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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Modular and complemented lattices are of special interest for applications in such diverse fields as probability theory, including ergodic theory and multiplicative processes, linear algebra, computer science, and several engineering disciplines. For example, the set L of all subspaces of ℝn is a complemented modular lattice. Here the orthogonal complement S⊥ of any subspace S satisfies S⊥∩S=∅ and S⊥∪S=ℝn. Also, by definition, a Boolean lattice is a complemented distributive lattice. The uses of Boolean algebras in computer science and engineering are manifold and range from the design of electrical networks to the theory of computing. Because of its importance in computer science and engineering we summarize the properties discussed above that define a Boolean lattice.
Fragments of quasi-Nelson: residuation
Published in Journal of Applied Non-Classical Logics, 2023
Another paper by the same author (Sendlewski, 1991) showed that, even if is not quite a Nelson algebra but a subreduct thereof (one that lacks, for instance, the implication connective), it may still be possible to represent as a twist-algebra over a subreduct of a Heyting algebra; in the case studied in Sendlewski (1991), the algebra was a pseudo-complemented distributive lattice (i.e. the -subreduct of a Heyting algebra). We shall see that, as one considers weaker and weaker fragments of Nelson logic (corresponding to poorer algebraic languages), establishing a twist representation becomes a harder puzzle, until one reaches a point where the very mechanism of the twist construction seems to break down (regarding this, see the second research direction mentioned in the concluding Section 10).
Generalized convex combination of triangular norms on bounded lattices
Published in International Journal of General Systems, 2020
Funda Karaçal, M. Nesibe Kesicioğlu, Ümit Ertuğrul
Let be a complemented lattice, be the linear combination of such that are t-norms, S is a t-conorm with n is a negation on L and . Consider the set defined by is a bounded partially ordered set. iff a is an idempotent element of T..
Triangular norm decompositions through methods using congruence relations
Published in International Journal of General Systems, 2022
Funda Karaçal, Samet Arpacı, Kübra Karacair
In the following proposition, we show that if L is a Boolean algebra, then is simple on L. But any t-norm on a Boolean algebra may not be simple. Indeed, let L be Boolean algebra and Then, T is directly decomposable with Theorem 4.10 since is ∨-distributive and all elements are complemented. For example, we consider the t-norm on any Boolean algebra , then it is easily shown that is not simple on L.