MacNeille transferability of fjnite latuices Frederik Mllerstrm - PowerPoint PPT Presentation

Universal sentences and forbidden confjgurations Examples by such clauses. 5. Any universal class of locally fjnite latuices can be axiomatised 4 Let τ ⊆ { 0 , 1 , ∧ , ∨} and let L be a fjnite latuice. Tien there exist a universal sentence ρ τ ( L ) such that K ̸| = ρ τ ( L ) ⇐ ⇒ L ֒ → τ K , for all τ -latuices K . Hence L is ideal transferable if and only if ρ ∧ , ∨ ( L ) is preserved by the operation K �→ Idl ( K ) . 1. Well-known examples ρ ∧ , ∨ ( N 5 ) and ρ ∧ , ∨ ( M 3 ) , 2. Join-irreducible top element ρ 1 , ∨ ( 2 × 2 ) , 3. No non-trivial complemented elements ρ 0 , 1 ∧ , ∨ ( 2 × 2 ) , 4. No doubly-irreducible elements ρ ∧ , ∨ ( D ) ,

Characterising ideal transferability Tieorem (Grätzer et al. 1970’ties) Let L be a fjnite latuice. Tien the following are equivalent: 1. L is ideal transferable, 2. L is sharply ideal transferable, 3. L is a sub-latuice of the free latuice on 3-generators, 4. L is (weakly) projective in the category of latuices, 5. L is semi-distributive and satisfjes Whitman’s condition (W). Hence ideal transferable latuices have no doubly reducible elements. Tieorem (Gaskill 1972 (1973), Nelson 1974) Any fjnite distributive latuice is sharply ideal transferable for the class of all distributive latuices. New results for ideal transferability of distributive latuices with respect to certain classes of modular latuices Wehrung 2018. 5

MacNeille transferability K K L K L : if for all K -MacNeille transferable for 2. L is sharply for all K L Defjnition K = L , if -MacNeille transferable for 1. A fjnite latuice L is -latuices. be a class of and let Let 6

MacNeille transferability 2. L is sharply K L K L : if for all K -MacNeille transferable for for all K Defjnition K L K = L , if -MacNeille transferable for 1. A fjnite latuice L is 6 Let τ ⊆ { 0 , 1 , ∧ , ∨} and let K be a class of τ -latuices.

MacNeille transferability -MacNeille transferable for K L K L : if for all K 2. L is sharply 6 Defjnition Let τ ⊆ { 0 , 1 , ∧ , ∨} and let K be a class of τ -latuices. 1. A fjnite latuice L is τ -MacNeille transferable for K , if → τ K = ⇒ L ֒ → τ K , for all K ∈ K , L ֒

MacNeille transferability Defjnition 6 Let τ ⊆ { 0 , 1 , ∧ , ∨} and let K be a class of τ -latuices. 1. A fjnite latuice L is τ -MacNeille transferable for K , if → τ K = ⇒ L ֒ → τ K , for all K ∈ K , L ֒ 2. L is sharply τ -MacNeille transferable for K if for all K ∈ K : ∀ h : L ֒ → τ K ∃ k : L ֒ → τ K ( x ≤ y ⇐ ⇒ k ( x ) ≤ h ( y )) .

Why is this interesting 1. Universal classes of latuices closed under MacNeille completions, 2. Canonicity of stable intermediate logics G. & N. Bezhanishvili & J. Ilin, 3. Connections with Algebraic Proof Tieory Ciabattoni, Galatos & Terui; Belardinelli, Jipsen & Ono, …, 4. Non-syntactic proof of the fact that universal -clauses are preserved under MacNeille completions of Heyting algebras Ciabattoni, Galatos & Terui 2011. 7

