tense mv algebras and related functions
play

Tense MV-algebras and related functions Jan Paseka Michal Botur - PowerPoint PPT Presentation

Nov 16, 2022 •370 likes •1.59k views

Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Tense MV-algebras

  1. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  2. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  3. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  4. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  5. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  6. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  7. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  8. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  9. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  10. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  11. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  12. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  13. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  14. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  15. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 12 / 42

  16. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Dyadic numbers and MV-terms The set D of dyadic numbers is the set of the rational numbers that can be written as a finite sum of power of 2. If a is a number of [ 0 , 1 ] , a dyadic decomposition of a is a sequence a ∗ = ( a i ) i ∈ N of elements of { 0 , 1 } such that a = ∑ ∞ i = 1 a i 2 − i . We denote by a ∗ i the i th element of any sequence (of length greater than i ) a ∗ . If a is a dyadic number of [ 0 , 1 ] , then a admits a unique finite dyadic decomposition, called the dyadic decomposition of a . If a ∗ is a dyadic decomposition of a real a and if k is a positive integer then we denote by � a ∗ � k the finite sequence ( a 1 ,..., a k ) defined by the first k elements of a ∗ and by � a ∗ � k the dyadic number ∑ k i = 1 a i 2 − i . Operators on MV-algebras 13 / 42

  17. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Dyadic numbers and MV-terms Definition (Ostermann,Teheux) We denote by f 0 ( x ) and f 1 ( x ) the terms x ⊕ x and x ⊙ x respectively, and by T D the clone generated by f 0 ( x ) and f 1 ( x ) . We also denote by g . the mapping between the set of finite sequences of elements of { 0 , 1 } (and thus of dyadic numbers in [ 0 , 1 ] ) and T D defined by: g ( a 1 ,..., a k ) = f a k ◦···◦ f a 1 i = 1 a i 2 − i , we for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } . If a = ∑ k sometimes write g a instead of g ( a 1 ,..., a k ) . We also denote, for a dyadic number a ∈ D ∩ [ 0 , 1 ) and a positive integer k ∈ N such that 2 − k ≤ 1 − a , by l ( a , k ) : [ a , a + 2 − k ] → [ 0 , 1 ] a linear function defined as follows l ( a , k )( x ) = 2 k ( x − a ) for all x ∈ [ a , a + 2 − k ] . Operators on MV-algebras 14 / 42

  18. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  19. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  20. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  21. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions t r on unit interval [ 0 , 1 ] Example ( 0 , 1 ) ( 0 . 5 , 1 ) ( 1 , 1 ) ( 0 , 1 ) ( 0 . 75 , 1 ) ( 1 , 1 ) ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ( 0 , 0 ) ( 1 , 0 ) ( 0 , 0 ) ( 0 . 5 , 0 ) ( 1 , 0 ) Function t 1 2 = g 0 Function t 3 4 = g 10 Operators on MV-algebras 16 / 42

  22. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 17 / 42

  23. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Lemma Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M such that s ( x ) = 1 . Then, for any n ∈ N , n > 1 , n × x = 1 . Proposition Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M . Then s ( x ) = 1 iff t r ( x ) = 1 for all r ∈ ( 0 , 1 ) ∩ D . Equivalently, s ( x ) < 1 iff there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) � = 1 . In this case, s ( x ) < r . Operators on MV-algebras 18 / 42

  24. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Lemma Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M such that s ( x ) = 1 . Then, for any n ∈ N , n > 1 , n × x = 1 . Proposition Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M . Then s ( x ) = 1 iff t r ( x ) = 1 for all r ∈ ( 0 , 1 ) ∩ D . Equivalently, s ( x ) < 1 iff there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) � = 1 . In this case, s ( x ) < r . Operators on MV-algebras 18 / 42

  25. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Proposition Let M be an MV-algebra, x ∈ M and F be any filter of M . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < 1 if and only if there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) / ∈ F . Corollary Let M be an MV-algebra, x ∈ M and F be any filter of M such that t r ( x ) / ∈ F for some dyadic number r ∈ ( 0 , 1 ) ∩ D . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < r < 1 . Operators on MV-algebras 19 / 42

  26. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Proposition Let M be an MV-algebra, x ∈ M and F be any filter of M . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < 1 if and only if there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) / ∈ F . Corollary Let M be an MV-algebra, x ∈ M and F be any filter of M such that t r ( x ) / ∈ F for some dyadic number r ∈ ( 0 , 1 ) ∩ D . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < r < 1 . Operators on MV-algebras 19 / 42

  27. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 20 / 42

  28. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  29. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  30. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  31. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  32. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  33. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  34. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  35. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  36. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  37. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  38. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  39. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  40. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  41. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  42. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  43. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  44. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  45. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  46. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  47. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  48. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  49. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  50. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  51. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  52. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 26 / 42

