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Abstract

Recently, the notions of EIMTL-filters, associative filters, t-filters and I-filters were introduced in MTL-algebras and residuated lattices, respectively. In this paper, we consider these filters in residuated lattices and study some properties of them. We prove that t-filters and I-filters coincide in residuated lattices. The conditions under which an EIMTL-filter is a t-filter, and under which a residuated lattice has a proper associative filter are established. We also study the existence of a proper EIMTL-filter and prove that every maximal filter is an EIMTL-filter, and hence all residuated lattices have proper EIMTL-filters.

Keywords

Residuated lattice EIMTL-filter associative filter

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References

Zadeh

L.A.

, Is there a need for fuzzy logic?, Inform Sci 178 (2008), 2751–2779.

Ward

and Dilworth

R.M.

, Residuated lattice, Trans Amer Math Soc 45 (1939), 335–354.

Esteva

and Godo

, Monoidal t-norm-based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst 124 (2001), 271–288.

Turunen

, Mathematics behind fuzzy logic, Physica-Verlag, 1999.

Dvurečenskij

and Rachůnek

, Probabilistic averaging in bounded commutative residuated ℓ-monoids, Discrete Math 306 (2006), 1317–1326.

Liu

L.Z.

and Li

K.T.

, Boolean filters and positive implicative filters of residuated lattices, Inform Sci 177 (2007), 5725–5738.

Víta

, Fuzzy t-filters and their properties, Fuzzy Sets Syst 247 (2014), 127–134.

Z.M.

and Hu

B.Q.

, Characterizations and new subclasses of I-filters in residuated lattices, Fuzzy Sets Syst 247 (2014), 92–107.

Borzooei

R.A.

, Shoar

S.K.

and Ameri

, Some types of filters in MTL-algebras, Fuzzy Sets Syst 187 (2012), 92–102.

10.

Kondo

, Characterization of extended filters in residuated lattices, Soft Computing 18 (2014), 427–432.

11.

Ciungu

L.C.

, Classes of residuated lattices, Annals of University of Craiova, Math Comp Sci Ser 33 (2006), 189–207.

12.

Burris

and Sankappanavar

H.P.

, A course in universal algebravolume 78 of Graduate Texts in Mathematics, Springer-Verlag, 1981.

13.

Saeid

A.B.

and Zahiri

, Radicals in MTL-algebras, Fuzzy Sets Syst 236 (2014), 91–103.

14.

Cignoli

and Torrell

A.T.

, Glivenko like theorems in natural expansions of BCK-logic, Math Log Quart 50 (2004), 111–125.

15.

Liu

L.Z.

, States on finite monoidal t-norm based algebras, Inform Sci 181 (2011), 1369–1383.

16.

Liu

L.Z.

and Zhang

X.Y.

, States on IMTL-algebras, Soft Comput 15 (2011), 2021–2028.

17.

Haveshki

, A note on some types of filters in MTL-algebras, Fuzzy Sets Syst 247 (2014), 135–137.