Join-Completions of Ordered Structures

Home , Semilattice

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

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

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

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

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

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 . 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

Join-completions C. Tsinakis - slide #9 Abstract Description of Join-Extensions

Note that the inclusion map P P Abstract i : →N ( ) Themes preserves all existing joins and meets. The Join-Completions Crawley completion C(P) – with canonical image Join-Extensions Meets consisting of all complete lower sets of P, that is, Lower Sets Representation lower sets that are closed with respect to any Abstract Descript. (1) Abstract Descript. (2) existing joins of their elements – is the largest Operators (1) Operators (2) join-completion with this property. Join-Completions (1)

Ordered Structures

Applications I

Applications II

Join-completions C. Tsinakis - slide #10 Closure Operators and Closure Retractions

Recall that a closure operator on a poset P is a Abstract Themes map γ : P → P with the usual properties of being Join-Completions order-preserving (x ≤ y ⇒ γ(x) ≤ γ(y)), enlarging Join-Extensions Meets (x ≤ γ(x)), and idempotent (γ(γ(x)) = γ(x)). It is Lower Sets Representation completely determined by its image Pγ by virtue of Abstract Descript. (1) Abstract Descript. (2) the formula Operators (1) Operators (2) γ(x) = min{c ∈ Pγ : x ≤ c}. Join-Completions (1)

Ordered Structures

Applications I

Applications II

Join-completions C. Tsinakis - slide #11 Closure Operators and Closure Retractions

An opposite direction also holds: Let us call a Ordered Structures closure retraction of P, a subposet C of P that Applications I satisﬁes: Applications II min{a ∈ C : x ≤ a} exists for all x ∈ P .

Join-completions C. Tsinakis - slide #11 Closure Operators and Closure Retractions

Abstract Bijective Correspondence Themes Join-Completions ⋆ If C is a closure retraction of P, then γC : P → P Join-Extensions Meets deﬁned by γC(x) = min{c ∈ C : x ≤ c} is a Lower Sets Representation closure operator. Abstract Descript. (1) Abstract Descript. (2) ⋆ Conversely, if γ : P → P is a closure operator, Operators (1) Operators (2) then Pγ = γ[P] is a closure retraction of P . Join-Completions (1) ⋆ Moreover, P = C and γ = γ. γC Pγ Ordered Structures

Applications I

Applications II

Join-completions C. Tsinakis - slide #12 Closure Operators and Closure Retractions

Applications I If Q is a join-completion of P, then Q is a closure Applications II retraction of L(P) and of ℘(P ). In what follows, we

write γQ for the associated closure operator on

L(P) and δQ for the one on ℘(P ).

Join-completions C. Tsinakis - slide #12 Back to Join-Completions

Abstract P"!#! Themes

δ "# Join-Completions Join-Extensions Meets Lower Sets "# Representation Abstract Descript. (1) !!! Abstract Descript. (2) !!!!! δ L"!#! !# Operators (1)

γ "# Operators (2) ! Join-Completions (1) L"!#! ! Ordered Structures

Applications I

Applications II

Join-completions C. Tsinakis - slide #13 Back to Join-Completions

Abstract P"!#! Themes

δ "# Join-Completions Join-Extensions Meets Lower Sets "# Representation Abstract Descript. (1) !!! Abstract Descript. (2) !!!!! δ L"!#! !# Operators (1)

γ "# Operators (2) ! Join-Completions (1) L"!#! ! Ordered Structures

Applications I

Applications II

There is a bijective correspondence between join-completions of P P and closure operators γ on L( ) with P ⊆L(P )γ

Join-completions C. Tsinakis - slide #13 Back to Join-Completions

Abstract P"!#! Themes

δ "# Join-Completions Join-Extensions Meets Lower Sets "# Representation Abstract Descript. (1) !!! Abstract Descript. (2) !!!!! δ L"!#! !# Operators (1)

γ "# Operators (2) ! Join-Completions (1) L"!#! ! Ordered Structures

Applications I

Applications II

There is a bijective correspondence between join-completions of P P and closure operators γ on L( ) with P ⊆L(P )γ

There is a bijective correspondence between join-completions of P

and closure operators δ on ℘(P ) with P ⊆ ℘(P )δ ⊆L(P ).