Why is this interesting 1. Universal classes of latuices closed under MacNeille completions, 2. Canonicity of stable intermediate logics G. & N. Bezhanishvili & J. Ilin, 3. Connections with Algebraic Proof Tieory Ciabattoni, Galatos & Terui; Belardinelli, Jipsen & Ono, …, preserved under MacNeille completions of Heyting algebras Ciabattoni, Galatos & Terui 2011. 7 4. Non-syntactic proof of the fact that universal { 0 , 1 , ∧} -clauses are

For any latuice L there exist distributive latuice D L such that D L Harding 1993. MacNeille transferability for latuices Tieorem A fjnite latuice -MacNeille transferable for a class of latuices containing all distributive latuices is necessarily distributive. Proof. L Remark Tiis can be seen as a generalisation of the fact that that latuice N is not -MacNeille transferable for the class of distributive latuices Funayama 1944. In particular, the class of distributive latuices is not closed under MacNeille completions 8

For any latuice L there exist distributive latuice D L such that D L Harding 1993. MacNeille transferability for latuices Tieorem containing all distributive latuices is necessarily distributive. Proof. L Remark Tiis can be seen as a generalisation of the fact that that latuice N is not -MacNeille transferable for the class of distributive latuices Funayama 1944. In particular, the class of distributive latuices is not closed under MacNeille completions 8 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices

MacNeille transferability for latuices Tieorem containing all distributive latuices is necessarily distributive. Proof. Remark Tiis can be seen as a generalisation of the fact that that latuice N is not -MacNeille transferable for the class of distributive latuices Funayama 1944. In particular, the class of distributive latuices is not closed under MacNeille completions 8 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices For any latuice L there exist distributive latuice D L such that L ֒ → ∧ , ∨ D L Harding 1993.

MacNeille transferability for latuices Tieorem containing all distributive latuices is necessarily distributive. Proof. Remark latuices Funayama 1944. In particular, the class of distributive latuices is not closed under MacNeille completions 8 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices For any latuice L there exist distributive latuice D L such that L ֒ → ∧ , ∨ D L Harding 1993. Tiis can be seen as a generalisation of the fact that that latuice N 5 is not {∧ , ∨} -MacNeille transferable for the class of distributive

MacNeille transferability for latuices Tieorem Proof. For K a bounded latuice we have that Idl K K K Gehrke, Harding & Venema 2006. So if L Idl K , then L K , by Łos’ Tieorem. 9 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices K closed under ultrapowers is also ideal transferable for K .

MacNeille transferability for latuices Tieorem Proof. For K a bounded latuice we have that Gehrke, Harding & Venema 2006. So if L Idl K , then L K , by Łos’ Tieorem. 9 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices K closed under ultrapowers is also ideal transferable for K . → ∧ , ∨ K δ ֒ → ∧ , ∨ K X / U, Idl ( K ) ֒

MacNeille transferability for latuices Tieorem Proof. For K a bounded latuice we have that 9 A fjnite latuice {∧ , ∨} -MacNeille transferable for a class of latuices K closed under ultrapowers is also ideal transferable for K . → ∧ , ∨ K δ ֒ → ∧ , ∨ K X / U, Idl ( K ) ֒ → ∧ , ∨ Idl ( K ) , then Gehrke, Harding & Venema 2006. So if L ֒ L ֒ → ∧ , ∨ K , by Łos’ Tieorem.

MacNeille transferability for latuices Proof. transferable for the class of all latuices? -MacNeille Does this exactly characterise the latuices Problem this form Galvin & Jónsson 1961. 3. Any distributive latuice without doubly reducible elements is of 2. In particular, L has no doubly-reducible elements. then L is distributive and ideal transferable. -MacNeille transferable for the class of all latuices 1. If L is for C a chain. Corollary C 2 or 2 2 2 1 latuices must be a linear sum of latuices isomorphic to: -MacNeille transferable for the class of all Any fjnite latuice 10

