A duality-theoretic approach to MTL-algebras

Transcription

1 A duality-theoretic approach to MTL-algebras Sara Ugolini (Joint work with W. Fussner) BLAST Denver, August 6th 2018

2 A commutative, integral residuated lattice, or CIRL, is a structure A = (A,,,,, 1) where: (i) (A,,, 1) is a lattice with top element 1, (ii) (A,, 1) is a commutative monoid, (iii) (, ) is a residuated pair, i.e. it holds for every x, y, z A: CIRLs constitute a variety, RL. x z y iff z x y. Examples: (Z, +,, min, max, 0), ideals of a commutative ring... Sara Ugolini 2/37

3 A bounded CIRL, or BCIRL, is a CIRL A = (A,,,,, 0, 1) with an extra constant 0 that is the least element of the lattice. Examples: Boolean algebras, Heyting algebras... In every BRL we can define further operations and abbreviations: x = x 0, x + y = ( x y), x 2 = x x. Totally ordered structures are called chains. Sara Ugolini 3/37

4 A bounded CIRL, or BCIRL, is a CIRL A = (A,,,,, 0, 1) with an extra constant 0 that is the least element of the lattice. Examples: Boolean algebras, Heyting algebras... In every BRL we can define further operations and abbreviations: x = x 0, x + y = ( x y), x 2 = x x. Totally ordered structures are called chains. A CIRL, or BCIRL, is semilinear (or prelinear, or representable) if it is a subdirect product of chains. We call semilinear CIRLs GMTL-algebras and semilinear BCIRLs MTL-algebras. They constitute varieties that we denote with GMTL and MTL. MTL-algebras are the semantics of Esteva and Godo s MTL, the fuzzy logic of left-continuous t-norms. Sara Ugolini 3/37

5 Priestley duality MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X,, τ), where (X, τ) is a compact topological space, (X, ) is a poset, and for any x y there exists a clopen U X with x U and y / U. Sara Ugolini 4/37

6 Priestley duality MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X,, τ), where (X, τ) is a compact topological space, (X, ) is a poset, and for any x y there exists a clopen U X with x U and y / U. S Pries BDL A Sara Ugolini 4/37

7 Priestley duality MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X,, τ), where (X, τ) is a compact topological space, (X, ) is a poset, and for any x y there exists a clopen U X with x U and y / U. S Pries BDL S(D): prime filters ordered by inclusion with topology generated by {ϕ(d) : d D} {ϕ(d) c : d D} A where ϕ(d) = {X prime filter of D : d X} Sara Ugolini 4/37

8 Priestley duality MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X,, τ), where (X, τ) is a compact topological space, (X, ) is a poset, and for any x y there exists a clopen U X with x U and y / U. S Pries BDL S(D): prime filters ordered by inclusion with topology generated by {ϕ(d) : d D} {ϕ(d) c : d D} A where ϕ(d) = {X prime filter of D : d X} A(X,, τ): collection of clopen upsets (Cl(X),,,, X) Sara Ugolini 4/37

9 Priestley duality Priestley duality admits numerous modifications. E.g. if one or both of the lattice bounds are dropped we obtain a dual category of pointed, or doubly-pointed, (i.e. bounded above or bounded) Priestley spaces. Sara Ugolini 5/37

10 Priestley duality Priestley duality admits numerous modifications. E.g. if one or both of the lattice bounds are dropped we obtain a dual category of pointed, or doubly-pointed, (i.e. bounded above or bounded) Priestley spaces. Moreover, Priestley duality can be extended to distributive residuated lattices. Our approach is essentially drawn from [Galatos, PhD thesis, 2003] and [Urquhart, 1996], however a similar approach to duals of MTL-algebras has been developed by Cabrer and Celani in Usually, the multiplication is dualized as a ternary relation on prime filters. Sara Ugolini 5/37