  53. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  54. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  55. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  56. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  57. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  58. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  59. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  60. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  61. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  62. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  63. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  64. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  65. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  66. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  67. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  68. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  69. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  70. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  71. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  72. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras I By a frame is meant a triple ( S , T , R ) where S , T are non-void sets and R ⊆ S × T . Having an MV-algebra M = ( M ; ⊕ , ⊙ , ¬ , 0 , 1 ) and a non-void set T , we can produce the direct power M T = ( M T ; ⊕ , ⊙ , ¬ , o , j ) where the operations ⊕ , ⊙ and ¬ are defined and evaluated on p , q ∈ M T componentwise. Moreover, o , j are such elements of M T that o ( t ) = 0 and j ( t ) = 1 for all t ∈ T . The direct power M T is again an MV-algebra. Operators on MV-algebras 30 / 42

  73. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras I By a frame is meant a triple ( S , T , R ) where S , T are non-void sets and R ⊆ S × T . Having an MV-algebra M = ( M ; ⊕ , ⊙ , ¬ , 0 , 1 ) and a non-void set T , we can produce the direct power M T = ( M T ; ⊕ , ⊙ , ¬ , o , j ) where the operations ⊕ , ⊙ and ¬ are defined and evaluated on p , q ∈ M T componentwise. Moreover, o , j are such elements of M T that o ( t ) = 0 and j ( t ) = 1 for all t ∈ T . The direct power M T is again an MV-algebra. Operators on MV-algebras 30 / 42

  74. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras II Theorem Let M be a linearly ordered complete MV-algebra, ( S , T , R ) be a frame and G ∗ be a map from M T into M S defined by G ∗ ( p )( s ) = � { p ( t ) | t ∈ T , sRt } , for all p ∈ M T and s ∈ S . Then G ∗ is a strong fm-function between MV-algebras which has a left adjoint P ∗ . In this case, for all q ∈ M S and t ∈ T , P ∗ ( q )( t ) = � { q ( s ) | s ∈ T , sRt } and P ∗ : ( M S ) op → ( M T ) op is a strong fm-function between MV-algebras. We say that G ∗ : M T → M S is the canonical strong fm-function between MV-algebras induced by the frame ( S , T , R ) and the MV-algebra M . Operators on MV-algebras 31 / 42

  75. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras II Theorem Let M be a linearly ordered complete MV-algebra, ( S , T , R ) be a frame and G ∗ be a map from M T into M S defined by G ∗ ( p )( s ) = � { p ( t ) | t ∈ T , sRt } , for all p ∈ M T and s ∈ S . Then G ∗ is a strong fm-function between MV-algebras which has a left adjoint P ∗ . In this case, for all q ∈ M S and t ∈ T , P ∗ ( q )( t ) = � { q ( s ) | s ∈ T , sRt } and P ∗ : ( M S ) op → ( M T ) op is a strong fm-function between MV-algebras. We say that G ∗ : M T → M S is the canonical strong fm-function between MV-algebras induced by the frame ( S , T , R ) and the MV-algebra M . Operators on MV-algebras 31 / 42

  76. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 32 / 42

  77. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  78. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  79. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  80. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  81. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  82. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  83. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  84. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  85. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  86. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

  87. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

  88. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

Recommend

(3) (1) (2) 2 TENSE STUDY VOCABULARY TENSE STUDY IMMEDIATE FUTURE USAGE

(3) (1) (2) 2 TENSE STUDY VOCABULARY TENSE STUDY IMMEDIATE FUTURE USAGE

TENSE (3) (1) (2) 2 TENSE STUDY VOCABULARY TENSE STUDY IMMEDIATE FUTURE USAGE STRUCTURE AUXILIARY EXAMPLES EXERCISES 3 4 um tempo verbal que expressa uma ao que poderia acontecer no futuro,

648 views • 16 slides
Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Notre Dame 2015 Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c University of Zagreb, Croatia Supported by CSF, grant. no. 2634 Conference on Lie algebras, vertex operator algebras and related

586 views • 39 slides
Connections between centrality and local monotonicity of certain functions on C -algebras

Connections between centrality and local monotonicity of certain functions on C -algebras

Introduction The main theorem Connections between centrality and local monotonicity of certain functions on C -algebras Dniel Virosztek Budapest University of Technology and Economics and MTA-DE &quot;Lendlet&quot; Functional Analysis

335 views • 21 slides
Algebras of Generalized Functions and Nonstandard Analysis Hans Vernaeve (joint work with Todor