Join-completions C. Tsinakis - slide #13 Join-Extensions of Ordered Structures

Join-completions C. Tsinakis - slide #14 Residuals in Partially Ordered Monoids

Given a partially ordered monoid – pom for short – Abstract Themes P, the following question arises: Which Join-Completions join-completions of P are residuated lattices with Ordered Structures respect to a, necessarily unique, multiplication that Poms (1) Poms (2) extends the multiplication of P? Poms (3) Nuclei & Subacts Main Theorem Lemma

Applications I

Applications II

Join-completions C. Tsinakis - slide #15 Residuals in Partially Ordered Monoids

x\z = max{y ∈ P : xy ≤ z}, and Applications I

z/x = max{y ∈ P : yx ≤ z}, Applications II

Join-completions C. Tsinakis - slide #15 Residuals in Partially Ordered Monoids

x\z = max{y ∈ P : xy ≤ z}, and Applications I

z/x = max{y ∈ P : yx ≤ z}, Applications II whenever these maxima exist.

Join-completions C. Tsinakis - slide #15 Residuals in Partially Ordered Monoids

x\z = max{y ∈ P : xy ≤ z}, and Applications I

z/x = max{y ∈ P : yx ≤ z}, Applications II whenever these maxima exist. x\z is read as “x under z” z/x is read as “z over x”

Join-completions C. Tsinakis - slide #15 Residuals in Partially Ordered Monoids

Applications I

Applications II

Join-completions C. Tsinakis - slide #16 Residuals in Partially Ordered Monoids

A residuated partially ordered monoid – residuated Abstract Themes pom – P is one in which all quotients x\z and z/x Join-Completions exist. In particular, the binary operations \ and / Ordered Structures are defined everywhere on P . Poms (1) Poms (2) Alternatively, a residuated pom P is one in which Poms (3) Nuclei & Subacts the binary operation · is residuated. This means Main Theorem Lemma that there exist binary operations \ and / on P such that for all x,y,z ∈ P , Applications I Applications II xy ≤ z iff x ≤ z/y iff y ≤ x\z.

Join-completions C. Tsinakis - slide #16 Residuals in Partially Ordered Monoids

Join-completions C. Tsinakis - slide #16 Residuals in Partially Ordered Monoids Let P be a monoid. Then ℘(P) = h℘(P ), ∩, ∪, ·, \, /, {1}i is a residuated Abstract Themes lattice where: Join-Completions

X · Y = {x · y | x ∈ X,y ∈ Y }, Ordered Structures Poms (1) X\Y = {z | X · {z}⊆ Y }, and Poms (2) Poms (3) Nuclei & Subacts Y/X = {z | {z} · X ⊆ Y }. Main Theorem Lemma

Applications I

Applications II

Join-completions C. Tsinakis - slide #17 Residuals in Partially Ordered Monoids Let P be a monoid. Then ℘(P) = h℘(P ), ∩, ∪, ·, \, /, {1}i is a residuated Abstract Themes lattice where: Join-Completions

X · Y = {x · y | x ∈ X,y ∈ Y }, Ordered Structures Poms (1) X\Y = {z | X · {z}⊆ Y }, and Poms (2) Poms (3) Nuclei & Subacts Y/X = {z | {z} · X ⊆ Y }. Main Theorem Lemma Let P be a pom. Then Applications I L(P) = hL(P ), ∩, ∪, ·, \, /, {1}i is a residuated lattice where: Applications II X · Y =↓ {x · y | x ∈ X,y ∈ Y }, X\Y = {z | X · {z}⊆ Y }, and Y/X = {z | {z} · X ⊆ Y }. Note: (↓ x) · (↓ y) =↓ (x · y); hence P˙ ,isa submonoid of L(P) Join-completions C. Tsinakis - slide #17 Nuclei and Subacts