11 Residuated spaces We call a structure (S, R, E) an unpointed residuated space if S is a Priestley space R is a ternary relation on S, E is a subset of S, for all x, y, z, w, x, y, z S and U, V A(S): (i) R(x, y, u) and R(u, z, w) for some u S if and only if R(y, z, v) and R(x, v, w) for some v S. (ii) If x x, y y, and z z, then R(x, y, z) implies R(x, y, z ). (iii) If R(x, y, z), then there exist U, V A(S) such that x U, y V, and z / R[U, V, ]. (iv) For all U, V A(S), each of R[U, V, ], {z S : R[z, V, ] U}, and {z P : R[B, z, ] U} are clopen. (v) E A(S) and for all U A(S) we have R[U, E, ] = R[E, U, ] = U. Where R[U, V, ] = {z S : ( x U)( y V )(R(x, y, z))}. Sara Ugolini 6/37

12 Residuated spaces If S 1 = (S 1, 1, τ 1, R 1, E 1) and S 2 = (S 2, 2, τ 2, R 2, E 2) are unpointed residuated spaces, a map α: S 1 S 2 is a bounded morphism if: (i) α is a continuous isotone map. (ii) If R 1(x, y, z), then R 2(α(x), α(y), α(z)). (iii) If R 2(u, v, α(z)), then there exist x, y S 1 such that u α(x), v α(y), and R 1(x, y, z). (iv) For all U, V A(S 2) and all x S 1, if R 1[x, α 1 [U], ] α 1 [V ], then R 2[α(x), U, ] V. (v) α 1 [E 2] E 1. We denote the category of unpointed residuated spaces and bounded morphisms by urs. The following is proven in [Galatos, PhD thesis]. Theorem The category of bounded residuated lattices with residuated lattice homomorphisms preserving the lattice bounds is dually equivalent to urs. Sara Ugolini 7/37

13 Extending functors Given an unpointed residuated space S = (S,, τ, R, E), we define A(S) = (A(S,, τ),,, E), where U V = R[U, V, ] U V = {x S : R[x, U, ] V } for U, V A(S,, τ), where R[U, V, ] = {z S : ( x U)( y V )(R(x, y, z))}. Sara Ugolini 8/37

14 Extending functors Given an unpointed residuated space S = (S,, τ, R, E), we define A(S) = (A(S,, τ),,, E), where U V = R[U, V, ] U V = {x S : R[x, U, ] V } for U, V A(S,, τ), where R[U, V, ] = {z S : ( x U)( y V )(R(x, y, z))}. Given a BCIRL A, we define a product on prime filters as the upset of the complex product : a b = (a b) = {z A : x a, y b, xy z} S(A) = (S(D), R, E), where for a bounded residuated lattice A with bounded lattice reduct D, we define a ternary relation R on S(D) and a subset of S(D) by R(a, b, c) iff a b c, E = {a S(D) : 1 a}. Sara Ugolini 8/37

15 Duality for MTL Let A = (A,,,, \, /, 1,, ) be a bounded residuated lattice and S = (S,, τ, R, E) its dual space. A is commutative iff for all x, y, z S, R(x, y, z) iff R(y, x, z). A is integral iff E = S. In the presence of integrality and commutativity, A is semilinear iff for all x, y, z, v, w S, if R(x, y, z) and R(x, v, w), then y w or v z. We denote by MTL τ the full subcategory of urs whose objects satisfy these three conditions. Theorem MTL τ is dually equivalent to MTL. Sara Ugolini 9/37

16 Duality for GMTL Let MTL div be the full subcategory of MTL-algebras without zero divisors and SMTL ind be the full subcategory of directly indecomposable SMTL-algebras. Theorem The categories GMTL, MTL div, and SMTL ind are equivalent. Sara Ugolini 10/37

17 Duality for GMTL Let MTL div be the full subcategory of MTL-algebras without zero divisors and SMTL ind be the full subcategory of directly indecomposable SMTL-algebras. Theorem The categories GMTL, MTL div, and SMTL ind are equivalent. We provide a duality for GMTL, using as a bridge the full subcategory MTL τ div, that is characterized by having a greatest element. Sara Ugolini 10/37