MacNeille transferability for latuices -MacNeille transferable for the class of all latuices transferable for the class of all latuices? -MacNeille Does this exactly characterise the latuices Problem this form Galvin & Jónsson 1961. 3. Any distributive latuice without doubly reducible elements is of 2. In particular, L has no doubly-reducible elements. then L is distributive and ideal transferable. 1. If L is Corollary Proof. for C a chain. or latuices must be a linear sum of latuices isomorphic to: 10 Any fjnite latuice {∧ , ∨} -MacNeille transferable for the class of all 2 × 2 × 2 , 2 × C , 1 ,

MacNeille transferability for latuices then L is distributive and ideal transferable. transferable for the class of all latuices? -MacNeille Does this exactly characterise the latuices Problem this form Galvin & Jónsson 1961. 3. Any distributive latuice without doubly reducible elements is of 2. In particular, L has no doubly-reducible elements. 10 Corollary Proof. for C a chain. or latuices must be a linear sum of latuices isomorphic to: Any fjnite latuice {∧ , ∨} -MacNeille transferable for the class of all 2 × 2 × 2 , 2 × C , 1 , 1. If L is {∧ , ∨} -MacNeille transferable for the class of all latuices

MacNeille transferability for latuices Corollary transferable for the class of all latuices? Problem this form Galvin & Jónsson 1961. 3. Any distributive latuice without doubly reducible elements is of 2. In particular, L has no doubly-reducible elements. then L is distributive and ideal transferable. Proof. for C a chain. or latuices must be a linear sum of latuices isomorphic to: 10 Any fjnite latuice {∧ , ∨} -MacNeille transferable for the class of all 2 × 2 × 2 , 2 × C , 1 , 1. If L is {∧ , ∨} -MacNeille transferable for the class of all latuices Does this exactly characterise the latuices {∧ , ∨} -MacNeille

Projective latuices A 1970). L is closed under meets (Balbes & Horn distributive latuices ifg 2. A fjnite distributive latuice L is projective in the category of category of meet-semilatuices (Horn & Kimura 1971), 1. Every fjnite distributive latuice (reduct) is projective in the Tieorem B P A making the following diagram commute Defjnition arrow P , there exist an B in A B and any surjection P arrow is (weakly) projective if for any An object P in a concrete category 11

Projective latuices Tieorem 1970). L is closed under meets (Balbes & Horn distributive latuices ifg 2. A fjnite distributive latuice L is projective in the category of category of meet-semilatuices (Horn & Kimura 1971), 1. Every fjnite distributive latuice (reduct) is projective in the 11 Defjnition B P A An object P in a concrete category C is (weakly) projective if for any arrow h : P → B and any surjection q : A ↠ B in C , there exist an arrow P → A making the following diagram commute ∃ q h

Projective latuices Defjnition 1970). 2. A fjnite distributive latuice L is projective in the category of category of meet-semilatuices (Horn & Kimura 1971), 1. Every fjnite distributive latuice (reduct) is projective in the Tieorem 11 B P A An object P in a concrete category C is (weakly) projective if for any arrow h : P → B and any surjection q : A ↠ B in C , there exist an arrow P → A making the following diagram commute ∃ q h distributive latuices ifg J 0 ( L ) is closed under meets (Balbes & Horn

MacNeille transferability for bounded latuices K closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Idl K K L S Tieorem Tiis is an application of Baker & Hales 1974: For Proof. -latuices. -MacNeille transferable for the class of all distributive latuice is . Tien any fjnite be such that Let 12

MacNeille transferability for bounded latuices K closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Idl K K L S Tieorem Tiis is an application of Baker & Hales 1974: For Proof. -latuices. -MacNeille transferable for the class of all distributive latuice is Tien any fjnite 12 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ .

MacNeille transferability for bounded latuices L closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Idl K K K Tieorem S Tiis is an application of Baker & Hales 1974: For Proof. 12 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ . Tien any fjnite distributive latuice is τ -MacNeille transferable for the class of all τ -latuices.

MacNeille transferability for bounded latuices L closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Idl K K K Tieorem S For Tiis is an application of Baker & Hales 1974: Proof. 12 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ . Tien any fjnite distributive latuice is τ -MacNeille transferable for the class of all τ -latuices.

