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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A number of papers he wrote in the 1930s, culminating in his monograph, Lattice Theory (1940; the third edition remains in print), turned lattice theory into a major branch of abstract algebra. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra.
This is about lattice theory.For other similarly named results, see Birkhoff's theorem (disambiguation).. In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets.
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
Likewise, although Jon Cohen has some quibbles with the ordering and selection of topics (particularly the inclusion of congruences at the expense of a category-theoretic view of the subject), he concludes that the book is "a wonderful and accessible introduction to lattice theory, of equal interest to both computer scientists and mathematicians".
A relatively complemented lattice is a lattice. (def) 19. A heyting algebra is distributive. [5] 20. A totally ordered set is a distributive lattice. 21. A metric lattice is modular. [6] 22. A modular lattice is semi-modular. [7] 23. A projective lattice is modular. [8] 24. A projective lattice is geometric. (def) 25. A geometric lattice is ...
This notation may clash with other notation, as in the case of the lattice (N, |), i.e., the non-negative integers ordered by divisibility. In this locally finite lattice, the infimal element denoted "0" for the lattice theory is the number 1 in the set N and the supremal element denoted "1" for the lattice theory is the number 0 in the set N.
This page was last edited on 8 February 2021, at 12:18 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
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