18 Duality for GMTL Let MTL div be the full subcategory of MTL-algebras without zero divisors and SMTL ind be the full subcategory of directly indecomposable SMTL-algebras. Theorem The categories GMTL, MTL div, and SMTL ind are equivalent. We provide a duality for GMTL, using as a bridge the full subcategory MTL τ div, that is characterized by having a greatest element. Thus, let GMTL τ be the category whose objects are of the form (S, R, E, ), where (S, R, E) is an object of MTL τ with a. The morphisms are bounded morphisms that preserve. Theorem GMTL and GMTL τ are dually equivalent. Sara Ugolini 10/37

19 Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation on the underlying Priestley dual of the lattice reduct. Sara Ugolini 11/37

20 Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL. Sara Ugolini 11/37

21 Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL. Lemma Let A be a GMTL-algebra or an MTL-algebra, and let a, b be nonempty filters of A. Then if either one of a or b is prime, we have that either a b is prime or a b = A. Sara Ugolini 11/37

22 Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL. Lemma Let A be a GMTL-algebra or an MTL-algebra, and let a, b be nonempty filters of A. Then if either one of a or b is prime, we have that either a b is prime or a b = A. Corollary If A is an GMTL-algebra, then is a binary operation on S(A). If A is an MTL-algebra, then is gives a partial operation on S(A) and is undefined for a, b S(A) only if a b = A. Sara Ugolini 11/37

23 The following provides a mechanism for defining on MTL τ and GMTL τ. Lemma Let S = (S,, τ, R, E) be an object of MTL τ. If x, y, z S with R(x, y, z), then there exists a least element z S such that R(x, y, z ). If S is in GMTL τ, then for any x, y S there exists a least z S with R(x, y, z ). x y = { min{z S : R(x, y, z)}, undefined, if {z S : R(x, y, z)} = otherwise Sara Ugolini 12/37

24 The following provides a mechanism for defining on MTL τ and GMTL τ. Lemma Let S = (S,, τ, R, E) be an object of MTL τ. If x, y, z S with R(x, y, z), then there exists a least element z S such that R(x, y, z ). If S is in GMTL τ, then for any x, y S there exists a least z S with R(x, y, z ). x y = { min{z S : R(x, y, z)}, undefined, if {z S : R(x, y, z)} = otherwise Lemma Let S be an object of MTL τ or GMTL τ, and let x, y, z S. (i) R(x, y, z) iff x y z. (ii) Each of the following holds (whenever is defined). 1 x (y z) = (x y) z. 2 x y = y x. 3 x y implies that x z y z and z x z y. Sara Ugolini 12/37

25 Moreover, possesses a partial residual. Proposition Let A be an MTL-algebra or GMTL-algebra, and suppose that b, c S(A) are such that there exists a S(A) with a b c. Then there is a greatest such a, and it is given by b c := {a S(A) : a b c}. Moreover, a b c if and only if a b c. Sara Ugolini 13/37

26 Moreover, possesses a partial residual. Proposition Let A be an MTL-algebra or GMTL-algebra, and suppose that b, c S(A) are such that there exists a S(A) with a b c. Then there is a greatest such a, and it is given by b c := {a S(A) : a b c}. Moreover, a b c if and only if a b c. Corollary Let S be an object of MTL τ or GMTL τ, and suppose that y, z S are such that there exists x S with R(x, y, z). Then there is a greatest such x, which we denote by y z. Moreover, x y z if and only if x y z. Sara Ugolini 13/37

27 In order to treat negation, we use the Routley star ([Routley, Meyer ,1973], [Urquhart, 1996]). If A is an MTL-algebra and a S(A), we define a = {a A : a a} It is easy to see that if a is a prime filter, then so is a. Moreover, is an order-reversing operation on prime filters. Sara Ugolini 14/37

28 In order to treat negation, we use the Routley star ([Routley, Meyer ,1973], [Urquhart, 1996]). If A is an MTL-algebra and a S(A), we define a = {a A : a a} It is easy to see that if a is a prime filter, then so is a. Moreover, is an order-reversing operation on prime filters. Lemma Let A be an MTL-algebra and let a S(A). Then a is the largest prime filter of A such that a a A. Corollary Let S be an object of MTL τ, and let x S. Then there exists a greatest y S such that R(x, y, z) for some z S. Equivalently, there exists a greatest y S such that x y is defined. In light of the previous corollary, for an abstract object S of MTL τ we define for any x S, x := max{y S : z S, R(x, y, z)}. Sara Ugolini 14/37

