Notes on exact meets and joins R. N. Ball, J. Picado and A. Pultr 1 - PDF document

. Notes on exact meets and joins R. N. Ball, J. Picado and A. Pultr 1

Exact meets and joins. Recall the following operations a ↓ b = { x | x ∧ a ≤ b } and a ↑ b = { x | x ∨ a ≥ b } . An element b is the exact meet of a subset A of a lattice L if • b is a lower bound of A , and • for any c < d , if A ⊆ c ↑ d then b ∈ c ↑ d . (the latter in detail: if a ∨ c ≥ d for all a ∈ A then b ∨ c ≥ d ). Dually, b is the exact join of a subset A if • b is an upper bound of A , and • for any c < d , if A ⊆ c ↓ d then b ∈ c ↓ d . (Bruns and Lakser speak of admissible joins.) 2

Open and closed sublocales. G-sets. In the co-frame S ℓ ( L ) of all sublocales of a locale L we have in particular – the open sublocale associated with the elements a ∈ L o ( a ) = { a → x | x ∈ L } = { x | a → x = x } . – and their complements, the closed sublocales c ( a ) = ↑ a. In the co-frame S ℓ ( L ) we have ∨ ∨ o ( a ) ∧ o ( b ) = o ( a ∧ b ) , o ( a i ) = o ( a i ) , i ∈ J i ∈ J ∧ ∨ c ( a i ) = c ( a i ) , c ( a ) ∨ c ( b ) = c ( a ∧ b ) . i ∈ J i ∈ J 3

For purposes of the discussion of more gen- eral (bounded) lattices L we will generalize the sublocales to the geometric subsets (briefly, G- subsets ) the subsets S ⊆ L such that ∧ ∧ if M ⊆ S and M ∈ S . M exists then The system of all G-subsets of a lattice L will be denoted by G ( L ) and following the situation from S ℓ ( L ) of a frame we speak of the subsets c ( a ) = ↑ a as of the closed G-subsets. 4

Proposition. For any lattice, G ( L ) ordered by inclusion is a complete lattice with the join ∨ ∧ ∪ ∧ S i = { M | M ⊆ M exists } . S i , i ∈ J i ∈ J Consequently, if L is a frame, the sublocale co- frame S ℓ ( L ) is a subset of G ( L ) closed under all joins. i ∈ J c ( a i ) in G ( L ) be closed. Proposition. Let ∨ Then a = ∧ i a i exists, and ∨ i ∈ J c ( a i ) = c ( a ) . Theorem. A meet ∧ A in L is exact if and only if the join S = ∨ { c ( a ) | a ∈ A } is closed. 5

Exact meets in frames Fact. In the co-frame S ℓ ( L ) we have for the pseudosupplement x # , ∨ c ( a i ) = c ( a ) if and only if ( ∩ o ( a i )) ## = o ( a ) . Theorem. TFAE: (1) The meet a = ∧ i a i is exact. (2) For all b ∈ L , ( ∧ i a i ) ∨ b = ∧ i ( a i ∨ b ) . (3) ∧ a i = a and ∨ i c ( a i ) = c ( a ) in S ℓ ( L ) . (4) ∧ a i = a and ∨ i c ( a i ) is closed. (5) If x ≥ ∧ i a i then there exist x i ≥ a i such that x = ∧ i x i . i o ( a i )) ## = ( ∩ i o ( a i )) ## = o ( a ) in S ℓ ( L ) . (6) ( ∧ i o ( a i )) ## = ( ∩ i o ( a i )) ## is an open sublo- (7) ( ∧ cale of L . 6

Strongly exact (free) meets. Instead of closed joins of closed sublocales, we re- quire the meets (intersections) of open sublocales to be open: ∧ ∩ o ( a i ) = o ( a i ) = o ( a ) . (s-exact) i ∈ J i ∈ J Theorem. The following facts about a meet a = ∧ i ∈ J a i in a frame L are equivalent. (1) The meet a = ∧ i a i is strongly exact. (2) ∧ i o ( a i ) = ∩ i o ( a i ) is open. (3) If a i → x = x for all i ∈ J then ∧ a i ) → x = x. ( i ∈ J 7

Strongly exact meets appeared, viewed from another perspective, as the free meets in the unpublished Thesis by Todd Wilson: the meets that are preserved by all frame homo- morphisms . Wilson proved, a.o., the equivalent of (s-exact) as one of the characteristics of the freeness. Wil- son’s characteristic, slightly modified: Theorem. TFAE: (1) ∧ i a i is strongly exact. (2) for every frame homomorphism h : L → M, h ( ∧ i a i ) = ∧ i h ( a i ) and it is strongly exact. (3) for every frame homomorphism h : L → M, h ( ∧ i a i ) = ∧ i h ( a i ) . (4) For every x ∈ L , ∧ a ∈ A ( a → x ) → x = (( ∧ A ) → x ) → x . 8

Note. N ( L ) = S ℓ ( L ) op is a frame and we have a frame homomorphism c L = ( a �→ o ( a )): L → N ( L ) starting the famous Assembly Tower. Another characteristic of the strongly exact (free) meets Todd Wilson presented was that ∧ ∧ c N ( L ) c L [ A ] = c N ( L ) c L ( A ) in N 2 ( L ). 9

