Characterizations and new subclasses of -filters in residuated lattices
Introduction
Residuated lattices [31] constitute the semantics of Höhle's Monoidal Logic (ML) [16], which are the basis for the majority of formal fuzzy logics, like Esteva and Godo's Monoidal T-norm based Logic (MTL) [8], Hájek's Basic Logic (BL) [11], Łukasiewicz Logic (LL) [24], Intuitionistic Logic (IL) [15], and Gödel Logic (GL) [7]. Corresponding to these special logics, particular subalgebras of residuated lattices such as MTL-algebras, BL-algebras, MV-algebras, Heyting algebras, Gödel algebras, NM-algebras and -algebras were also proposed.
In studying these algebras and the completeness of the corresponding non-classical logics, filters play a vital role. A filter is also called a deductive system [30]. From logical point of view, filters correspond to sets of provable formulae. In order to investigate logical algebras by filters, the extension property of some types of filters was also widely investigated in [4], [12], [38]. At present, various filters and their characterizations under the context of different logical algebras have been studied [3], [4], [10], [12], [14], [20], [29], [30], [36], [37], [38]. Among these characterizations of special filters, some of them become invalid varying with the algebraic background. In BL-algebras [11], [12], [20], [30], prime filters, fantastic filters (called MV3-filters in [37]), implicative filters and Boolean filters were introduced. It was shown in [37] that Boolean filters coincide with implicative filters in -algebras. In [4], [20], fantastic filters, EIMTL-filters and IMTL-filters are proved to be equivalent in BL-algebras. When extended to MTL-algebras, they do not coincide and hence three kinds of filters arise [4]. However, in [4], associative filters were also defined, but in [14], this notion was proved to be useless. In [3], the relationships between the notions of various types of filters and radicals of filters in MTL-algebras were investigated. In [36], topological properties of prime filters in MTL-algebras were studied. Since both BL-algebras and MTL-algebras are residuated lattices, it is natural to define these filters in residuated lattices. At this time, prime filters and their alternative definitions [11], [29], [30], [36] do not coincide as well as fantastic filters and their characterizations [20], so prime filters of the second kind [10] and regular filters [38] were proposed.
It is remarkable that an abstract extension property of filters was defined in [10] which provides us with a uniform way to study quotients of logical algebras by filters. In [20], it is pointed out that several types of filters in BL-algebras could be formally defined not only by quasi-identity forms [12], [30], [38], but also by identity forms, then new characterizations with identity forms of implicative filters, fantastic filters and Boolean filters were given. At this time, the extension property is obvious. But we find that this identity form is too unspecific to reflect more common features of these filters. The aim of the present paper is to propose a residuum-based identity form which is specific and convenient to study the common features of these filters.
This paper is structured as follows: In Section 2, some basic properties, well-known subalgebras and special filters of residuated lattices are recalled. In Sections 3 and 4, -filters are introduced, which possess a more specific form than that in [20] and can be considered as a formal definition of several famous types of filters, and then a series of trivial characterizations of -filters, non-trivial characterizations of special classes of -filters such as implicative filters, regular filters, fantastic filters and Boolean filters, and some characterizations of special -algebras are obtained. In Section 5, three new types of -filters called divisible, strong and n-contractive filters are introduced. In particular, it is shown that a BL-algebra is n-fold implicative if and only if it is n-contractive. In Section 6, the relationships between these -filters are investigated. Particularly, it is pointed out that fantastic filters and divisible regular filters coincide.
Section snippets
Preliminaries
In this section, we recall some basic definitions and results, which will be frequently used in the following parts.
Definition 2.1 (See [11], [31].) A residuated lattice is an algebra such that for all is a bounded lattice, is a commutative monoid, forms an adjoint pair, i.e. iff .
We denote as ¬x and then get the following properties:
Lemma 2.2 (See [2].) Let be a residuated lattice. Then for all , ,
-filters and their characterizations
In this section, we will propose a formal definition and characterizations for several classes of filters.
Definition 3.1 Let F be a filter of . For all , if there exist terms of such that and , then F is called an -filter w.r.t. , and -filter for short.
-filter can be considered as a formal definition for some types of filters such as implicative filters, regular filters, fantastic filters, Boolean filters and prime filters of the third
Characterizations of subclasses of -filters
In this section, we mainly focus on the non-trivial characterizations of several subclasses of -filters.
New subclasses of -filters
Here, we will introduce three new subclasses of -filters in residuated lattices.
The relationships between particular -filters
In this section, the relationships between particular -filters in residuated lattices will be established.
As it has been proved in [38], Boolean filers are fantastic filters, and fantastic filters are regular filters.
Theorem 6.1 Let F be an implicative filter of . Then F is a divisible filter, but the converse does not always hold.
Proof Assume that F is an implicative filter. By Lemma 2.2(1), Theorem 4.1(2) and the isotonicity of the second variable of →, we have , that is, F is a
Conclusions
As it has been pointed out earlier, filters play a vital role in studying the logical systems. In this study, we proposed the concept of -filters to give a formal description for several types of filters by an identity including the operation “residuum”. By this means, we introduced the concept of divisible, strong and n-contractive filters and characterized them. At last, we established the relationships between these filters except n-contractive filters, because we left it as a further
Acknowledgements
The authors thank the anonymous reviewers and the Editor-in-Chief for their valuable suggestions in improving this paper. This research was supported by AMEP of Linyi University, the Natural Science Foundation of Shandong Province (Grant No. ZR2011FL017), the National Nature Science Foundation of China (Grant No. 61179038).
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