7] Esteva, F., Godo, L.: Monoidal $t-$norm based logic: towards a logic for left-continnuos $t-$norms. Fuzzy Sets and Systems 124 (2001), 3, 271-288. | DOI | MR

[8] Lele, C., Nganou, J. B.: MV-algebras derived from ideals in BL-algebra. Fuzzy Sets and Systems 218 (2013), 103-113. | DOI | MR

[9] Maroof, F. G., Saeid, A. B., Eslami, E.: On co-annihilators in residuated lattices. J. Intelligent Fuzzy Systems 31 (2016), 1263-1270. | DOI

[10] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: an Algebraic Glimpse at Substructural Logics. Studies in Logics and the Foundations of Mathematics, Elsevier 2007. | MR

[11] Holdon, L. C., Niţu, L. M., Chiriac, G.: Distributive residuated lattices. Ann. University of Craiova-Mathematics and Computer Science Series 39 (2012), 100-109. | MR

[12] Mureşan, C.: Co-stone Residuated Lattices. Ann. University of Craiova-Mathematics and Computer Science Series 40 (2013), 52-75. | DOI | MR

[13] Iorgulescu, A.: Algebras of Logic as BCK-algebras. Academy of Economic Studies Bucharest, Romania 2008. | MR

[14] Leuştean, L.: Baer extensions of BL-algebras. J. Multiple-Valued Logic Soft Computing 12 (2006), 321-335. | MR

[15] Piciu, D.: Algebras of Fuzzy Logic. Editura Universitaria, Craiova 2007.

[16] Rachunek, J., Salounova, D.: Ideals and involutive filters in residuated lattices. In: SSAOS 2014, Stara Lesna.

[17] Turunen, E.: Mathematics Behind Fuzzy logic. Physica-Verlag Heidelberg, New York 1999. | MR

[18] Zou, Y. X., Xin, X. L., He, P. F.: On annihilators in BL-algebras. Open Math. 14 (2016), 324-337. | DOI | MR

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