Overview Classical examples In many examples but not all, T is a monoid. 1 A is an integral domain and T = A \ { 0 } ; 2 A is a graded algebra, and T is the monoid of homogeneous elements. 3 A is a vector space with base T . 4 More specifically, A is an algebra with a multiplicative base T . This could be viewed in terms of the previous slide. For example, A could be the group algebra of a group T . 5 A is a Hopf algebra and T is a special subset (such as the group-like elements or primitive elements). 6 A is the set of class functions from a finite group to a field F ; T 0 is the sub-semiring of characters. (This can be generalized to table algebras.) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 11 / 75
Overview Two non-classical examples Our interest however was stimulated by examples outside of classical algebra. Before delving into the theory, we consider two of the main examples, postponing the others until we develop some theory: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 12 / 75
Overview The max-plus algebra The parent structure in tropical algebra, which also arises in varied contexts in applied mathematics, is the well-known max-plus algebra on an ordered monoid, where multiplication is the old addition, and addition is the maximum. We append the subscript max to indicate the corresponding max-plus algebra, e.g., N max or Q max . Specifically, ordered groups, such as ( Q , +) or ( R , +), are viewed at once as max-plus semifields † , generalizing to the following elegant observation of Green: (To emphasize the algebraic structure we still use the usual algebraic notation of · and + throughout.) Any ordered monoid ( M , · ) gives rise to a bipotent semiring † , where we define a + b to be max { a , b } . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 13 / 75
Overview Puiseux series and tropicalization For a structure A of a given signature in universal algebra, one can define the set A = A{{ t }} of Puiseux series on the variable t , which is the set of formal series of the form f = � ∞ k = ℓ c k t k / N where N ∈ N , ℓ ∈ Z , and c k ∈ S , with the convolution product. Then one has the Puiseux valuation val : A{{ t }} \ { 0 } → Q ⊂ R defined by val( f ) = − min c k � =0 { k / N } , (1) which we also call tropicalization . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 14 / 75
Overview Puiseux series and tropicalization For a structure A of a given signature in universal algebra, one can define the set A = A{{ t }} of Puiseux series on the variable t , which is the set of formal series of the form f = � ∞ k = ℓ c k t k / N where N ∈ N , ℓ ∈ Z , and c k ∈ S , with the convolution product. Then one has the Puiseux valuation val : A{{ t }} \ { 0 } → Q ⊂ R defined by val( f ) = − min c k � =0 { k / N } , (1) which we also call tropicalization . Customarily the target Q of − val has been viewed as the max-plus algebra, but this is inaccurate. Although − val( f ) − val( g ) = max {− val( f ) , − val( g ) } when − val( f ) � = − val( g ) , this can fail when − val( f ) = − val( g ), due to cancelation in the lowest terms of f and g . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 14 / 75
Overview Puiseux series and tropicalization For a structure A of a given signature in universal algebra, one can define the set A = A{{ t }} of Puiseux series on the variable t , which is the set of formal series of the form f = � ∞ k = ℓ c k t k / N where N ∈ N , ℓ ∈ Z , and c k ∈ S , with the convolution product. Then one has the Puiseux valuation val : A{{ t }} \ { 0 } → Q ⊂ R defined by val( f ) = − min c k � =0 { k / N } , (1) which we also call tropicalization . Customarily the target Q of − val has been viewed as the max-plus algebra, but this is inaccurate. Although − val( f ) − val( g ) = max {− val( f ) , − val( g ) } when − val( f ) � = − val( g ) , this can fail when − val( f ) = − val( g ), due to cancelation in the lowest terms of f and g . For example, if f = 2 λ 2 + 7 λ 4 and g = − 2 λ 2 + 5 λ 3 + 7 λ 4 then f + g = 5 λ 3 + 14 λ 4 and v ( f + g ) = 3 > 2 . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 14 / 75
Overview Thus, valuations behave like the min-plus algebra EXCEPT perhaps when evaluated on elements having the same value. Hence, tropicalization is not functorial! We need a replacement to the max-plus which is almost bipotent, in the sense that a + b = max { a , b } except for a = b . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 15 / 75