A nucleus on a pom P is a closure operator γ on P Abstract such that γ(a)γ(b) ≤ γ(ab) (equivalently, Themes

γ(γ(a)γ(b)) = γ(ab)), for all a, b ∈ P . Join-Completions

Ordered Structures Poms (1) Poms (2) Poms (3) Nuclei & Subacts Main Theorem Lemma

Applications I

Applications II

Join-completions C. Tsinakis - slide #18 Nuclei and Subacts

A nucleus on a pom P is a closure operator γ on P Abstract such that γ(a)γ(b) ≤ γ(ab) (equivalently, Themes

γ(γ(a)γ(b)) = γ(ab)), for all a, b ∈ P . Join-Completions

A closure retraction C of a residuated poset P is Ordered Structures Poms (1) called a subact of P if x/y, y\x ∈ C, for all x ∈ C Poms (2) Poms (3) and y ∈ P . Nuclei & Subacts Main Theorem Lemma

Applications I

Applications II

Join-completions C. Tsinakis - slide #18 Nuclei and Subacts

A nucleus on a pom P is a closure operator γ on P Abstract such that γ(a)γ(b) ≤ γ(ab) (equivalently, Themes

γ(γ(a)γ(b)) = γ(ab)), for all a, b ∈ P . Join-Completions

A closure retraction C of a residuated poset P is Ordered Structures Poms (1) called a subact of P if x/y, y\x ∈ C, for all x ∈ C Poms (2) Poms (3) and y ∈ P . Nuclei & Subacts Main Theorem Let γ be a closure operator on a residuated pom Lemma

P, and let Pγ be the closure retraction associated Applications I

with γ. Then γ is a nucleus iff Pγ is a subact of P. Applications II

Join-completions C. Tsinakis - slide #18 Nuclei and Subacts

A nucleus on a pom P is a closure operator γ on P Abstract such that γ(a)γ(b) ≤ γ(ab) (equivalently, Themes

γ(γ(a)γ(b)) = γ(ab)), for all a, b ∈ P . Join-Completions

P, and let Pγ be the closure retraction associated Applications I

with γ. Then γ is a nucleus iff Pγ is a subact of P. Applications II

A subact Pγ of a residuated pom P is a residuated pom. The product of two elements x,y ∈ Pγ is given by x ◦γ y = γ(x · y), and the residuals are the restrictions on Pγ of the residuals of P. In particular, if P is a residuated lattice, then so is Pγ, with x ∨γ y = γ(x ∨ y) and x ∧γ y = x ∧ y

Join-completions C. Tsinakis - slide #18 Main Theorem

Let Q be a join completion of a pom P, let γQ be Abstract the associated closure operator on L(P), and let Themes

δQ be the one on ℘(P). TFAE: Join-Completions

(1) Q is a residuated lattice with respect to a Ordered Structures Poms (1) multiplication extends the multiplication of P. Poms (2) Poms (3) Nuclei & Subacts (2) a\L b ∈ Q and b/L a ∈ Q, for all a ∈ P and (P) (P) Main Theorem b ∈ Q. Lemma

Applications I (3) γQ is a nucleus on L(P) (equivalently, Q is a subact of L(P)). Applications II

(4) δQ is a nucleus on ℘(P) (equivalently, Q is a subact of ℘(P)). Furthermore, if the preceding conditions are sat- isﬁed, then the inclusion map P ֒→ Q preserves multiplication, all meets and all existing residuals.