Conservative subsets. (1) In our language, a subset A ⊆ L is conser- vative iff B is exact for all B ⊆ A . Dowker and Papert (1975) and Chen (1992) used conservative subsets of frames in the study of paracompact- ness. (2) Exact meets are also related with the concepts of interior-preserving and closure-preserving families of sublocales of Plewe. A family S = { S i | i ∈ I } ⊆ S ℓ ( L ) is closure-preserving if for all J ⊆ I , ∨ ∨ cl ( S i ) = cl ( S i ) . i ∈ J i ∈ J Dually, S is interior-preserving if for all J ⊆ I , ∧ ∧ int ( S i ) = int ( S i ) . i ∈ J i ∈ J Then, a subset A of L is said to be interior- preserving (resp. closure-preserving ) if { o ( a ) | a ∈ A } is interior-preserving (resp. { c ( a ) | a ∈ A } is closure-preserving). 10

Interior-preserving covers play a decisive role in the construction of (canonical) examples of tran- sitive quasi-uniformities for frames (Ferreira and Picado). Any interior-preserving cover of L is closure- preserving but somewhat surprising, contrarily to what happens in the classical case, the converse does not hold in general (the cover N of the frame L = ( ω + 1) op = {∞ < · · · < 2 < 1 } is such an example). 11

Lemma. Let A ⊆ L . Then: (1) A is interior-preserving iff ∧ ∧ o ( b ) = o ( B ) b ∈ B for every B ⊆ A. (2) A is closure-preserving iff ∨ ∧ c ( b ) = c ( B ) b ∈ B for every B ⊆ A. Corollary. A subset A of a frame L is con- servative if and only if it is closure-preserving. This gives an example of an A ⊆ L for which any B ⊆ A is exact but, being not interior- preserving, such that there is some B ⊆ A which is not strong exact. Thus strong exactness is indeed a stronger prop- erty than exactness. 12

Exact and strongly exact in spaces. A space X is T D if ∀ x ∈ X ∃ U ∋ x open such that U � { x } is open. (Aull and Thron 1963, also Bruns 1962.) Proposition. A space is T D iff there holds the equivalence ( ∀ A open , int ( U ∪ A ) = int U ∪ A ) U is open . iff Corollary. A space is T D − 0 iff for every ∼ - set U there holds the equivalence ( ∀ A open , int ( U ∪ A ) = int U ∪ A ) U is open . iff Lemma. In any space X , ∧ int U = { X � { x } | x / ∈ U } . 13

Theorem. TFAE for a topological space X . (1) X is T D − 0 . (2) A meet ∧ U i is exact in Ω( X ) iff ∩ U i is open. What this says about intersections of open sub- sets in spaces: A subset A of a topological space X induces a congruence E A = { ( U, V ) | U ∩ A = V ∩ A } . Write A ∼ B for E A = E B . For T D -spaces we have A ∼ B iff A = B , and this fact holds only in T D -spaces. In fact, one needs T D even for the special case when one of the A, B is open. Thus the facts above can be di- rectly interpreted in spaces only as the statement that if X is a T D -space then the meet ∧ U i in Ω( X ) is strongly exact iff ∧ o ( U i ) is open. 14

BUT : Lemma. Let X be an arbitrary space, A, W subsets, A ∼ W and W open. Then for each open subset U ⊆ X we have A ⊆ U W ⊆ U. iff Corollary. For any topological space X the meet ∧ U i in Ω( X ) is strongly exact iff ∩ U i is open. Corollary. If X is not T D − 0 then the exact- ness and strong exactness in Ω( X ) differ. Theorem. A spatial L is T D -spatial iff strongly exact meets and exact meets in L coincide. 15

Note: Exact meets of meet irreducible elements. View Σ L as the set of all meet-irreducible ele- ments p ∈ L and take the natural isomorphism for L spatial, ( a �→ Σ a ): L ∼ = ΩΣ L. We obtain that a = ∧ i a i is strongly exact iff Σ a = ∧ i Σ a i is strongly exact. Hence a = ∧ i a i is strongly exact iff Σ a = ∩ i Σ a i . i a i ≤ p iff ∃ j, a j ≤ p , which reduces to Thus, ∧ the implication ∧ a i ≤ p ⇒ ∃ j, a j ≤ p. i For an L ∼ = Ω( X ) with a T D -space X , this is then another criterion of exactness. 16

Strongly exact in Scott topology The set of all up-sets will be denoted by U ( X ) . The Scott topology σ X on a lattice X consists of the U ∈ U ( X ) such that ∨ D ∈ U ⇒ D ∩ U ̸ = ∅ for any directed D ⊆ X . Now the spectrum of a frame L will be rep- resented as the set Σ ′ L of all completely prime filters P in L endowed with the topology consist- ing of the open sets Σ ′ a = { P | a ∈ P } , a ∈ L . Each P ∈ Σ ′ L is Scott open in L . More generally, in a lattice L we will consider the pre-topology Σ ′ L = { Σ ′ Σ ′ x | x ∈ L } , x = { U ∈ U ( L ) | x ∈ U } . 17

One of the important facts needed in the proof of the Hofmann-Lawson duality is that an intersection ∩ P of a set of completely prime filters is Scott open ( that is, ∧ P is strongly exact in σ L ) iff P is a compact sub- set of Σ ′ L . This is a part of a more general fact. A subset U of U ( L ) is d-compact if one can choose in every directed cover of U by the element of Σ ′ L an element covering U . 18

Proposition. Let a set U of Scott open sets L . Then ∩ U is Scott open, be d-compact in Σ ′ and hence ∧ U is strongly exact in σ L . Proposition. Let X = ( X, ≤ ) be a complete lattice. Let U be a set of Scott open sets in X and let ∩ U be Scott open. Then U is d- compact in Σ ′ X . Proposition. Let L be a complete lattice. Then a meet ∧ U in the Scott topology σ L is strongly exact iff U is d-compact in the pre- topology Σ ′ L on U ( L ) . 19