Overview Supertropical semirings † and supertropical domains † To remedy this, we recall some basics of supertropical algebra. Definition: A ν -semiring † is a quadruple R := ( R , T , G , ν ) where R is a semiring † , T is a submonoid, and G ⊂ R is a semiring † ideal, with a multiplicative monoid homomorphism ν : R → G , satisfying ν 2 = ν as well as the condition: a + b = ν ( a ) whenever ν ( a ) = ν ( b ) . R is called a supertropical semiring † when ν is onto, G is ordered, and a + b = a whenever ν ( a ) > ν ( b ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 16 / 75
Overview Supertropical semirings † and supertropical domains † To remedy this, we recall some basics of supertropical algebra. Definition: A ν -semiring † is a quadruple R := ( R , T , G , ν ) where R is a semiring † , T is a submonoid, and G ⊂ R is a semiring † ideal, with a multiplicative monoid homomorphism ν : R → G , satisfying ν 2 = ν as well as the condition: a + b = ν ( a ) whenever ν ( a ) = ν ( b ) . R is called a supertropical semiring † when ν is onto, G is ordered, and a + b = a whenever ν ( a ) > ν ( b ) . (Attention focuses on supertropical semirings † , but the more general definition of ν -semiring † enables one to work with polynomials and matrices.) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 16 / 75
Overview The elements of G are called ghost elements and ν : R → G is called the ghost map . T is the monoid of tangible elements , and encapsulates the tropical aspect. A supertropical semiring † R is called a supertropical domain † when the multiplicative monoid ( R , · ) is commutative, ν | T is 1:1, and R is cancellative. In this case ν | T : T → G is a monoid isomorphism, and T inherits the order from G . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 17 / 75
Overview The standard supertropical semifield † is A := T ∪ T ν (where customarily T = Q max or R max ). Addition is now given by ν ( a ) whenever a = b , a + b = a whenever a > b , b whenever a < b . The standard supertropical semifield is the standard supertropical semifield † with 0 adjoined. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 18 / 75
Overview The standard supertropical semifield † is A := T ∪ T ν (where customarily T = Q max or R max ). Addition is now given by ν ( a ) whenever a = b , a + b = a whenever a > b , b whenever a < b . The standard supertropical semifield is the standard supertropical semifield † with 0 adjoined. Thus, we start with T and pass to the standard supertropical semifield † A . This is our main model for the tropical theory. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 18 / 75
Overview Our overall goal in this talk is to provide an algebraic umbrella, especially to tropical mathematics and related areas, in a general framework which includes as much of the classical theory as possible, with the goal of addressing the following basic questions: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 19 / 75
Overview What is the basic algebraic structure on which to pin our theory? What is a variety in this framework? (We would want an algebraic definition that matches geometric intuition.) Can the definition be made natural, in the sense that it commutes with tropicalization? What is the dimension of a variety? How can we develop linear algebra to obtain analogs of the main theorems of classical matrix theory? How does one algebraically define basic geometric invariants such as resultants, discriminants, genus, etc.? How should representation theory take shape? What are the analogs of the classical groups, exterior algebras, and Lie algebras for example? Is there a version of module theory which handles direct sum decompositions of submodules of free modules, that could support a homological theory? Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 20 / 75
Overview Negation maps There is a basic difficulty in developing the structure theory of semirings in place of rings: Cosets need not be disjoint (this fact relying on additive cancelation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 21 / 75
Overview Negation maps There is a basic difficulty in developing the structure theory of semirings in place of rings: Cosets need not be disjoint (this fact relying on additive cancelation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). In order to overcome partially the lack of negatives, we introduce a formal negation map a �→ ( − ) a which satisfies all of the properties of negation except a + (( − ) a ) = 0. A negation map ( − ) is an additive homomorphism ( − ) : ( A , +) → ( A , +) of order ≤ 2 , written a �→ ( − ) a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 21 / 75