29 We now focus on a special class of MTL-algebras: srdl-algebras, that are the axiomatic extension of MTL by: (x + x) 2 = x 2 + x 2 (DL) (x 2 ) ( x x) = 1 (r) Relevant subvarieties are: product algebras, Gödel algebras, the variety generated by perfect MV-algebras, nilpotent minimum without negation fixpoint, pseudocomplemented MTL algebras... [Cignoli, Torrens ], [Aguzzoli, Flaminio, U ]: srdl is generated by δ-rotations of GMTL-algebras. Sara Ugolini 15/37

30 δ-rotation Let R = (R,,,,, 1) be a GMTL-algebra, and let δ : R R be a wdl-admissible operator: closure operator; δ(x) δ(y) δ(x y) (nucleus on R); δ(x y) = δ(x) δ(y), δ(x y) = δ(x) δ(y). Sara Ugolini 16/37

31 δ-rotation Let R = (R,,,,, 1) be a GMTL-algebra, and let δ : R R be a wdl-admissible operator: closure operator; δ(x) δ(y) δ(x y) (nucleus on R); δ(x y) = δ(x) δ(y), δ(x y) = δ(x) δ(y). Examples: δ D = id, δ L = 1, i.e. δ L(x) = 1 for every x R. Sara Ugolini 16/37

32 δ-rotation Let R = (R,,,,, 1) be a GMTL-algebra, and let δ : R R be a wdl-admissible operator: closure operator; δ(x) δ(y) δ(x y) (nucleus on R); δ(x y) = δ(x) δ(y), δ(x y) = δ(x) δ(y). Examples: δ D = id, δ L = 1, i.e. δ L(x) = 1 for every x R. We define the δ-rotation R δ (R) as the structure with domain ({1} R) ({0} δ[r]) and suitably defined operations. Sara Ugolini 16/37

33 Notable subvarieties With δ = id we get directly indecomposable sidl-algebras (introduced as IBP 0-algebras in [Noguera, Esteva, Gispert ]), i.e. disconnected rotation (as defined by Jenei) of GMTL-algebras. Examples: the variety generated by perfect MV-algebras, NM of nilpotent minimum without negation fixpoint. Sara Ugolini 17/37

34 Notable subvarieties With δ = 1 we get directly indecomposable pseudocomplemented MTL-algebras: algebras A isomorphic to 2 R for some GMTL R. Examples: Gödel algebras (prelinear Heyting algebras), Product algebras. Sara Ugolini 18/37

35 srdl-algebras [Aguzzoli, Flaminio, U ]: srdl-algebras are equivalent to categories whose objects are quadruples (B, R, e, δ): B is a Boolean algebra, R is a GMTL-algebra, δ : R R is wdl-admissible, e : B R R is an external join: (V1) For fixed b B and c R: ν b (x) = b e x is an endomorphism of R, γ c(x) = x e c is a lattice homomorphism from B to R. (V2) ν 0 is the identity on R, ν 1 is constantly equal to 1. (V3) For all b, b B and for all c, c R, ν b (c) ν b (c ) = ν b b (c c ) = ν b (h b (c c )). Sara Ugolini 19/37

36 BRL MVR n MTL rdl HA IMTL srdl widl SRL sidl SMTL NR SHA DLMV P NM G B Sara Ugolini 20/37

37 Dualized construction In analogy to the algebraic side, we expect that if A is an srdl-algebra, S(A) can be reconstructed by: the Stone space associated to its Boolean skeleton B(A); the object in GMTL τ associated to Rad(A); the maps induced on the dual side by the (external) join; the dual of the wdl-admissible operator δ ( of A). Sara Ugolini 21/37

