Goldblatt-Thomason for LE-logics Apostolos Tzimoulis joint work with W. Conradie and A. Palmigiano SYSMICS 2018 Orange, California
Goldblatt-Thomason theorem for modal logic Theorem Let L be a modal signature and let K be a class of Kripke L -frames that is closed under taking ultrapowers. Then K is L -definable if and only if K is closed under p-morphic images, generated subframes and disjoint unions, and reflects ultrafilter extensions. 2 / 20
LE-logics The logics algebraically captured by varieties of normal lattice expansions. φ :: = p | ⊥ | ⊤ | φ ∧ φ | φ ∨ φ | f ( φ ) | g ( φ ) where p ∈ AtProp, f ∈ F , g ∈ G . Normality ◮ Every f ∈ F is finitely join-preserving in positive coordinates and finitely meet-reversing in negative coordinates. ◮ Every g ∈ G is finitely meet-preserving in positive coordinates and finitely join-reversing in negative coordinates. Examples: substructural, Lambek, Lambek-Grishin, Orthologic... 3 / 20
Goldblatt-Thomason theorem for LE-logics Theorem Let L be an LE signature and let K be a class of L -frames that is closed under taking ultrapowers. Then K is L -definable if and only if K is closed under p-morphic images, generated subframes and co-products, and reflects filter-ideal extensions. 4 / 20
LE frames Definition An L -frame is a tuple F = ( W , R F , R G ) such that W = ( W , U , N ) is a polarity, R F = { R f | f ∈ F } , and R G = { R g | g ∈ G} such that for each f ∈ F and g ∈ G , the symbols R f and R g respectively denote ( n f + 1) -ary and ( n g + 1) -ary relations on W , R f ⊆ U × W ǫ f and R g ⊆ W × U ǫ g , (1) In addition, we assume that the following sets are Galois-stable (from now on abbreviated as stable ) for all w 0 ∈ W , u 0 ∈ U , w ∈ W ǫ f , and u ∈ U ǫ g : R (0) f [ w ] and R ( i ) f [ u 0 , w i ] (2) R (0) g [ u ] and R ( i ) g [ w 0 , u i ] (3) 5 / 20
co-product for LE frames Let L = { � } , i.e. R � ⊆ W × U : x 1 y 1 x 2 y 2 x 1 y 1 x 2 y 2 a 1 a 2 a 1 a 2 b 1 b 2 b 1 b 2 F 1 F 2 F 1 ⊎ F 2 6 / 20
p-morphisms for LE logics Definition A p-morphism of L -frames, F 1 = ( W 1 , R 1 F , R 1 G ) and F 2 = ( W 2 , R 2 F , R 2 G ) , is a pair ( S , T ) : F 1 → F 2 such that: p1. S ⊆ W 1 × U 2 and T ⊆ U 1 × W 2 ; p2. S (0) [ u ] , S (1) [ w ] , T (0) [ w ] and T (1) [ u ] are Galois stable sets; p3. ( T (0) [ w ]) ↓ ⊆ S (0) [ w ↑ ] for every w ∈ W 2 ; p4. T (0) [( S (1) [ w ]) ↓ ] ⊆ w ↑ for every w ∈ W 1 ; f ) (0) [(( T ǫ f ) (0) [ w ]) ∂ ] for every R i p5. T (0) [(( R 2 f ) (0) [ w ]) ↓ ] = ( R 1 f ∈ R i F , where T 1 = T and T ∂ = S ; p6. S (0) [(( R 2 g ) (0) [ u ]) ↑ ] = ( R 1 g ) (0) [(( S ǫ g ) (0) [ u ]) ∂ ] for every R i g ∈ R i G , where S 1 = S and S ∂ = T . 7 / 20
p-morphisms for LE logics Definition A p-morphism of L -frames, F 1 = ( W 1 , R 1 ♦ , R 1 � ) and F 2 = ( W 2 , R 2 ♦ , R 2 � ) , is a pair ( S , T ) : F 1 → F 2 such that: p1. S ⊆ W 1 × U 2 and T ⊆ U 1 × W 2 ; p2. S (0) [ u ] , S (1) [ w ] , T (0) [ w ] and T (1) [ u ] are Galois stable sets; p3. ( T (0) [ w ]) ↓ ⊆ S (0) [ w ↑ ] for every w ∈ W 2 ; p4. T (0) [( S (1) [ w ]) ↓ ] ⊆ w ↑ for every w ∈ W 1 ; ♦ ) (0) [ w ]) ↓ ] = ( R 1 ♦ ) (0) [(( T ) (0) [ w ]) ↓ ] ; p5. T (0) [(( R 2 p6. S (0) [(( R 2 � ) (0) [ u ]) ↑ ] = ( R 1 � ) (0) [(( S ) (0) [ u ]) ↑ ] . 8 / 20