Overview Negation maps There is a basic difficulty in developing the structure theory of semirings in place of rings: Cosets need not be disjoint (this fact relying on additive cancelation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). In order to overcome partially the lack of negatives, we introduce a formal negation map a �→ ( − ) a which satisfies all of the properties of negation except a + (( − ) a ) = 0. A negation map ( − ) is an additive homomorphism ( − ) : ( A , +) → ( A , +) of order ≤ 2 , written a �→ ( − ) a . When A has multiplication we also require ( − )( a 1 a 2 ) = (( − ) a 1 ) a 2 = a 1 (( − ) a 2 ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 21 / 75
Overview Negation maps There is a basic difficulty in developing the structure theory of semirings in place of rings: Cosets need not be disjoint (this fact relying on additive cancelation, which fails in the max-plus algebra, since 1 + 3 = 2 + 3 = 3). In order to overcome partially the lack of negatives, we introduce a formal negation map a �→ ( − ) a which satisfies all of the properties of negation except a + (( − ) a ) = 0. A negation map ( − ) is an additive homomorphism ( − ) : ( A , +) → ( A , +) of order ≤ 2 , written a �→ ( − ) a . When A has multiplication we also require ( − )( a 1 a 2 ) = (( − ) a 1 ) a 2 = a 1 (( − ) a 2 ) . We view ( − ) as a unary operator in universal algebra, and require that it preserves the other linear operators. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 21 / 75
Overview The usual negation in classical algebra is clearly a negation map, but in non-classical situations we lack negatives. In particular, negation is notably absent in the tropical theory, but is circumvented in two main ways: The identity itself is a perfectly valid negation map (since one just erases the minus signs). One can introduce a negation map through the process of “symmetrization,” based on the classical way of constructing Z from N , by taking ordered pairs ( m , n ) and modding out the equivalence identifying ( m 1 , n 1 ) and ( m 2 , n 2 ) when m 1 + n 2 = m 2 + n 1 . Here we exploit the same equivalence but do not mod out by it (since everything would degenerate). Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 22 / 75
Overview Define e = 1 ◦ = 1( − )1 , e ′ = e + 1 . (2) Also we define 1 = 1 , and inductively n + 1 = n + 1. The negation map ( − ) is said to be of the first kind if ( − )1 = 1 (and thus ( − ) is the identity), and of the second kind if ( − ) a � = a for all a ∈ T . When we have cancelation, it is enough to check whether or not ( − )1 = 1. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 23 / 75
Overview Define e = 1 ◦ = 1( − )1 , e ′ = e + 1 . (2) Also we define 1 = 1 , and inductively n + 1 = n + 1. The negation map ( − ) is said to be of the first kind if ( − )1 = 1 (and thus ( − ) is the identity), and of the second kind if ( − ) a � = a for all a ∈ T . When we have cancelation, it is enough to check whether or not ( − )1 = 1. We write a ( − ) b for a + (( − ) b ) , and a = ( ± b ) when a = b or a = ( − ) b . Given a ∈ A we define the quasi-zero a ◦ := a ( − ) a , and A ◦ = { a ◦ : a ∈ A} . ( − ) a is called the quasi-negative of a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 23 / 75
Overview Define e = 1 ◦ = 1( − )1 , e ′ = e + 1 . (2) Also we define 1 = 1 , and inductively n + 1 = n + 1. The negation map ( − ) is said to be of the first kind if ( − )1 = 1 (and thus ( − ) is the identity), and of the second kind if ( − ) a � = a for all a ∈ T . When we have cancelation, it is enough to check whether or not ( − )1 = 1. We write a ( − ) b for a + (( − ) b ) , and a = ( ± b ) when a = b or a = ( − ) b . Given a ∈ A we define the quasi-zero a ◦ := a ( − ) a , and A ◦ = { a ◦ : a ∈ A} . ( − ) a is called the quasi-negative of a . A semigroup ( A , +) has characteristic k > 0 if k + 1 = 1 with k ≥ 1 minimal. A has characteristic 0 if A does not have characteristic k for any k ≥ 1 . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 23 / 75
Overview Define e = 1 ◦ = 1( − )1 , e ′ = e + 1 . (2) Also we define 1 = 1 , and inductively n + 1 = n + 1. The negation map ( − ) is said to be of the first kind if ( − )1 = 1 (and thus ( − ) is the identity), and of the second kind if ( − ) a � = a for all a ∈ T . When we have cancelation, it is enough to check whether or not ( − )1 = 1. We write a ( − ) b for a + (( − ) b ) , and a = ( ± b ) when a = b or a = ( − ) b . Given a ∈ A we define the quasi-zero a ◦ := a ( − ) a , and A ◦ = { a ◦ : a ∈ A} . ( − ) a is called the quasi-negative of a . A semigroup ( A , +) has characteristic k > 0 if k + 1 = 1 with k ≥ 1 minimal. A has characteristic 0 if A does not have characteristic k for any k ≥ 1 . Any idempotent algebra has “characteristic 1,” leading to the notion of “ F 1 geometry.’ Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 23 / 75
