Join-Completions of Ordered Structures Constantine Tsinakis Vanderbilt University October 29, 2010 Join-completions C. Tsinakis - slide #1 Abstract-Motivation The aim of this talk is to present a systematic study Abstract of join-extensions of ordered structures. Themes Join-Completions Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #2 Abstract-Motivation The aim of this talk is to present a systematic study Abstract of join-extensions of ordered structures. Themes Join-Completions H. Ono, Completions of Algebras and Completeness of Modal and Substructural Ordered Structures Logics, Advances in Modal Logics 4 (2002), Applications I 1- 20. Applications II H. Ono, Closure Operators and Complete Embeddings of Residuated Lattices, Studia Logica 74 (3) (2003), 427 - 440. Join-completions C. Tsinakis - slide #2 Featured Themes Abstract (1) Abstract treatment of join-extensions Themes Join-Completions Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #3 Featured Themes Abstract (1) Abstract treatment of join-extensions Themes Join-Completions Ordered Structures (2) The role of nuclei and co-nuclei in logic Applications I Applications II Join-completions C. Tsinakis - slide #3 Featured Themes Abstract (1) Abstract treatment of join-extensions Themes Join-Completions Ordered Structures (2) The role of nuclei and co-nuclei in logic Applications I Applications II (3) Interaction of residuals with join extensions Join-completions C. Tsinakis - slide #3 Featured Themes Abstract (1) Abstract treatment of join-extensions Themes Join-Completions Ordered Structures (2) The role of nuclei and co-nuclei in logic Applications I Applications II (3) Interaction of residuals with join extensions (4) Applications to important classes of ordered structures Join-completions C. Tsinakis - slide #3 Join-Completions Join-completions C. Tsinakis - slide #4 Join-Extensions and Join-Completions A poset Q is called be an extension of a poset P Abstract provided P ⊆ Q and the order of Q restricts to that Themes of P. In case every element of Q is a join (in Q) of Join-Completions Join-Extensions elements of P, we say that Q is a join-extension of Meets Lower Sets P and that P is join-dense in Q. We use the term Representation Abstract Descript. (1) join-completion for a join-extension that is a Abstract Descript. (2) Operators (1) complete lattice. The concepts of a meet-extension Operators (2) and a meet-completion are defined dually. Join-Completions (1) Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #5 Join-Extensions and Join-Completions A poset Q is called be an extension of a poset P Abstract provided P ⊆ Q and the order of Q restricts to that Themes of P. In case every element of Q is a join (in Q) of Join-Completions Join-Extensions elements of P, we say that Q is a join-extension of Meets Lower Sets P and that P is join-dense in Q. We use the term Representation Abstract Descript. (1) join-completion for a join-extension that is a Abstract Descript. (2) Operators (1) complete lattice. The concepts of a meet-extension Operators (2) and a meet-completion are defined dually. Join-Completions (1) Join-completions, introduced by B. Banaschewski, Ordered Structures are intimately related to representations of Applications I complete lattices studied systematically by J.R. Applications II Büchi. B. Banaschewski, Hüllensysteme und Erweiterungen von Quasi-Ordunungen, Z. math. Logik Grundl. Math. 2 (1956), 117 - 130. J. R. Büchi, Representations of complete lattices by sets, Portugaliae Math. 11 (1952), 151 - 167. Join-completions C. Tsinakis - slide #5 Preservation of Meets If Q is a join-extension of P, then the inclusion Abstract Themes map i : P → Q preserves all existing meets. Join-Completions P Q Join-Extensions Equivalently, if X ⊆ P and V X exists, then V X Meets P Q Lower Sets exists and V X = V X Representation Abstract Descript. (1) Abstract Descript. (2) Operators (1) Operators (2) Join-Completions (1) Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #6 Preservation of Meets If Q is a join-extension of P, then the inclusion Abstract Themes map i : P → Q preserves all existing meets. Join-Completions P Q Join-Extensions Equivalently, if X ⊆ P and V X exists, then V X Meets P Q Lower Sets exists and V X = V X Representation Abstract Descript. (1) Abstract Descript. (2) Dually, If Q is a meet-extension of P, then the Operators (1) Operators (2) inclusion map i : P → Q preserves all existing Join-Completions (1) joins. Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #6 Lower and Upper Sets A subset I of a poset P is said to be a lower set of Abstract P if whenever y ∈ P , x ∈ I, and y ≤ x, then y ∈ I. Themes Note that the empty set ∅ is a lower set. Join-Completions Join-Extensions A principal lower set is a lower set of the form Meets Lower Sets ↓ a = {x ∈ P | x ≤ a} (a ∈ P ). Representation Abstract Descript. (1) For A ⊆ P , Abstract Descript. (2) ↓ A = {x ∈ P | x ≤ a, for some a ∈ A} Operators (1) Operators (2) denotes the smallest lower set containing A. Join-Completions (1) The set L(P) of lower sets of P ordered by set Ordered Structures inclusion is a complete lattice; the join is the Applications I set-union, and the meet is the set-intersection. Applications II Join-completions C. Tsinakis - slide #7 Lower and Upper Sets A subset I of a poset P is said to be a lower set of Abstract P if whenever y ∈ P , x ∈ I, and y ≤ x, then y ∈ I. Themes Note that the empty set ∅ is a lower set. Join-Completions Join-Extensions A principal lower set is a lower set of the form Meets Lower Sets ↓ a = {x ∈ P | x ≤ a} (a ∈ P ). Representation Abstract Descript. (1) For A ⊆ P , Abstract Descript. (2) ↓ A = {x ∈ P | x ≤ a, for some a ∈ A} Operators (1) Operators (2) denotes the smallest lower set containing A. Join-Completions (1) The set L(P) of lower sets of P ordered by set Ordered Structures inclusion is a complete lattice; the join is the Applications I set-union, and the meet is the set-intersection. Applications II The upper sets of P are defined dually. Further, we write U(P) for the lattice of upper sets of P, ↑ a = {x ∈ P | a ≤ x} for a ∈ P , and ↑ A = {x ∈ P | a ≤ x, for some a ∈ A} for A ⊆ P . Join-completions C. Tsinakis - slide #7 Canonical Representation of Join-Extensions Each join-extension Q of a poset P can be Abstract identified with its canonical image Q˙ : Themes Q˙ = {↓ x ∩ P : x ∈ Q}. Join-Completions Join-Extensions ˙ Meets In particular, P can be identified with the poset P Lower Sets Representation of its principal lower sets: Abstract Descript. (1) ˙ Abstract Descript. (2) P = {↓ x : x ∈ P }. Operators (1) Operators (2) Join-Completions (1) Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #8 Canonical Representation of Join-Extensions Each join-extension Q of a poset P can be Abstract identified with its canonical image Q˙ : Themes Q˙ = {↓ x ∩ P : x ∈ Q}. Join-Completions Join-Extensions ˙ Meets In particular, P can be identified with the poset P Lower Sets Representation of its principal lower sets: Abstract Descript. (1) ˙ Abstract Descript. (2) P = {↓ x : x ∈ P }. Operators (1) Operators (2) The largest join-extension of P is L(P). Thus, for Join-Completions (1) any join-extension Q of P, we have Ordered Structures P˙ ⊆ Q˙ ⊆L(P). Applications I Applications II Join-completions C. Tsinakis - slide #8 Canonical Representation of Join-Extensions Each join-extension Q of a poset P can be Abstract identified with its canonical image Q˙ : Themes Q˙ = {↓ x ∩ P : x ∈ Q}. Join-Completions Join-Extensions ˙ Meets In particular, P can be identified with the poset P Lower Sets Representation of its principal lower sets: Abstract Descript. (1) ˙ Abstract Descript. (2) P = {↓ x : x ∈ P }. Operators (1) Operators (2) The largest join-extension of P is L(P). Thus, for Join-Completions (1) any join-extension Q of P, we have Ordered Structures P˙ ⊆ Q˙ ⊆L(P). Applications I The smallest join-completion of P, is the so called Applications II Dedekind-MacNeille completion N (P). Its canonical image consists of all lower sets that are intersections of principal lower sets. Lastly, we will have an occasion to consider the ideal completion I(P) of a join-semilattice P. Join-completions C. Tsinakis - slide #8 Abstract Description of Join-Extensions We often wish to look at the elements a join- Abstract extension of P as just elements – such as the ele- Themes ments of P itself – rather than certain lower sets of Join-Completions P Join-Extensions . Meets Lower Sets Representation Abstract Descript. (1) Abstract Descript. (2) Operators (1) Operators (2) Join-Completions (1) Ordered Structures Applications I Applications II Join-completions C. Tsinakis - slide #9 Abstract Description of Join-Extensions We often wish to look at the elements a join- Abstract extension of P as just elements – such as the ele- Themes ments of P itself – rather than certain lower sets of Join-Completions P Join-Extensions . Meets Lower Sets E.g., L(P) can be described abstractly as a Representation Abstract Descript. (1) bi-algebraic distributive lattice whose poset of Abstract Descript. (2) Operators (1) completely join-prime elements is isomorphic to P. Operators (2) N (P), has an equally satisfying abstract Join-Completions (1) description due to Banaschewski: it is the only join Ordered Structures and meet-completion of P .