MacNeille transferability for bounded latuices L closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Idl K K K Tieorem S Proof. 12 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ . Tien any fjnite distributive latuice is τ -MacNeille transferable for the class of all τ -latuices. Tiis is an application of Baker & Hales 1974: For ∧ ∈ τ ⊆ { 0 , 1 , ∧}

MacNeille transferability for bounded latuices L closed under MacNeille completions Ciabattoni et al. 2011. -clauses is Tiis entails that any class of HAs axiomatised by Remark Tieorem K 12 Proof. S Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ . Tien any fjnite distributive latuice is τ -MacNeille transferable for the class of all τ -latuices. Tiis is an application of Baker & Hales 1974: For ∧ ∈ τ ⊆ { 0 , 1 , ∧} ∧ , ∨ K X / U ∧ ∧ , ∨ 0 , 1 , ∧ ∧ Idl ( K )

MacNeille transferability for bounded latuices S closed under MacNeille completions Ciabattoni et al. 2011. Remark Tieorem L K Proof. 12 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that {∧ , ∨} ̸⊆ τ . Tien any fjnite distributive latuice is τ -MacNeille transferable for the class of all τ -latuices. Tiis is an application of Baker & Hales 1974: For ∧ ∈ τ ⊆ { 0 , 1 , ∧} ∧ , ∨ K X / U ∧ ∧ , ∨ 0 , 1 , ∧ ∧ Idl ( K ) Tiis entails that any class of HAs axiomatised by { 0 , 1 , ∧} -clauses is

MacNeille transferability for distributive latuices Tiere is a fjnite distributive latuice L not -MacNeille transferable for the class of latuices whose MacNeille completions are distributive. K L Note that K is not a Heyting algebra. 13

MacNeille transferability for distributive latuices transferable for the class of latuices whose MacNeille completions are distributive. K L Note that K is not a Heyting algebra. 13 Tiere is a fjnite distributive latuice L not {∧ , ∨} -MacNeille

MacNeille transferability for distributive latuices K . Since P is a fjnite projective Balbes & Horn 1970. P for K P K = P distributive latuice we have that K then P Tieorem If P Proof. the class of all distributive -latuices. -MacNeille transferable for projective distributive latuice. Tien P is and let P be a fjnite be such that Let 14

MacNeille transferability for distributive latuices distributive latuice we have that Balbes & Horn 1970. P for K P K = P K . Since P is a fjnite projective Tieorem K then P If P Proof. the class of all distributive -latuices. -MacNeille transferable for Tien P is projective distributive latuice. 14 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that { 0 , 1 } ̸⊆ τ and let P be a fjnite

MacNeille transferability for distributive latuices P Balbes & Horn 1970. P for K P K = distributive latuice we have that Tieorem K . Since P is a fjnite projective K then P If P Proof. 14 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that { 0 , 1 } ̸⊆ τ and let P be a fjnite projective distributive latuice. Tien P is τ -MacNeille transferable for the class of all distributive τ -latuices.

MacNeille transferability for distributive latuices distributive latuice we have that Balbes & Horn 1970. P for K P K = P Since P is a fjnite projective Tieorem Proof. 14 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that { 0 , 1 } ̸⊆ τ and let P be a fjnite projective distributive latuice. Tien P is τ -MacNeille transferable for the class of all distributive τ -latuices. If P ֒ → 0 , ∧ , ∨ K then P ֒ → 0 , ∧ K .