Overview Symmetrized T -monoid modules Although the max-plus algebra and its modules initially lack negation, one obtains negation maps of second kind for them through the next main idea, the symmetrization process, obtained by Gaubert (1992) in his dissertation, where an algebraic structure is embedded into a doubled structure with a natural negation map. A to be A (2) = A × A , with Given any T -monoid module A , define � componentwise addition. Also define � T = ( T × { 0 } ) ∪ ( { 0 } × T ) with multiplication � T × � A → � A given by ( a 0 , a 1 )( b 0 , b 1 ) = ( a 0 b 0 + a 1 b 1 , a 0 b 1 + a 0 b 1 ) . We also define a negation map given by the “switch” ( − )( a 0 , a 1 ) = ( a 1 , a 0 ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 24 / 75
Overview Symmetrized T -monoid modules Although the max-plus algebra and its modules initially lack negation, one obtains negation maps of second kind for them through the next main idea, the symmetrization process, obtained by Gaubert (1992) in his dissertation, where an algebraic structure is embedded into a doubled structure with a natural negation map. A to be A (2) = A × A , with Given any T -monoid module A , define � componentwise addition. Also define � T = ( T × { 0 } ) ∪ ( { 0 } × T ) with multiplication � T × � A → � A given by ( a 0 , a 1 )( b 0 , b 1 ) = ( a 0 b 0 + a 1 b 1 , a 0 b 1 + a 0 b 1 ) . We also define a negation map given by the “switch” ( − )( a 0 , a 1 ) = ( a 1 , a 0 ) . � T is a monoid (resp. group) whenever T is. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 24 / 75
Overview This is reminiscent of the familiar construction of Z from N , where ( m , n ) is identified with − ( n , m ). Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 25 / 75
Overview This is reminiscent of the familiar construction of Z from N , where ( m , n ) is identified with − ( n , m ). In particular, � N is itself a semiring with negation given by ( − )( m , n ) = ( n , m ), which we call Z . The difference from the construction of Z from N , is that here we distinguish ( m , n ) from ( m + k , n + k ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 25 / 75
Overview This is reminiscent of the familiar construction of Z from N , where ( m , n ) is identified with − ( n , m ). In particular, � N is itself a semiring with negation given by ( − )( m , n ) = ( n , m ), which we call Z . The difference from the construction of Z from N , is that here we distinguish ( m , n ) from ( m + k , n + k ) . When A has multiplication, � A looks like a superalgebra, in the sense that one defines multiplication ( a 0 , a 1 )( b 0 , b 1 ) = ( a 0 b 0 + a 1 b 1 , a 0 b 1 + a 1 b 1 ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 25 / 75
Overview This is reminiscent of the familiar construction of Z from N , where ( m , n ) is identified with − ( n , m ). In particular, � N is itself a semiring with negation given by ( − )( m , n ) = ( n , m ), which we call Z . The difference from the construction of Z from N , is that here we distinguish ( m , n ) from ( m + k , n + k ) . When A has multiplication, � A looks like a superalgebra, in the sense that one defines multiplication ( a 0 , a 1 )( b 0 , b 1 ) = ( a 0 b 0 + a 1 b 1 , a 0 b 1 + a 1 b 1 ) . Any congruence can be viewed naturally as a substructure of � A . D. Joo and K. Mincheva have used this to good effect in defining prime congruences, and their definition generalizes to triples. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 25 / 75
Overview Modified symmetrized T -monoid modules Here is an alternative version, due to Gaubert. Given any ordered monoid ( G , · ), define � G to be the subset of ( G ∪ { 0 } ) × ( G ∪ { 0 } ) generated by G × { 0 } , { 0 } × G and G × G , with componentwise multiplication and addition dominated by the larger component. For example, � ( a , a ) if a ≥ b ( a , a ) + ( b , 0) = ( b , 0) if a < b . Define T = G× , { 0 } , { 0 } × G , and ( − )( a , 0) = (0 , a ) , ( − )( a , a ) = ( a , a ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 26 / 75
Overview Digression: Imposing distributivity There is a cheap but useful way to give A a distributive multiplication, in cases where distributivity is lacking (as we shall see in some hyperfields). Theorem: Any T -module A can be made (uniquely) into a semiring † via the multiplication � �� � � = a i b j a i b j . i j i , j For the proof, it suffices to show that this is well-defined, i.e., if � i a i = � i then � i , j a i b j = � i a ′ i , j a ′ i b j (and likewise for b j , b ′ j ). But �� � �� � � � � � � � = a ′ a ′ a i b j = a i b j a i b j = b j = i b j . i i , j i j j i j i i , j Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 27 / 75