38 Dualized construction In analogy to the algebraic side, we expect that if A is an srdl-algebra, S(A) can be reconstructed by: the Stone space associated to its Boolean skeleton B(A); the object in GMTL τ associated to Rad(A); the maps induced on the dual side by the (external) join; the dual of the wdl-admissible operator δ ( of A). Indeed, we can reconstruct S(A) by F A : sets of pairs (u, y) such that (i) u is an ultrafilter of B(A), (ii) y is a prime lattice filter of Rad(A), (iii) for b B(A), c Rad(A), if b c y, then b u or c y. plus the information given by. Sara Ugolini 21/37

39 Dualized construction Let u, v, w S(B(A)). Aaa Sara Ugolini 22/37

40 Dualized construction Below u: x S(R(A)) that respect the external primality condition wrt u, ordered by inclusion. Sara Ugolini 23/37

41 Dualized construction Same for v, w,... Aaa Sara Ugolini 24/37

42 Dualized construction Rotate upwards the δ-images of the elements below u. Aaa Sara Ugolini 25/37

43 Dualized construction The dualized rotation construction obtained, that we denote SB(A) Υ SR(A) (where = δ 1 and Υ = {ν 1 b } b B(A) ), is isomorphic to S(A). Sara Ugolini 26/37

44 Dualized construction Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Sara Ugolini 27/37

45 Dualized construction Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srdl-algebra A contains a unique ultrafilter of its Boolean skeleton. Sara Ugolini 27/37

46 Dualized construction Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srdl-algebra A contains a unique ultrafilter of its Boolean skeleton. Indeed, let a S(A), then for each u B(A) we have that u u = 1 a. But a is prime, thus u a or u a. We denote the ultrafilter associated to a by u a. Sara Ugolini 27/37

47 Dualized construction Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srdl-algebra A contains a unique ultrafilter of its Boolean skeleton. Indeed, let a S(A), then for each u B(A) we have that u u = 1 a. But a is prime, thus u a or u a. We denote the ultrafilter associated to a by u a. For each ultrafilter u, set S u = {a S(A) : u a = u} and call this the site at u. Sara Ugolini 27/37

48 Dualized construction Let u be an ultrafilter of B(A), y S(Rad(A)). We say that u fixes y if there exists a S u such that y = a Rad(A). Sara Ugolini 28/37

49 Dualized construction Let u be an ultrafilter of B(A), y S(Rad(A)). We say that u fixes y if there exists a S u such that y = a Rad(A). For every b B(A), let µ b (y) = ν 1 b [y] = {x Rad(A) : b x y}. Sara Ugolini 28/37

50 Dualized construction Let u be an ultrafilter of B(A), y S(Rad(A)). We say that u fixes y if there exists a S u such that y = a Rad(A). For every b B(A), let µ b (y) = ν 1 b [y] = {x Rad(A) : b x y}. Lemma Let A be an srdl-algebra, u be an ultrafilter of B(A), and y S(Rad(A)). The following are equivalent. For every b B(A) and c Rad(A), if b c y, then b u or c y. u fixes y. µ b (y) = y for each b / u. Sara Ugolini 28/37

51 Example Let us consider Chang MV-algebra C, with domain C = {0, c,..., nc,..., 1 nc,..., 1 c, 1}. C + = {1, 1 c,..., 1 nc,...} is isomorphic to Z, and C is isomorphic to the disconnected rotation of Z. C generates the variety of perfect MV-algebras. Sara Ugolini 29/37

52 Example Let us consider Chang MV-algebra C, with domain C = {0, c,..., nc,..., 1 nc,..., 1 c, 1}. C + = {1, 1 c,..., 1 nc,...} is isomorphic to Z, and C is isomorphic to the disconnected rotation of Z. C generates the variety of perfect MV-algebras. Consider C 2 = C C: (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (mc, 0) (0, nc) (0, 0) Sara Ugolini 29/37

53 (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (mc, 0) (0, nc) (0, 0) The Boolean skeleton of C 2 is the Boolean algebra of 4 elements B(C 2 ) = {(0, 0), (0, 1), (1, 0), (1, 1)}. The radical R(C 2 ) is isomorphic to Z Z, and is the upper square. Sara Ugolini 30/37