MacNeille transferability for distributive latuices Proof. Tieorem distributive latuice we have that 14 Let τ ⊆ { 0 , 1 , ∧ , ∨} be such that { 0 , 1 } ̸⊆ τ and let P be a fjnite projective distributive latuice. Tien P is τ -MacNeille transferable for the class of all distributive τ -latuices. If P ֒ → 0 , ∧ , ∨ K then P ֒ → 0 , ∧ K . Since P is a fjnite projective ⇒ ˆ h : P ֒ → 0 , ∧ K = h : P ֒ → 0 , ∧ , ∨ K , for ˆ h ( x ) := ∨ { h ( a ) : a ∈ J 0 ( P ) ∩ ↓ x } Balbes & Horn 1970.

algebras. Note: Tie latuice D also plays a central role in Wehrung MacNeille transferability for distributive latuices Tie latuice D is -MacNeille transferable for the class of distributive latuices but not projective in the category of distributive latuices. D However, D is not sharply -MacNeille transferable for the class of distributive latuices. Not even for the class of Heyting 2018. 15

algebras. Note: Tie latuice D also plays a central role in Wehrung MacNeille transferability for distributive latuices latuices. D However, D is not sharply -MacNeille transferable for the class of distributive latuices. Not even for the class of Heyting 2018. 15 Tie latuice D is {∧ , ∨} -MacNeille transferable for the class of distributive latuices but not projective in the category of distributive

algebras. Note: Tie latuice D also plays a central role in Wehrung MacNeille transferability for distributive latuices latuices. D class of distributive latuices. Not even for the class of Heyting 2018. 15 Tie latuice D is {∧ , ∨} -MacNeille transferable for the class of distributive latuices but not projective in the category of distributive However, D is not sharply {∧ , ∨} -MacNeille transferable for the

MacNeille transferability for distributive latuices latuices. D class of distributive latuices. Not even for the class of Heyting algebras. Note: Tie latuice D also plays a central role in Wehrung 2018. 15 Tie latuice D is {∧ , ∨} -MacNeille transferable for the class of distributive latuices but not projective in the category of distributive However, D is not sharply {∧ , ∨} -MacNeille transferable for the

MacNeille transferability for distributive latuices latuices. D class of distributive latuices. Not even for the class of Heyting 2018. 15 Tie latuice D is {∧ , ∨} -MacNeille transferable for the class of distributive latuices but not projective in the category of distributive However, D is not sharply {∧ , ∨} -MacNeille transferable for the algebras. Note: Tie latuice D also plays a central role in Wehrung

MacNeille transferability for Heyting algebras P doubly-reducible element. latuices, and D is the seven element distributive latuice with a where P is a fjnite latuice projective in the category of distributive 1 D D 1 1 D 1 1 P 1 1 P Lemma 1 is ( Let be a class of -latuices closed under principal ideals. If L is -MacNeille transferable for the L )-MacNeille 1 transferable for . Similar, mutatis mutandis, for principal fjlters. Tieorem Tie following latuices are all -MacNeille transferable for the class of Heyting algebras: 16

MacNeille transferability for Heyting algebras P doubly-reducible element. latuices, and D is the seven element distributive latuice with a where P is a fjnite latuice projective in the category of distributive 1 D D 1 1 D 1 1 1 Lemma 1 P P 1 class of Heyting algebras: -MacNeille transferable for the Tie following latuices are all Tieorem 16 Let K be a class of ( τ ∪ { 1 } ) -latuices closed under principal ideals. If L is τ -MacNeille transferable for K the L ⊕ 1 is ( τ ∪ { 1 } )-MacNeille transferable for K . Similar, mutatis mutandis, for principal fjlters.

MacNeille transferability for Heyting algebras 1 doubly-reducible element. latuices, and D is the seven element distributive latuice with a where P is a fjnite latuice projective in the category of distributive 1 D D 1 1 D 1 P Lemma 1 1 P P 1 class of Heyting algebras: Tieorem 16 Let K be a class of ( τ ∪ { 1 } ) -latuices closed under principal ideals. If L is τ -MacNeille transferable for K the L ⊕ 1 is ( τ ∪ { 1 } )-MacNeille transferable for K . Similar, mutatis mutandis, for principal fjlters. Tie following latuices are all { 0 , 1 , ∧ , ∨} -MacNeille transferable for the