Overview Triples ( A , T , ( − )) together (where ( − ) is a negation map with ( − ) T = T ) is called a pseudo-triple ; ( A , T , ( − )) is a triple when T generates ( A , +) additively. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 28 / 75
Overview Triples ( A , T , ( − )) together (where ( − ) is a negation map with ( − ) T = T ) is called a pseudo-triple ; ( A , T , ( − )) is a triple when T generates ( A , +) additively. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 28 / 75
Overview Triples ( A , T , ( − )) together (where ( − ) is a negation map with ( − ) T = T ) is called a pseudo-triple ; ( A , T , ( − )) is a triple when T generates ( A , +) additively. We usually require that T ∩ A ◦ = ∅ . (In particular 0 / ∈ T . We write T 0 for T ∪ { 0 } . ) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 28 / 75
Overview Triples ( A , T , ( − )) together (where ( − ) is a negation map with ( − ) T = T ) is called a pseudo-triple ; ( A , T , ( − )) is a triple when T generates ( A , +) additively. We usually require that T ∩ A ◦ = ∅ . (In particular 0 / ∈ T . We write T 0 for T ∪ { 0 } . ) ( A , T , ( − )) is called a T - group module triple when T is a multiplicative group. A triple ( A , T , ( − )) is a T - semiring triple if A is a semiring. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 28 / 75
Overview Uniquely negated triples One of the key concepts: A triple ( A , T , ( − )) is uniquely negated if a + b ∈ A ◦ for a , b ∈ T implies b = ( − ) a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 29 / 75
Overview Uniquely negated triples One of the key concepts: A triple ( A , T , ( − )) is uniquely negated if a + b ∈ A ◦ for a , b ∈ T implies b = ( − ) a . Unique negation fails in idempotent semirings in which negation is of the first kind, such as the max-plus, since any a ∈ T satisfies a = a + a = a ◦ ∈ T ∩ A ◦ = ∅ . IMPORTANT: There is a big difference in taking a + b for a = ( − ) b , in which case it is a ◦ , and for a � = ( − ) b . Accordingly, we need to exclude quasi-negatives from our criterion for bipotence. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 29 / 75
Overview Bipotent and meta-tangible triples A triple ( A , T , ( − )) is ( − ) -bipotent if a + b ∈ { a , b } whenever a , b ∈ T with b � = ( − ) a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 30 / 75
Overview Bipotent and meta-tangible triples A triple ( A , T , ( − )) is ( − ) -bipotent if a + b ∈ { a , b } whenever a , b ∈ T with b � = ( − ) a . The triples used in tropicalization (related to the max-plus algebra) are all ( − )-bipotent, thereby motivating us to develop the algebraic theory of such triples. Any ( − )-bipotent triple of the second kind is idempotent since ( − ) a � = a implies a + a = max { a , a } = a . Conversely, any idempotent triple satisfying is of the second kind. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 30 / 75
Overview Bipotent and meta-tangible triples A triple ( A , T , ( − )) is ( − ) -bipotent if a + b ∈ { a , b } whenever a , b ∈ T with b � = ( − ) a . The triples used in tropicalization (related to the max-plus algebra) are all ( − )-bipotent, thereby motivating us to develop the algebraic theory of such triples. Any ( − )-bipotent triple of the second kind is idempotent since ( − ) a � = a implies a + a = max { a , a } = a . Conversely, any idempotent triple satisfying is of the second kind. The triple ( � A , � T , ( − )) is uniquely negated but not ( − )-bipotent. The modified symmetrized T -monoid module is ( − )-bipotent, which is why it is more useful at times. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 30 / 75
Overview Bipotent and meta-tangible triples A triple ( A , T , ( − )) is ( − ) -bipotent if a + b ∈ { a , b } whenever a , b ∈ T with b � = ( − ) a . The triples used in tropicalization (related to the max-plus algebra) are all ( − )-bipotent, thereby motivating us to develop the algebraic theory of such triples. Any ( − )-bipotent triple of the second kind is idempotent since ( − ) a � = a implies a + a = max { a , a } = a . Conversely, any idempotent triple satisfying is of the second kind. The triple ( � A , � T , ( − )) is uniquely negated but not ( − )-bipotent. The modified symmetrized T -monoid module is ( − )-bipotent, which is why it is more useful at times. The following property, weaker than ( − )-bipotence, actually is enough to carry the theory: A meta-tangible triple is a uniquely negated triple for which a + b ∈ T for any a � = ( − ) b in T . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 30 / 75