Algebras of Generalized Functions and Nonstandard Analysis Hans Vernaeve (joint work with Todor

Algebras of Generalized Functions and Nonstandard Analysis Hans Vernaeve (joint work with Todor Todorov) University of Innsbruck June 2008 Generalized Functions and N.S.A. . (Hans Vernaeve) 1 / 15 Generalized functions: introduction and

380 views • 21 slides
Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun

Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun

Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras Yuqun Chen (Joint work with L.A. Bokut) South China Normal University, China The 4th International Workshop on Differential Algebra and Related Topics,

1.21k views • 87 slides
Past tense, present tense or future? 1. Mr Grinling is rowing to the lighthouse. 2. Mr Grinling

Past tense, present tense or future? 1. Mr Grinling is rowing to the lighthouse. 2. Mr Grinling

Day 3 past and present tense.notebook June 04, 2020 Past tense, present tense or future? 1. Mr Grinling is rowing to the lighthouse. 2. Mr Grinling rowed to the lighthouse. 3. Tomorrow Mr Grinling will row to the lighthouse. Day 3 past and

622 views • 5 slides
FIXED POINT AND RELATED GEOMETRIC PROPERTIES ON THE FOURIER AND FOURIER STIELTJES ALGEBRAS OF

FIXED POINT AND RELATED GEOMETRIC PROPERTIES ON THE FOURIER AND FOURIER STIELTJES ALGEBRAS OF

FIXED POINT AND RELATED GEOMETRIC PROPERTIES ON THE FOURIER AND FOURIER STIELTJES ALGEBRAS OF LOCALLY COMPACT GROUPS Anthony To-Ming Lau (University of Alberta) International Conference on Abstract Harmonic Analysis Granada, Spain May 20 -

806 views • 44 slides
In search of meaning: Semantic effects on past-tense inflection Rebecca Butler, Karalyn Patterson

In search of meaning: Semantic effects on past-tense inflection Rebecca Butler, Karalyn Patterson

In search of meaning: Semantic effects on past-tense inflection Rebecca Butler, Karalyn Patterson and Anna M. Woollams -- Nehchal Jindal (Y9366) Basics Past-tense inflection (or simply inflection) Given present-tense verb, convert it to

400 views • 9 slides
Amalgamated algebras along an ideal: a class of ring extensions related to Nagatas

Amalgamated algebras along an ideal: a class of ring extensions related to Nagatas

1 2 3 4 5 6 7 8 Amalgamated algebras along an ideal: a class of ring extensions related to Nagatas idealization Marco Fontana Dipartimento di Matematica

1.67k views • 124 slides
Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March 2018 1 We consider finiteness conditions and ques- tions of growth of noncommutative algebras, known as A con s. They appear in M.Wemyss work on

214 views • 17 slides
Presentation Exercise: Chapter 10 What are the three components of a present-tense verb in the

Presentation Exercise: Chapter 10 What are the three components of a present-tense verb in the

Presentation Exercise: Chapter 10 What are the three components of a present-tense verb in the fourth conjugation? ____________________ ______________________ ____________________________ Fill in the chart below with the present-tense forms of

441 views • 5 slides
Hardy Algebras, Berezin Transform and Taylors Taylor Series Paul Muhly and Baruch Solel

Hardy Algebras, Berezin Transform and Taylors Taylor Series Paul Muhly and Baruch Solel

Introduction The algebras Representations Family of functions without a generator Maps Hardy Algebras, Berezin Transform and Taylors Taylor Series Paul Muhly and Baruch Solel Banach Algebras 2013, Goteborg, Sweden Introduction The

235 views • 22 slides
Functions as relations A relation is a function iff each element of its domain is related to at

Functions as relations A relation is a function iff each element of its domain is related to at

Functions 1 Myrto Arapinis School of Informatics University of Edinburgh September 29, 2014 1 Slides mainly borrowed from Richard Mayr 1 / 15 Functions as relations A relation is a function iff each element of its domain is related to at

1k views • 65 slides
Annotating tense in a tenseless language Nianwen Xue, Hua Zhong and Kai-Yun Chen

Annotating tense in a tenseless language Nianwen Xue, Hua Zhong and Kai-Yun Chen

Annotating tense in a tenseless language Nianwen Xue, Hua Zhong and Kai-Yun Chen Computational Language and Education Center University of Colorado at Boulder May 28, 2008 LREC 2008 Marrakech, Morocco 2 Inferring Chinese tense

1k views • 21 slides
Today Morphology of verbs, contd Derivational vs. Inflectional Tense &amp; Aspect

Today Morphology of verbs, contd Derivational vs. Inflectional Tense &amp; Aspect