54 We want pairs (u, x) such that for every b / u, (1, 1) {y R(C 2 ) : b y x} = x r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (mc, 0) (0, nc) (0, 0) Sara Ugolini 31/37

55 We want pairs (u, x) such that for every b / u, (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) {y R(C 2 ) : b y x} = x Let u 1 be the Boolean ultrafilter generated by (1, 0), u 2 the one generated by (0, 1), Z 1 = {(1, y) : y C + }, and Z 2 = {(x, 1) : x C + } (mc, 0) (0, nc) (0, 0) Sara Ugolini 31/37

56 We want pairs (u, x) such that for every b / u, (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) {y R(C 2 ) : b y x} = x Let u 1 be the Boolean ultrafilter generated by (1, 0), u 2 the one generated by (0, 1), Z 1 = {(1, y) : y C + }, and Z 2 = {(x, 1) : x C + } We get: (u 2, [r n)) and (u 1, [r m)). (mc, 0) (0, nc) (0, 0) Sara Ugolini 31/37

57 We want pairs (u, x) such that for every b / u, (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) {y R(C 2 ) : b y x} = x Let u 1 be the Boolean ultrafilter generated by (1, 0), u 2 the one generated by (0, 1), Z 1 = {(1, y) : y C + }, and Z 2 = {(x, 1) : x C + } We get: (u 2, [r n)) and (u 1, [r m)). (mc, 0) (0, nc) (u 1, R(C 2 )) (u 2, R(C 2 )) (0, 0) (u 1, [r m)) (u 2, [r n)) (u 1, Z 1) (u 2, Z 2) Sara Ugolini 31/37

58 (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (u 1, δ[z 1]) (u 1, δ[[r m)]) (u 1, R(C 2 )) (u 1, [r m)) (u 2, δ[z 2]) (u 2, δ[[r n)]) (u 2, R(C 2 )) (u 2, [r n)) (mc, 0) (0, nc) (u 1, Z 1) (u 2, Z 2) (0, 0) Sara Ugolini 32/37

59 (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (u 1, δ[z 1]) (u 1, δ[[r m)]) (u 1, R(C 2 )) (u 1, [r m)) (u 2, δ[z 2]) (u 2, δ[[r n)]) (u 2, R(C 2 )) (u 2, [r n)) (mc, 0) (0, nc) (u 1, Z 1) (u 2, Z 2) (0, 0) Sara Ugolini 33/37

60 (1, 1) r m r n (1, 0) (0, 1) (1 mc, 0) (0, 1 nc) (u 1, δ[z 1]) (u 1, δ[[r m)]) (u 1, R(C 2 )) (u 1, [r m)) (u 2, δ[z 2]) (u 2, δ[[r n)]) (u 2, R(C 2 )) (u 2, [r n)) (mc, 0) (0, nc) (u 1, Z 1) (u 2, Z 2) (0, 0) Sara Ugolini 34/37

61 Dualized quadruples Let a dual quadruple be a structure (S, X, Υ, ) where (i) S is a Stone space; (ii) X is in GMTL τ ; (iii) Υ = {υ U } U A(S) is an indexed family of GMTL τ -morphisms υ U : X X such that the map e : A(S) A(X) A(X) defined by is an external join; e(u, X) = υ 1 U [X] (iv) : X X is a continuous closure operator such that R(x, y, z) implies R( x, y, z). Sara Ugolini 35/37

62 Dualized quadruples Given a dual quadruple (S, X, Υ, ), let S Υ X be the structure obtained by the dual rotation construction, with suitably defined topology and multiplication. Theorem Let (S, X, Υ, ) be a dual quadruple. Then S Υ X is the extended Priestley dual of some srdl-algebra. Theorem Let Y be the extended Priestley dual of an srdl-algebra. Then there exists a dual quadruple (S, X, Υ, ) such that Y = S Υ X. Sara Ugolini 36/37

63 Thank you! Sara Ugolini 37/37