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A positive implicative filter of L is a subset F of L such that• 1 ∈ F,• for all x, y and z in F: if (x∗y) ⇒ z ∈ F and x ⇒ y ∈ F, then x ⇒ z ∈ F.An alternative definition for a filter F (see e.g. [23]) of a residuated latticeL = (L, ⊓, ⊔, ∗, ⇒,0, 1) is the following:• 1 ∈ F,• for all x and y in L: if x ∈ F and x ⇒ y ∈ F, then y ∈ F.An alternative definition for a Boolean filter F of a residuated lattice L =(L, ⊓, ⊔, ∗, ⇒,0, 1) is the following: F is a filter of L satisfying• for all x, y and z in L: if (x∗¬z) ⇒ y ∈ F and y ⇒ z ∈ F, then x ⇒ z ∈ F.This was proven in [17], where these filters were called implicative filters. Itwas also shown in [17] that a Boolean filter is always a positive implicativefilter and that also the converse is true in residuated lattices with an involutivenegation ¬. A positive implicative filter of a residuated lattice is always a filterof that residuated lattice. [17]2 Filters of residuated lattices2.1 Finite filters and finite prime filters of the second kindIf L = (L, ⊓, ⊔, ∗, ⇒, 0, 1) is a residuated lattice and l ∈ L such that l ∗ l = l,then it is not difficult to see that F l := {x ∈ L|l ≤ x} is a filter of L. In fact,finite filters of residuated lattices are always of this form.Proposition 3 Let L = (L, ⊓, ⊔, ∗, ⇒, 0, 1) be a residuated lattice and F afinite filter of L. Then F = F l for some idempotent l in L.PROOF. If F is a finite filter of L, then the product m of all its elements(a finite number) must also be an element of F. So m ∗ m ∈ F and thereforem (as the product of all elements of F) is smaller than or equal to m ∗ m(because of Proposition 1(1)), which implies m ∗ m = m. ✷Remark that this property does not hold for infinite filters. For example, ]0, 1]is a filter of ([0, 1], min, max, min, ⇒ min , 0, 1), but ]0, 1] is not of the form F lfor any l ∈ [0, 1].4
Finite prime filters of the second kind of residuated lattices are always of theform F l , with l idempotent and ⊔-irreducible. Conversely, filters (even infiniteones) of this form will always be prime filters of the second kind.Proposition 4 Let L = (L, ⊓, ⊔, ∗, ⇒, 0, 1) be a residuated lattice. If l is anidempotent and ⊔-irreducible element of L, then F l is a prime filter of thesecond kind of L.PROOF. Suppose a⊔b ∈ F l . Then l ≤ a⊔b and therefore, using Proposition1(13), l = l ∗ l ≤ l ∗ (a ⊔ b) = (l ∗ a) ⊔ (l ∗ b) ≤ l. So l = (l ∗ a) ⊔ (l ∗ b), whichimplies l = l ∗a or l = l ∗b. So l ≤ a or l ≤ b, which means exactly that a ∈ F lor b ∈ F l . ✷2.2 Boolean filters of the second kind and prime filters of the third kindSome particular kinds of filters appear quite frequently in our investigation ofthe connections between different kinds of filters. Therefore we gave them aname and introduce them in the next definition.Definition 5 A Boolean filter of the second kind (BF2) of a residuated latticeL = (L, ⊓, ⊔, ∗, ⇒,0, 1) is a filter of L satisfying• for all x in L: x ∈ F or ¬x ∈ F (or both).A prime filter of the third kind (PF3) of a residuated lattice L = (L, ⊓, ⊔, ∗, ⇒,0, 1)is a filter of L satisfying• for all x and y in L: (x ⇒ y) ⊔ (y ⇒ x) ∈ F.It can easily be seen there is always a trivial example of each kind of filterdefined before, namely L; and that every filter always contains the greatestelement 1.Definition 6 Let S be a set and C be a subset of its powerset P(S).We say C satisfies the intersection property iff C is closed under intersections.We say C satisfies the monotonicity property iff C is closed under supersets,i.e., iff S 2 ∈ C whenever S 1 ∈ C and S 1 ⊆ S 2 ⊆ S.Straightforward to verify is that filters, prime filters of the third kind andBoolean filters satisfy the intersection property. We will see later that it alsoholds for positive implicative filters. For PF, PF2 and BF2 the intersectionproperty does not hold, as will be shown in some of the examples.From the definitions it is also clear that filters, prime filters, prime filters of5
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Page 3: (10) (x ⊔ y) ⇒ z = (x ⇒ z)
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Page 9 and 10: 1vabuFig. 5. In some of the residua
Page 11 and 12: • |B| = 1. Say, B = {b}. Then for
Page 13 and 14: PROOF. Suppose F is a positive impl
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Page 19 and 20: (F.2) if x,y ∈ F, then x ∗ y
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Page 23 and 24: BIF/BIF2IBF2/PIFIPFIBF/PIF3IPF2/PIF
Page 25 and 26: These two conditions are also suffi
Page 27 and 28: 5 Conclusion and future workIn this
Page 29: [25] B. Van Gasse, C. Cornelis, G.

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