Overview Height We define the height of an element c ∈ A as the minimal t such that c = � t i =1 a i with each a i ∈ T . (We say that 0 has height 0.) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 31 / 75
Overview Height We define the height of an element c ∈ A as the minimal t such that c = � t i =1 a i with each a i ∈ T . (We say that 0 has height 0.) The height of A is the maximal height of its elements. Thus A has height 2 iff A = T 0 ∪ ( T + T ). Most systems arising in tropical mathematics have height 2, but height 3 provides new interesting examples. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 31 / 75
Overview Supertropical matrix theory – First Pass Assume R = ( R , G , ν ) is a commutative supertropical domain † . One defines the matrix semiring † M n ( R ) in the usual way. Since − 1 is not available in tropical mathematics, we make do with the permanent, suggestively notated as | A | , and defined for any matrix A = ( a i , j ) as � | A | = a π (1) , 1 · · · a π ( n ) , n . π ∈ S n . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 32 / 75
Overview Definition: An n × n matrix A is singular if | A | is tangible; A is singular when | A | ∈ G 0 . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 33 / 75
Overview Definition: An n × n matrix A is singular if | A | is tangible; A is singular when | A | ∈ G 0 . Theorem: | AB | = | A | | B | for n × n matrices over a supertropical semiring, whenever AB is nonsingular. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 33 / 75
Overview Definition: An n × n matrix A is singular if | A | is tangible; A is singular when | A | ∈ G 0 . Theorem: | AB | = | A | | B | for n × n matrices over a supertropical semiring, whenever AB is nonsingular. � 0 � 0 The assertion fails for AB nonsingular. For example, take A = . 1 2 � 1 � � A 2 � � 2 � = 5 ν � = 4 = | A | 2 . | A | = 2 , but A 2 = , so 3 4 � 0 � 0 is nonsingular, whereas A 2 is singular. Here A = 1 2 Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 33 / 75
Overview Definition: = a , if b = a + c ν for some c . We say b ghost surpasses a , written b | G The correct theorem: Theorem: For any n × n matrices over a supertropical semiring R, we have | AB | | = | A | | B | . G In particular, | AB | = | A | | B | whenever AB is singular. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 34 / 75
Overview The surpassing relation The last theorem suggests that we want to generalize equality on T to a relation on A which is not symmetric! Definition: A surpassing relation on A , denoted � , is a partial pre-order satisfying the following, for elements of A : 1 0 � a . 2 a � b whenever a + c ◦ = b for some c ∈ A ◦ . 3 If a � b then ( − ) a � ( − ) b . 4 If a i � b i for i = 1 , 2 then a 1 + a 2 � b 1 + b 2 . 5 If a � b for a , b ∈ T , then a = b . 6 a ◦ �� b for any b ∈ T . A surpassing PO on A is a surpassing relation � that restricts to a PO on A ◦ . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 35 / 75 One other property that one often wants is that a � a ◦ , which holds in all
Overview Definition: The ◦ - relation � ◦ is the relation given by a � ◦ b iff b = a + c for some c ∈ A ◦ . One can check that � ◦ is indeed a surpassing relation in any meta-tangible triple. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 36 / 75
Overview Definition: The ◦ - relation � ◦ is the relation given by a � ◦ b iff b = a + c for some c ∈ A ◦ . One can check that � ◦ is indeed a surpassing relation in any meta-tangible triple. Let us see why the conditions of the definition of surpassing relation are desired for � to parallel equality. (2) shows that � refines � ◦ , and shows how the quasi-zeros behave like 0 under � . (3), (4) are needed for considerations in universal algebra. (5) enables us to view � as equality for tangible elements. (6) underlines the dichotomy between tangible elements and quasi-zeros. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 36 / 75
Overview Systems A system ( A , T , ( − ) , � ) is a uniquely negated triple ( A , T , ( − )) together with a T -surpassing relation � , which often is a PO. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 37 / 75
Overview Systems A system ( A , T , ( − ) , � ) is a uniquely negated triple ( A , T , ( − )) together with a T -surpassing relation � , which often is a PO. Here is a convenient way for building up triples and systems, based on our previous construction. Given triples ( A ℓ , T ℓ , ( − )) for ℓ ∈ L we form their direct sum ⊕ ℓ ∈ L A ℓ . This has been denoted A ( L ) when each A ℓ = A . There are several natural options for T ⊕A ℓ , which should be clear according to the context, for c ℓ ∈ A ℓ : Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 37 / 75
