Introduction to Lattices and Order

Davey, B. A., & Priestley, H. A. (2002). Introduction to Lattices and Order. Cambridge University Press.

Type

Book

Intellectual base

Textbook

Year

2002

Pages cited

239-240

DOI

10.1017/cbo9780511809088

ISBN

9780521784511

DOI registrant

Cambridge University Press

Link

https://doi.org/10.1017/CBO9780511809088

Cited from

mathematics

Cited by

12 citations across 11 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Equivalence Relation
  • An equivalence relation is not the same as a partial order, a structural duality that Davey and Priestley (2002) develop as the canonical contrast between symmetric and antisymmetric reflexive-transitive relations.

    This sourceContrasts equivalence relations (reflexive, symmetric, transitive) with partial orders (reflexive, antisymmetric, transitive) and develops the preorder decomposition into an equivalence plus a partial order on the quotient.

• Order
  • ranking stability, treated comprehensively in Davey and Priestley's textbook (2002)

    This sourceStandard order-theory reference: contrasts equivalence relations (reflexive, symmetric, transitive) with partial orders (reflexive, antisymmetric, transitive) and develops the preorder decomposition into an equivalence and a partial order on the quotient.

Domain-specific

• Bounded complete poset
  • … observable transfers from A complete lattice is bounded complete because every subset has a supremum, while a bounded-complete poset need not supply suprema for unbounded subsets. to A domain-theoretic proof checks empty-set conventions and whether directed, finite or arbitrary bounded subsets are intended..
• Closed Preordered Set
  • This is the inductive hypothesis used by maximality and fixed-point principles such as Zorn-style and Bourbaki–Witt arguments.
• Completely distributive lattice
  • A completely distributive lattice satisfies that the meet over j of the join over k in K_j of x_jk equals the join over all choice functions f of the meet over j of x_j,f(j).
• Duality (order theory)
  • Dual posets, dual isomorphism and self-duality differ, and theorem dualization must consistently exchange all order-theoretic notions such as least and greatest, meet and join.
  • … that the poset and original order, reversed relation, dual notation, translation of definitions and theorem, order-reversing isomorphism if claimed and self-duality condition are explicit. to An applied instance preserves the same invariant under changed notation, scale, jurisdiction, dataset, or implementation..
• Inclusion (Boolean algebra)
  • … that Boolean-algebra operations and zero and one, chosen order orientation, equivalent meet, join, and complement equations, strictness, and representation as sets where invoked are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Join and meet
  • … bounds, leastness and greatestness, existence and uniqueness, join and meet notation, empty-subset conventions, binary versus arbitrary completeness, duality and algebraic laws are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Monotonic Function
  • The convention must therefore be stated rather than guessed from the title.
• Strong antichain
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Symmetric closure
  • … constitutive identity and a convenient observable transfers from A canonical example satisfies the full defining rule of Symmetric closure with all parameters and conventions explicit. to A careful use of Symmetric closure tests the constitutive rule and nearest confusable rather than relying on topical resemblance..

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