Injective and surjective p-morphisms Definition For every p-morphism ( S , T ) : F 1 → F 2 , 1. ( S , T ) : F 1 ։ F 2 , if a � b implies S (0) [( )] � S (0) [( [ a ] [ b ] )] , for every a , b ∈ ( F 2 ) + . In this case we say that F 2 is a p-morphic image of F 1 . 2. ( S , T ) : F 1 ֒ → F 2 , if for every a ∈ ( F 1 ) + there exists b ∈ ( F 2 ) + such that S (0) [( [ b ] )] = [ [ a ] ] . In this case we say that F 1 is a generated subframe of F 2 . 9 / 20
Example: generated subframe x 2 y 2 x 1 y 1 a 2 a 1 b 1 F 2 F 1 F 2 is a generated subframe of F 1 . 10 / 20
Example: p-morphic image y 1 x 1 x 2 a 1 a 2 b 1 F 1 F 2 ( ∅ , ∅ ) = ( S , T ) : F 1 → F 2 . F 2 is a p-morphic image of F 1 . 11 / 20
(Counter)example x 1 y 1 x 2 a 1 a 2 b 1 F 1 F 2 12 / 20
Filter-ideal extensions Definition The filter-ideal frame of an L -algebra A is A ⋆ = ( F A , I A , N ⋆ , R ⋆ F , R ⋆ G ) defined as follows: 1. F A = { F ⊆ A | F is a filter } ; 2. I A = { I ⊆ A | I is an ideal } ; 3. FN ⋆ I if and only if F ∩ I � ∅ ; ǫ f , R ⋆ 4. for any f ∈ F and any F ∈ F f ( I , F ) if and only f ( a ) ∈ I for some a ∈ F ; ǫ g , R ⋆ 5. for any g ∈ G and any I ∈ I g ( F , I ) if and only if g ( a ) ∈ F for some a ∈ I . Definition Let F be an L -frame. The filter-ideal extension of F is the L -frame ( F + ) ⋆ . 13 / 20
Ultraproducts of LE-frames ◮ L -frames as (multi-sorted) first-order structures. ◮ Given a family { F i | j ∈ J } of L -frames and an ultrafilter U over J , the ultraproduct ( � i ∈ I F i ) / U is defined as usual. ◮ ( � i ∈ I F i ) / U is an L -frame, by Łos Theorem. ◮ Let F J / U be the ultrapower of F . 14 / 20
Enlargement property Theorem (Enlargement property) There exists a surjective p-morphism ( S , T ) : F J / U ։ ( F + ) ⋆ for some set J and some ultrafilter U over J . s − 1 [[ sS I ⇐⇒ [ c ] ]] ∈ U for some c ∈ I (4) t − 1 [( tTF ⇐⇒ [ c ] )] ∈ U for some c ∈ F . (5) 15 / 20
Goldblatt-Thomason theorem for LE-logics Theorem Let L be an LE signature and let K be a class of L -frames that is closed under taking ultrapowers. Then K is L -definable if and only if K is closed under p-morphic images, generated subframes and co-products, and reflects filter-ideal extensions. Proof. Let F be an L -frame validating the L -theory of K. By Birkhoff’s Theorem: F + և A ֒ → ( � F i ) + . i ∈ I This gives � � F i ) J / U . ( F + ) ⋆ ֒ → A ⋆ և (( F i ) + ) ⋆ և ( i ∈ I i ∈ I � 16 / 20
Examples revisited: Difference The first-order condition R � = N c is not L -definable: x 1 y 1 x 2 y 2 x 1 y 1 x 2 y 2 a 1 a 2 a 1 a 2 b 1 b 2 b 1 b 2 F 1 F 2 F 1 ⊎ F 2 17 / 20
Examples revisited: Irreflexivity The first-order condition R c ⊆ N is not L -definable: x 1 y 1 x 2 a 1 a 2 b 1 F 1 F 2 18 / 20
Examples revisited: Every point has a predecessor The following first-order condition ∀ u ∃ w ( ¬ wRu ) is not L -definable: x 2 y 2 x 1 y 1 a 2 a 1 b 1 F 2 F 1 19 / 20
Thank you! 20 / 20
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