![]() An extension of relative pseudocomplementation to non-distributive lattices. From a ≤ e ≤ f we have a ≤ f which implies Θ ≤ ′ Θ = Θ by (i), proving transitivity of ≤ ′. Then, by (ii) there exists some e ∈ Θ with a ≤ e and because of Θ = Θ ≤ ′ Θ some f ∈ Θ with e ≤ f. Finally, let c ∈ P and assume Θ ≤ ′ Θ and Θ ≤ ′ Θ. Therefore Θ = Θ = Θ which proves antisymmetry of ≤ ′. Since a ≤ c ≤ d, a, d ∈ Θ and ( Θ, ≤ ′ ) is convex we conclude c ∈ Θ. Because of Θ = Θ ≤ ′ Θ there exists some d ∈ Θ with c ≤ d. Then, by (ii), there exists some c ∈ Θ with a ≤ c. If, conversely, there exists some c ∈ Θ with a ≤ c, then according to (i) we have Θ ≤ ′ Θ = Θ. (ii) If Θ ≤ ′ Θ, then a ∗ b Θ 1 and hence a ≤ ( a ∗ b ) ∗ b ∈ Θ = Θ.This is the reason why we introduce the following property. Namely, our concept of a congruence on a strongly sectionally pseudocomplemented poset should respect also some aspects of the partial order relation. However, this condition is rather weak and we cannot expect to obtain a natural relationship between congruences and congruence kernels similar to that obtained for sectionally pseudocomplemented lattices in the previous section. Since a sectionally pseudocomplemented poset P has only one operation, namely ∗, a congruence on P should satisfy the substitution property with respect to ∗. It should be noted that there are sectionally pseudocomplemented posets which are not strongly sectionally pseudocomplemented, see, e.g., Chajda et al. This poset is not relatively pseudocomplemented since the relative pseudocomplement of c with respect to a does not exist. ∗ 0 a b c d e 1 0 1 1 1 1 1 1 1 a b 1 b 1 1 1 1 b c a 1 c 1 1 1 c b a b 1 1 1 1 d 0 a b c 1 e 1 e 0 a b c d 1 1 1 0 a b c d e 1 Sectionally pseudocomplemented lattices having 0 and their ideals will be the topic of one of our next studies. The situation with sectionally pseudocomplemented posets is a bit more complicated due to the fact that such a poset in general cannot be extended to a sectionally pseudocomplemented lattice by means of the Dedekind–MacNeille completion, see Chajda et al. ![]() ( 2012)) because sectionally pseudocomplemented lattices form a variety which is congruence permutable, congruence distributive and weakly regular. For lattices we can use the machinery of universal algebra (see, e.g., Chajda et al. In the present paper we focus on congruences and filters on sectionally pseudocomplemented lattices and posets. Later on, the concept of sectional pseudocomplementation was extended also to posets, see Chajda et al. This was realized by the first author in Chajda ( 2003) by introducing sectional pseudocomplementation. Because not every non-classical propositional calculus is necessarily distributive (for instance, the logic of quantum mechanics), it was a question whether the concept of relative pseudocomplementation can be extended in a reasonable way to non-distributive lattices. However, every relative pseudocomplemented lattice is distributive, see, e.g., Birkhoff ( 1979) and Lakser ( 1971). It was used in several branches of mathematics, e.g., as an algebraic axiomatization of intuitionistic logic (by Heyting and Brouwer) where the relative pseudocomplement is interpreted as the logical connective implication. Please help to improve this article by introducing more precise citations.The concept of a relative pseudocomplemented lattice was introduced by R. P.Dilworth (Dilworth ( 1939)). This article includes a list of general references, but it lacks sufficient corresponding inline citations. ![]()
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