Join-completions C. Tsinakis - slide #19 Crucial Lemma

Let P be a pom and let Q be a join-completion of Abstract Themes P that is a pom with respect to a multiplication that Join-Completions extends the multiplication of P. Then for all Ordered Structures a, b ∈ P , if a\P b exists, then a\Q b exists and Poms (1) Poms (2) Poms (3) a\P b = a\Q b = a\L(P) b = a\℘(P) b. Nuclei & Subacts Main Theorem Likewise for the other residual. Lemma Applications I

Applications II

Join-completions C. Tsinakis - slide #20 Applications I

Join-completions C. Tsinakis - slide #21 The Dedekind-MacNeille Completion

Abstract If P be a residuated pom, then its Dedekind- Themes

MacNeille completion N (P) is a residuated lat- Join-Completions -tice. Moreover, the inclusion map P ֒→ N (P) pre Ordered Structures serves products, residuals, and all existing meets Applications I and joins. DM Completion js-Monoids (1) js-Monoids (2)

Applications II

Join-completions C. Tsinakis - slide #22 The Dedekind-MacNeille Completion

Abstract If P be a residuated pom, then its Dedekind- Themes

All we need to prove is that if a ∈ P and b ∈N (P ), Applications II

then a\L(P) b ∈N (P ) and b/L(P) a ∈N (P ).

Join-completions C. Tsinakis - slide #22 The Dedekind-MacNeille Completion

Abstract If P be a residuated pom, then its Dedekind- Themes

All we need to prove is that if a ∈ P and b ∈N (P ), Applications II

then a\L(P) b ∈N (P ) and b/L(P) a ∈N (P ). Let a ∈ P and b ∈N (P ). Since P is meet-dense in N (P), there exists a family (bj | j ∈ J) of elements of P such that b = Vj∈J bj. We have:

a\L(P) b = a\L(P) Vj∈J bj = Vj∈J (a\L(P) bj).

Thus, a\L(P) b ∈N (P ), since a\L(P) bj = a\P bj.

Likewise, b/L(P) a ∈N (P ).

Join-completions C. Tsinakis - slide #22 Join-Semilattice Monoids

A join-semilattice monoid, or simply js-monoid,isa Abstract Themes pom that satisﬁes the equation Join-Completions x(y ∨ z)w ≈ (xyw) ∨ (xzw). Ordered Structures

Note that if a js-monoid P has a least element 0, Applications I DM Completion then x0=0=0x, for all x ∈ P. js-Monoids (1) js-Monoids (2)

Applications II

Join-completions C. Tsinakis - slide #23 Join-Semilattice Monoids

A join-semilattice monoid, or simply js-monoid,isa Abstract Themes pom that satisﬁes the equation Join-Completions x(y ∨ z)w ≈ (xyw) ∨ (xzw). Ordered Structures

Note that if a js-monoid P has a least element 0, Applications I DM Completion then x0=0=0x, for all x ∈ P. js-Monoids (1) js-Monoids (2)

Think of the ideal completion I(P) of P as an Applications II algebraic lattice whose join-semilattice of compact elements is P.

Join-completions C. Tsinakis - slide #23 Join-Semilattice Monoids

Applications II

Join-completions C. Tsinakis - slide #24 Join-Semilattice Monoids

If P is a js-monoid, then its ideal completion I(P ) Abstract is a residuated lattice. Moreover, the inclusion map Themes preserves products, residuals, all existing meets Join-Completions and finite joins. Ordered Structures Applications I DM Completion By Main Theorem, we need only need show that js-Monoids (1) js-Monoids (2) a\L(P) b ∈I(P ) and b/L(P) a ∈I(P ), whenever a ∈ P Applications II and b ∈I(P ). This is again simple: Let a and b as above. Define I < a, b >= {x ∈ P : a · x ≤ b}, c = W (P) . We remark that the product a · x in the definition of the set < a, b > takes place in P – or in L(P), since P is a submonoid of L(P). Is is almost immediate

that < a, b >=↓ c ∩ P, and so c = a\L(P) b.