Overview Systems A system ( A , T , ( − ) , � ) is a uniquely negated triple ( A , T , ( − )) together with a T -surpassing relation � , which often is a PO. Here is a convenient way for building up triples and systems, based on our previous construction. Given triples ( A ℓ , T ℓ , ( − )) for ℓ ∈ L we form their direct sum ⊕ ℓ ∈ L A ℓ . This has been denoted A ( L ) when each A ℓ = A . There are several natural options for T ⊕A ℓ , which should be clear according to the context, for c ℓ ∈ A ℓ : 1 T ⊕A ℓ = T , with the diagonal action a ( c ℓ ) = ( ac ℓ ) for a ∈ T . (This is useful in linear algebra, since we want to view T as scalars. This provides a quasi-triple but not a triple since it does not generate ⊕A ℓ .) 2 T ⊕A ℓ = ∪T ℓ . The action is defined componentwise, i.e., a k ( c ℓ ) = a k c k for a ℓ ∈ T ℓ . The negation map also is defined componentwise. 3 Same as in (2), but now T ⊕A ℓ = � ℓ T ℓ (which is generated by ∪T ℓ ). The action is defined componentwise, i.e., ( a ℓ )( c ℓ ) = ( a ℓ c ℓ ) for a ℓ ∈ T ℓ . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 37 / 75
Overview One can pretty well characterize the meta-tangible systems and recover the main examples in tropical mathematics, as well as some major examples in hyperfields, to be discussed: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 38 / 75
Overview Theorem: Any metatangible group module system ( A , T , ( − ) , � ) must satisfy one of the following: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 39 / 75
Overview Theorem: Any metatangible group module system ( A , T , ( − ) , � ) must satisfy one of the following: ( − ) is of the first kind. A = ∪ m ∈ N m T , and e ′ = 3 . 3 � = 1 . Then T is ( − ) -bipotent, and ( A , T , ( − ) , � ) is isomorphic to a layered system (layered by N in characteristic 0, and Z / k in characteristic k > 0 ). 3 = 1 . Hence ( A , T , − , � ) has characteristic 2 . One example is the classical algebra of characteristic 2 , but one also has other examples. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 39 / 75
Overview Theorem: Any metatangible group module system ( A , T , ( − ) , � ) must satisfy one of the following: ( − ) is of the first kind. A = ∪ m ∈ N m T , and e ′ = 3 . 3 � = 1 . Then T is ( − ) -bipotent, and ( A , T , ( − ) , � ) is isomorphic to a layered system (layered by N in characteristic 0, and Z / k in characteristic k > 0 ). 3 = 1 . Hence ( A , T , − , � ) has characteristic 2 . One example is the classical algebra of characteristic 2 , but one also has other examples. ( − ) is of the second kind. There are two possibilities: T is ( − ) -bipotent, and T (and thus A ) is idempotent. Taking the congruence identifying a with ( − ) , A / ≡ is a ( − ) -bipotent system of the first kind, under the induced addition and multiplication. T is not ( − ) -bipotent. Then the system is “classical.” Furthermore 3 = 1 . Hence A = T ∩ T ◦ . Either N ⊆ T , or ( A , T , − , � ) has characteristic k for some k ≥ 1 . In the latter case, ( A , T , − , � ) is layered by Z / k . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 39 / 75
Overview Examples of systems First the T -Semiring † and T -semifield † systems that we discussed. Classical algebra was considered above. Here the quasi-negative is the usual negative, which is unique, and A ◦ = { 0 } . a � ◦ b iff b = a + 0 = a , so we have the T -system ( A , T , − , =) , which is meta-tangible. The negation map is of second kind unless A has characteristic 2, in which case ( − ) is of the first kind. This helps to “explain” why the theory of meta-tangible T -systems of first kind often has the flavor of characteristic 2. In the max-plus algebra the quasi-negatives are far from unique, since whenever b < a we have a + b = a = a ◦ . Height 2. These provide tropical structures designed to refine the max-plus algebra. All of them are ( − )-bipotent T -systems, to be studied in depth. The familiar examples have characteristic 0, although some constructions can also be replicated in positive characteristic. Supertropical semirings † and the “symmetrized” T -system were Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 40 / 75
Overview Layered semirings † “Layered semirings” are of the form A = L × G , where L is the “layering semiring,” which has its own negation map that we designate as − , and ( G , · ) is an ordered monoid. In fact, associativity of multiplication in G is irrelevant, so we will call them “layered semialgebras.”. Addition is given by: ( ℓ 1 , a 1 ) if a 1 > a 2 ; ( ℓ 1 , a 1 ) + ( ℓ 2 , a 2 ) = ( ℓ 2 , a 2 ) if a 1 < a 2 ; . ( ℓ 1 + ℓ 2 , a 1 ) if a 1 = a 2 . T = {± 1 } × G . 1 A = (1 , 1) ∈ T , and by induction, for k ∈ N , k = ( k , 1) = ( k − 1 , 1) + ( k , 1) = 1 + · · · + 1 , taken k times. The ( k , 1) generate a sub-semiring with negation map, and A = ∪ k ∈ L ( k , 1) T . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 41 / 75
