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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.
Structure
Published in Michael H. Albert, Richard J. Nowakowski, David Wolfe, Lessons in Play, 2019
Michael H. Albert, Richard J. Nowakowski, David Wolfe
Definition 6.22. A distributive lattice is a lattice in which a meet distributes over a join (or, equivalently, a join distributes over a meet). That is, a∧(b∨c)=(a∧b)∨(a∧c).
Lattices and Boolean Algebra
Published in B. V. Senthil Kumar, Hemen Dutta, Discrete Mathematical Structures, 2019
B. V. Senthil Kumar, Hemen Dutta
Let 〈L, ∧, ∨〉 be a distributive lattice and a, b, c ∈ L. If a ∧ b = a ∧ c and a ∨ b = a ∨ c, then b = c. or
A variety of algebras closely related to subordination algebras
Published in Journal of Applied Non-Classical Logics, 2022
Note that if is a pseudo-subordination algebra, then its reduct is a bounded distributive lattice with an implication according to Cabrer and Celani (2006, Def. 2.1), and if is a bounded distributive lattice with an implication whose lattice reduct is a Boolean lattice, then a pseudo-subordination algebra is obtained by displaying the complement operation. Moreover, as we prove in Proposition 4.15, the bounded distributive lattices with an implication are the subalgebras of the reducts to the language of the pseudo-subordination algebras.
A family of genuine and non-algebraisable C-systems
Published in Journal of Applied Non-Classical Logics, 2021
Mauricio Osorio, Aldo Figallo-Orellano, Miguel Pérez-Gaspar
Now, recall that an algebra is said to be a Heyting algebra if the reduct is a bounded distributive lattice and the condition iff (*) holds. Besides, an algebra is said to be generalised Heyting algebra if the reduct is a distributive lattice and * is verified.