Join-completions C. Tsinakis - slide #24 Applications II

Join-completions C. Tsinakis - slide #25 Involutive Residuated Lattices An element 0 of a residuated pom P is called cyclic Abstract if for all x ∈ P, 0/x = x\0; it is called dualizing if it Themes

satisﬁes 0/(x\0) = (0/x)\0 = x, for all x ∈ P. If 0 is Join-Completions a cyclic element, we will write x → 0 for 0/x = x\0. Ordered Structures

Applications I

Applications II Involutive RLs (1) Involutive RLs (2) DM Completion ℓ-Groups (1) ℓ-Groups (2)

Join-completions C. Tsinakis - slide #26 Involutive Residuated Lattices An element 0 of a residuated pom P is called cyclic Abstract if for all x ∈ P, 0/x = x\0; it is called dualizing if it Themes

satisﬁes 0/(x\0) = (0/x)\0 = x, for all x ∈ P. If 0 is Join-Completions a cyclic element, we will write x → 0 for 0/x = x\0. Ordered Structures

An involutive residuated lattice is an algebra Applications I

Q = hQ, ∧, ∨, ·, \, /, 1, 0i such that Applications II ′ Q = hL, ∧, ∨, ·, \, /, 1i is a residuated lattice and 0 Involutive RLs (1) ′ Involutive RLs (2) is a cyclic dualizing element of Q . DM Completion ℓ-Groups (1) If in the preceding deﬁnition we replace the term ℓ-Groups (2) ‘lattice’ by the term ‘poset’, we obtain the concept of an involutive residuated pom Q = hQ, ≤, ·, \, /, 1, 0i.

Join-completions C. Tsinakis - slide #26 Involutive Residuated Lattices An element 0 of a residuated pom P is called cyclic Abstract if for all x ∈ P, 0/x = x\0; it is called dualizing if it Themes

satisﬁes 0/(x\0) = (0/x)\0 = x, for all x ∈ P. If 0 is Join-Completions a cyclic element, we will write x → 0 for 0/x = x\0. Ordered Structures

An involutive residuated lattice is an algebra Applications I

Join-completions C. Tsinakis - slide #26 Involutive Residuated Lattices

An involutive residuated lattice Q is term equivalent Abstract Themes to an algebra Q¯ = hQ, ∧, ∨, ·, 1,′ i satisfying: Join-Completions

(1) hQ, ·, 1i is a monoid; Ordered Structures ′ (2) hQ, ∧, ∨, i is an involutive lattice; and Applications I

′ ′ ′ ′ Applications II (3) xy ≤ z ⇐⇒ y ≤ (z x) ⇐⇒ x ≤ (yz ) , for all Involutive RLs (1) Involutive RLs (2) x,y,z ∈ Q. DM Completion ℓ-Groups (1) ℓ-Groups (2)

Join-completions C. Tsinakis - slide #27 Involutive Residuated Lattices

An involutive residuated lattice Q is term equivalent Abstract Themes to an algebra Q¯ = hQ, ∧, ∨, ·, 1,′ i satisfying: Join-Completions

(1) hQ, ·, 1i is a monoid; Ordered Structures ′ (2) hQ, ∧, ∨, i is an involutive lattice; and Applications I

′ ′ ′ ′ Applications II (3) xy ≤ z ⇐⇒ y ≤ (z x) ⇐⇒ x ≤ (yz ) , for all Involutive RLs (1) Involutive RLs (2) x,y,z ∈ Q. DM Completion ℓ-Groups (1) ℓ The Glivenko-Frink Theorem for Involutive RLs -Groups (2) A subact C of a residuated pom P is an involutive pom iff C = P for some cyclic element 0 of P. γ0 Recall that for a cyclic element 0 of a residuated pom P, γ0 denotes the nucleus x 7→ (x → 0) → 0 on P.

Join-completions C. Tsinakis - slide #27 The Join-Completion of an Inv. Res. Pom Let P be a residuated poset and let Q be a Abstract join-completion of P that is a residuated lattice with Themes

Join-completions C. Tsinakis - slide #28 The Join-Completion of an Inv. Res. Pom Let P be a residuated poset and let Q be a Abstract join-completion of P that is a residuated lattice with Themes

respect to a multiplication that extends the Join-Completions multiplication of P. Then for every cyclic element 0 Ordered Structures of P, Qγ is the Dedekind-MacNeille completion 0 Applications I N (Pγ ) of Pγ . 0 0 Applications II Involutive RLs (1) In particular: Involutive RLs (2) DM Completion ℓ-Groups (1) (1) L(P) is the Dedekind-MacNeille completion ℓ-Groups (2) γ0 of P . γ0 (2) The Dedekind-MacNeille completion N (P) of an involutive residuated pom P is an involutive residuated lattice, and the only join-completion of P. Furthermore, the inclusion map P ֒→N (P) preserves products, residuals, and all existing meets and joins.