MacNeille transferability for Heyting algebras class of Heyting algebras: doubly-reducible element. latuices, and D is the seven element distributive latuice with a where P is a fjnite latuice projective in the category of distributive Lemma 16 Tieorem Let K be a class of ( τ ∪ { 1 } ) -latuices closed under principal ideals. If L is τ -MacNeille transferable for K the L ⊕ 1 is ( τ ∪ { 1 } )-MacNeille transferable for K . Similar, mutatis mutandis, for principal fjlters. Tie following latuices are all { 0 , 1 , ∧ , ∨} -MacNeille transferable for the 1 ⊕ P , P ⊕ 1 , 1 ⊕ P ⊕ 1 , 1 ⊕ D ⊕ 1 , 1 ⊕ D , D ⊕ 1 ,

MacNeille transferability for Heyting algebras B , with the property C . A , for non-trivial C C latuice C . However, C B , for any fjnite directly indecomposable distributive that C B Tieorem we have that A ClpUp So for A Proof. -MacNeille transferable for the class of Heyting algebras. No fjnite and directly decomposable distributive latuice is 17

MacNeille transferability for Heyting algebras B , with the property C . A , for non-trivial C C latuice C . However, C B , for any fjnite directly indecomposable distributive that C B Tieorem we have that A ClpUp So for A Proof. No fjnite and directly decomposable distributive latuice is 17 { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting algebras.

MacNeille transferability for Heyting algebras B C . A , for non-trivial C C latuice C . However, C B , for any fjnite directly indecomposable distributive that C B , with the property we have that A Tieorem ClpUp So for A Proof. No fjnite and directly decomposable distributive latuice is 17 { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting algebras. . . . . . . X

MacNeille transferability for Heyting algebras ClpUp C . A , for non-trivial C C latuice C . However, C B , for any fjnite directly indecomposable distributive that C B , with the property B we have that A So for A Tieorem Proof. No fjnite and directly decomposable distributive latuice is 17 { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting algebras. . . . . . . X . . . . . . X

MacNeille transferability for Heyting algebras Tieorem C . A , for non-trivial C C However, C latuice C . 17 No fjnite and directly decomposable distributive latuice is Proof. { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting algebras. . . . . . . X . . . . . . X So for A := ClpUp ( X ) we have that A = B × B , with the property → 0 , 1 , ∧ , ∨ B , for any fjnite directly indecomposable distributive that C ֒

MacNeille transferability for Heyting algebras Proof. Tieorem 17 No fjnite and directly decomposable distributive latuice is { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting algebras. . . . . . . X . . . . . . X So for A := ClpUp ( X ) we have that A = B × B , with the property → 0 , 1 , ∧ , ∨ B , for any fjnite directly indecomposable distributive that C ֒ latuice C . However, C 1 × C 2 ̸ ֒ → 0 , 1 , ∧ , ∨ A , for non-trivial C 1 , C 2 .

Problem 1. Is every fjnite distributive latuice -MacNeille transferable for the class of Heyting algebras? 2. Is every fjnite and directly indecomposable distributive latuice -MacNeille transferable for the class of Heyting algebras? 3. Must every fjnite distributive of the form L 1 (or 1 L ) be -MacNeille transferable for the class of Heyting algebras? Remark Note that a positive answer to 3 will entail that every stable intermediate logic is canonical. 18

Problem for the class of Heyting algebras? 2. Is every fjnite and directly indecomposable distributive latuice -MacNeille transferable for the class of Heyting algebras? 3. Must every fjnite distributive of the form L 1 (or 1 L ) be -MacNeille transferable for the class of Heyting algebras? Remark Note that a positive answer to 3 will entail that every stable intermediate logic is canonical. 18 1. Is every fjnite distributive latuice {∧ , ∨} -MacNeille transferable

