Lattice Theory¶
Birkhoff, G. (1940). Lattice Theory.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Asymmetry
- The mathematical treatment isolates the swap-test as the defining diagnostic, as Birkhoff (1940) formalized in his treatment of order relations.
This sourceFoundational monograph on order and lattice theory: formalizes partial orders and their order relations (reflexivity, antisymmetry, transitivity), isolating the relational (swap-sensitive) structure that distinguishes orderings from symmetric/equivalence relations.
- The mathematical treatment isolates the swap-test as the defining diagnostic, as Birkhoff (1940) formalized in his treatment of order relations.
- Closure
- Algebraic closure of fields (\(\mathbb{C}\) as the algebraic closure of \(\mathbb{R}\); \(\bar{\mathbb{Q}}\) as the algebraic closure of \(\mathbb{Q}\)), affine and convex closure of subsets of vector spaces, transitive closure of binary relations, and saturation closures of model-theoretic structures are all closure-operator instances on the appropriate underlying lattice — a unifying lens that Birkhoff (1940) systematises in his treatment of closure operators on complete lattices.
This sourceFoundational lattice-theory monograph; among the first to exhibit the lattice structure of closure operators and to develop closure operators / closed-set systems on complete lattices — the framework unifying topological, algebraic, transitive, and convex closure.
- Algebraic closure of fields (\(\mathbb{C}\) as the algebraic closure of \(\mathbb{R}\); \(\bar{\mathbb{Q}}\) as the algebraic closure of \(\mathbb{Q}\)), affine and convex closure of subsets of vector spaces, transitive closure of binary relations, and saturation closures of model-theoretic structures are all closure-operator instances on the appropriate underlying lattice — a unifying lens that Birkhoff (1940) systematises in his treatment of closure operators on complete lattices.
- Completeness
- Birkhoff (1937) developed lattice theory and the notion of a complete lattice — a lattice in which arbitrary meets and joins exist — which became the canonical order-theoretic completeness notion.
This sourceAMS, Providence, RI. The first comprehensive treatment of lattice theory; defines the complete lattice (a lattice in which arbitrary meets and joins exist) as the canonical order-theoretic completeness notion and treats the MacNeille completion.
- Birkhoff (1937) developed lattice theory and the notion of a complete lattice — a lattice in which arbitrary meets and joins exist — which became the canonical order-theoretic completeness notion.
- Equivalence Relation
- Manufacturing and design use interchangeability classes — partitions of physical parts into equivalence classes whose members are functionally identical for the purposes of an assembly or repair context, an instance of the partition-lattice machinery Birkhoff (1940) develops in Lattice Theory.
This sourceDevelops the lattice of equivalence relations (equivalently, partitions) on a fixed carrier under the refinement order, with meet and join operations.
- Manufacturing and design use interchangeability classes — partitions of physical parts into equivalence classes whose members are functionally identical for the purposes of an assembly or repair context, an instance of the partition-lattice machinery Birkhoff (1940) develops in Lattice Theory.
- Order
- consolidated lattice theory as a discipline
This sourceFoundational lattice-theory monograph: develops the lattice of equivalence relations on a fixed carrier under the refinement order, establishing the partition-lattice machinery that underlies multi-criterion classification in mathematics, manufacturing, and data engineering.
- consolidated lattice theory as a discipline
- Relation
- Order relations
This sourceFoundational lattice-theory monograph: develops the lattice of equivalence relations on a fixed carrier under the refinement order, establishing the partition-lattice machinery that underlies multi-criterion classification in mathematics, manufacturing, and data engineering.
- Order relations
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