Join-completions C. Tsinakis - slide #28 ℓ-Groups A partially ordered group G is an involutive Abstract residuated pom, and hence its Dedekind-MacNeille Themes

completion N (G) is an involutive residuated lattice. Join-Completions

Join-completions C. Tsinakis - slide #29 ℓ-Groups A partially ordered group G is an involutive Abstract residuated pom, and hence its Dedekind-MacNeille Themes

completion N (G) is an involutive residuated lattice. Join-Completions

Join-completions C. Tsinakis - slide #29 ℓ-Groups A partially ordered group G is an involutive Abstract residuated pom, and hence its Dedekind-MacNeille Themes

completion N (G) is an involutive residuated lattice. Join-Completions

This completion is of little interest, since the Ordered Structures presence of least and greatest elements prevents Applications I N (G) from being a partially ordered group. Applications II ♯ Involutive RLs (1) Let N (G) denote the involutive pom obtained from Involutive RLs (2) DM Completion N (G) by removing its least and greatest elements. ℓ-Groups (1) ℓ Note that N (G)♯ is conditionally complete, and it is -Groups (2) a lattice precisely when G is directed. It is well-known that N (G)♯ is a conditionally complete ℓ-group iff G is integrally closed (Krull, Lorenzen, Clifford, Everett and Ulam). [G is said to be integrally closed if whenever x,y ∈ G such that y ≤ xn (n = 1, 2,... ), then x ≥ 1.]

Join-completions C. Tsinakis - slide #29 ℓ-Groups It is easy to verify that if N (G)♯ is conditionally Abstract complete, then G is integrally closed (Kantorovitch). Themes

Join-Completions

Ordered Structures

Applications I

Applications II Involutive RLs (1) Involutive RLs (2) DM Completion ℓ-Groups (1) ℓ-Groups (2)

Join-completions C. Tsinakis - slide #30 ℓ-Groups It is easy to verify that if N (G)♯ is conditionally Abstract complete, then G is integrally closed (Kantorovitch). Themes

Applications II Involutive RLs (1) Involutive RLs (2) DM Completion ℓ-Groups (1) ℓ-Groups (2)

Join-completions C. Tsinakis - slide #30 ℓ-Groups It is easy to verify that if N (G)♯ is conditionally Abstract complete, then G is integrally closed (Kantorovitch). Themes

Suppose next that G is integrally closed. We Join-Completions ′ observed that the map x 7→ x = x → 1 is an Ordered Structures ♯ involution of N (G) . Set Lx =↓ x ∩ G, Ux =↑ x ∩ G. Applications I ♯ We wish to show that each a ∈N (G) is invertible, Applications II ′ ♯ Involutive RLs (1) or equivalently that 1 ≤ aa , for all a ∈N (G) . This Involutive RLs (2) DM Completion reduces to 1 ≤ x, ∀x ∈ Uaa′ . Fix a and x. We have ℓ-Groups (1) ′ −1 ′ ℓ-Groups (2) a = V Ua, and hence, a = W Ua . Now aa ≤ −1 −1 x ⇒ (a W Ua ≤ x ⇒ au ≤ x, ∀u ∈ Ua ⇒ a ≤ xu, ∀u ∈ Ua ⇒ a ≤ V{xu | u ∈ Ua} = x(V Ua) = xa (since multiplication by an invertible element is an order-automorphism of N (G)♯). We have a ≤ xa, n and hence a ≤ x a (for n = 1, 2,... ). Fix u ∈ Ua n −1 n and w ∈ La. Then w ≤ x u, and so wu ≤ x (for n = 1, 2,... ). Since G is integrally closed, x ≥ 1. Join-completions C. Tsinakis - slide #30