Today Morphology of verbs, contd Derivational vs. Inflectional Tense &amp; Aspect present vs. past perfect vs. progressive Semantic vs. syntactic tense/aspect Reading: pp. 117-122 LING 100 -- McGarrity 1

614 views • 26 slides
A betting metaphor for belief functions on MV-algebras and fuzzy epistemic states Tommaso Flaminio

A betting metaphor for belief functions on MV-algebras and fuzzy epistemic states Tommaso Flaminio

A betting metaphor for belief functions on MV-algebras and fuzzy epistemic states Tommaso Flaminio 1 Lluis Godo 2 M ANY V AL 2013 1 DiSTA, University of Insubria, Italy. tommaso.flaminio@uninsubria.it 2 IIIA - CSIC, Campus de la UAB, Spain.

446 views • 27 slides
Current algebras and higher genus CFT partition functions Roberto Volpato Institute for

Current algebras and higher genus CFT partition functions Roberto Volpato Institute for

Current algebras and higher genus CFT partition functions Roberto Volpato Institute for Theoretical Physics ETH Zurich ZURICH, RTN Network 2009 Based on: M. Gaberdiel and R.V., arXiv: 0903.4107 [hep-th] M. Gaberdiel, C. Keller, R.V.,

608 views • 28 slides
Chapter 12: The Perfect Tense Chapter 12 covers the following: the formation of verbs in the

Chapter 12: The Perfect Tense Chapter 12 covers the following: the formation of verbs in the

Chapter 12: The Perfect Tense Chapter 12 covers the following: the formation of verbs in the perfect-tense system, the four principal parts of Latin verbs, and at the end of the lesson, well review the vocabulary which you should memorize in

301 views • 7 slides
Monk algebras, completions, and atom structures Robin Hirsch and Ian Hodkinson Aim To show that

Monk algebras, completions, and atom structures Robin Hirsch and Ian Hodkinson Aim To show that

Monk algebras, completions, and atom structures Robin Hirsch and Ian Hodkinson Aim To show that relation algebras can shed light on duality-related matters. To nail more nails in the cof fi n of the hope that RRA is docile. But also perhaps, to

268 views • 25 slides
Algebras of partial functions Brett McLean Laboratoire J. A. Dieudonn e, CNRS, Universit e

Algebras of partial functions Brett McLean Laboratoire J. A. Dieudonn e, CNRS, Universit e

Algebras of partial functions Brett McLean Laboratoire J. A. Dieudonn e, CNRS, Universit e C ote dAzur brett.mclean@unice.fr 18 October 2018 18 October 2018 1 / 20 Overview of talk Introductory part Complete representations

245 views • 21 slides
An Algebraic Framework for Pseudorandom Functions and Applications to Related-Key Security Michel

An Algebraic Framework for Pseudorandom Functions and Applications to Related-Key Security Michel

An Algebraic Framework for Pseudorandom Functions and Applications to Related-Key Security Michel Abdalla, Fabrice Benhamouda, Alain Passelgue Pseudorandom functions [GGM86] - efficiently computable function : -

1.01k views • 56 slides
QUAIS VERBOS ESTO PRESENTES NA CITAO?QUAL A TRADUO DA CITAO? 2 TENSE STUDY

QUAIS VERBOS ESTO PRESENTES NA CITAO?QUAL A TRADUO DA CITAO? 2 TENSE STUDY

QUAIS VERBOS ESTO PRESENTES NA CITAO?QUAL A TRADUO DA CITAO? 2 TENSE STUDY VOCABULARY TENSE STUDY SIMPLE PAST USAGE STRUCTURE AUXILIARY EXAMPLES EXERCISES ATIVIDADE COMPLEMENTAR 3 4

476 views • 24 slides
Cross-linguistic annotation of tense and aspect syntax and semantics Mark-Matthias Zymla

Cross-linguistic annotation of tense and aspect syntax and semantics Mark-Matthias Zymla

Introduction Temporal annotation A quick overview Comprehensive annotation of the category tense References Cross-linguistic annotation of tense and aspect syntax and semantics Mark-Matthias Zymla University of Konstanz November 22nd,

730 views • 59 slides
Annotation of Tense &amp; Aspect Semantics for Sentential AMR Lucia Donatelli 1 , Michael Regan 2

Annotation of Tense &amp; Aspect Semantics for Sentential AMR Lucia Donatelli 1 , Michael Regan 2

Annotation of Tense &amp; Aspect Semantics for Sentential AMR Lucia Donatelli 1 , Michael Regan 2 , William Croft 2 , &amp; Nathan Schneider 1 Georgetown University 1 , University of New Mexico 2 LAW-MWE-CxG 2018 26 August 2018 Why tense and

849 views • 59 slides

More recommend