Problem for the class of Heyting algebras? 2. Is every fjnite and directly indecomposable distributive latuice algebras? 3. Must every fjnite distributive of the form L 1 (or 1 L ) be -MacNeille transferable for the class of Heyting algebras? Remark Note that a positive answer to 3 will entail that every stable intermediate logic is canonical. 18 1. Is every fjnite distributive latuice {∧ , ∨} -MacNeille transferable { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting

Problem for the class of Heyting algebras? 2. Is every fjnite and directly indecomposable distributive latuice algebras? algebras? Remark Note that a positive answer to 3 will entail that every stable intermediate logic is canonical. 18 1. Is every fjnite distributive latuice {∧ , ∨} -MacNeille transferable { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting 3. Must every fjnite distributive of the form L ⊕ 1 (or 1 ⊕ L ) be { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of Heyting

MacNeille transferability for bi-Heyting algebras -MacNeille transferable for the class of all bi-Heyting bi-Heyting algebras. -MacNeille transferable for the class of all 1 is L 3. 1 algebras, 2. L is Tieorem bi-Heyting algebras of fjnite width, -MacNeille transferable for the class of all 1. L is sharply Let L be a fjnite distributive latuice. Tien, Tieorem Idl A . If A is a bi-Heyting algebra of fjnite width then A 19

MacNeille transferability for bi-Heyting algebras -MacNeille transferable for the class of all bi-Heyting bi-Heyting algebras. -MacNeille transferable for the class of all 1 is L 3. 1 algebras, 2. L is Tieorem bi-Heyting algebras of fjnite width, -MacNeille transferable for the class of all 1. L is sharply Let L be a fjnite distributive latuice. Tien, Tieorem 19 If A is a bi-Heyting algebra of fjnite width then A ֒ → 0 , 1 , ∧ , ∨ Idl ( A ) .

MacNeille transferability for bi-Heyting algebras -MacNeille transferable for the class of all bi-Heyting bi-Heyting algebras. -MacNeille transferable for the class of all 1 is L 3. 1 algebras, 2. L is Tieorem bi-Heyting algebras of fjnite width, Let L be a fjnite distributive latuice. Tien, Tieorem 19 If A is a bi-Heyting algebra of fjnite width then A ֒ → 0 , 1 , ∧ , ∨ Idl ( A ) . 1. L is sharply {∧ , ∨} -MacNeille transferable for the class of all

MacNeille transferability for bi-Heyting algebras Tieorem Tieorem Let L be a fjnite distributive latuice. Tien, bi-Heyting algebras of fjnite width, algebras, 3. 1 L 1 is -MacNeille transferable for the class of all bi-Heyting algebras. 19 If A is a bi-Heyting algebra of fjnite width then A ֒ → 0 , 1 , ∧ , ∨ Idl ( A ) . 1. L is sharply {∧ , ∨} -MacNeille transferable for the class of all 2. L is {∧ , ∨} -MacNeille transferable for the class of all bi-Heyting

MacNeille transferability for bi-Heyting algebras Tieorem Tieorem Let L be a fjnite distributive latuice. Tien, bi-Heyting algebras of fjnite width, algebras, bi-Heyting algebras. 19 If A is a bi-Heyting algebra of fjnite width then A ֒ → 0 , 1 , ∧ , ∨ Idl ( A ) . 1. L is sharply {∧ , ∨} -MacNeille transferable for the class of all 2. L is {∧ , ∨} -MacNeille transferable for the class of all bi-Heyting 3. 1 ⊕ L ⊕ 1 is { 0 , 1 , ∧ , ∨} -MacNeille transferable for the class of all

Future work and 4. Syntax? Cf., Grätzer 1966/1970, Baker & Hales 1974. intermediate notion of transferability. K , as an L = K 3. Investigate -transferability, L . with -MacNeille transferability for the class of Heyting algebras 2. the class of all bi-Heyting algebras, 1.5 1. Complete characterisation of -MacNeille transferability for the class of all Heyting algebras, and 1.4 the class of all Heyting algebras, and 1.3 the class of all bounded latuices, and 1.2 the class of all latuices, and 1.1 : 20

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