Overview Layered semirings † “Layered semirings” are of the form A = L × G , where L is the “layering semiring,” which has its own negation map that we designate as − , and ( G , · ) is an ordered monoid. In fact, associativity of multiplication in G is irrelevant, so we will call them “layered semialgebras.”. Addition is given by: ( ℓ 1 , a 1 ) if a 1 > a 2 ; ( ℓ 1 , a 1 ) + ( ℓ 2 , a 2 ) = ( ℓ 2 , a 2 ) if a 1 < a 2 ; . ( ℓ 1 + ℓ 2 , a 1 ) if a 1 = a 2 . T = {± 1 } × G . 1 A = (1 , 1) ∈ T , and by induction, for k ∈ N , k = ( k , 1) = ( k − 1 , 1) + ( k , 1) = 1 + · · · + 1 , taken k times. The ( k , 1) generate a sub-semiring with negation map, and A = ∪ k ∈ L ( k , 1) T . The negation map is given by ( − )( k , a ) = ( − k , a ) . Thus the quasi-zeros will be of level 1 − 1. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 41 / 75
Overview Here are some natural explicit examples of layered semialgebras: L = N , formally with − ℓ = ℓ , T = { ( ℓ, a ) ∈ L × G : ℓ = 1 } , and ( − ) is the identity (thus of the first kind). T ◦ is the layer 2. (The higher levels, if they exist, are neither tangible nor in T ◦ . In fact e ′ = 1 + 1 + 1 has layer 3.) This is useful for tropical differentiation. It has height equal to the cardinality of the submonoid of L generated by 1. It often provides counterexamples to assertions that hold in height 2. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 42 / 75
Overview Here are some natural explicit examples of layered semialgebras: L = N , formally with − ℓ = ℓ , T = { ( ℓ, a ) ∈ L × G : ℓ = 1 } , and ( − ) is the identity (thus of the first kind). T ◦ is the layer 2. (The higher levels, if they exist, are neither tangible nor in T ◦ . In fact e ′ = 1 + 1 + 1 has layer 3.) This is useful for tropical differentiation. It has height equal to the cardinality of the submonoid of L generated by 1. It often provides counterexamples to assertions that hold in height 2. L = Z with the usual negation, T = { ( ℓ, a ) ∈ L × G : ℓ = ± 1 } , and ( − )( ℓ, a ) = ( − ℓ, a ) , of the second kind. This is useful for tropical integration. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 42 / 75
Overview Here are some natural explicit examples of layered semialgebras: L = N , formally with − ℓ = ℓ , T = { ( ℓ, a ) ∈ L × G : ℓ = 1 } , and ( − ) is the identity (thus of the first kind). T ◦ is the layer 2. (The higher levels, if they exist, are neither tangible nor in T ◦ . In fact e ′ = 1 + 1 + 1 has layer 3.) This is useful for tropical differentiation. It has height equal to the cardinality of the submonoid of L generated by 1. It often provides counterexamples to assertions that hold in height 2. L = Z with the usual negation, T = { ( ℓ, a ) ∈ L × G : ℓ = ± 1 } , and ( − )( ℓ, a ) = ( − ℓ, a ) , of the second kind. This is useful for tropical integration. L is the residue ring of a valuation, where now T = { ( ℓ, a ) ∈ L × G : ℓ � = 0 } . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 42 / 75
Overview Here are some natural explicit examples of layered semialgebras: L = N , formally with − ℓ = ℓ , T = { ( ℓ, a ) ∈ L × G : ℓ = 1 } , and ( − ) is the identity (thus of the first kind). T ◦ is the layer 2. (The higher levels, if they exist, are neither tangible nor in T ◦ . In fact e ′ = 1 + 1 + 1 has layer 3.) This is useful for tropical differentiation. It has height equal to the cardinality of the submonoid of L generated by 1. It often provides counterexamples to assertions that hold in height 2. L = Z with the usual negation, T = { ( ℓ, a ) ∈ L × G : ℓ = ± 1 } , and ( − )( ℓ, a ) = ( − ℓ, a ) , of the second kind. This is useful for tropical integration. L is the residue ring of a valuation, where now T = { ( ℓ, a ) ∈ L × G : ℓ � = 0 } . L is a finite field of characteristic 2, where T = { ( ℓ, a ) ∈ L × G : ℓ � = 0 } , and ( − ) is the identity. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 42 / 75
Overview A somewhat more esoteric example from the tropical standpoint, but quite significant algebraically. Fixing n > 0, taking L = Z n , identify each level modulo n . (This has height n and characteristic n .) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 43 / 75
Overview A somewhat more esoteric example from the tropical standpoint, but quite significant algebraically. Fixing n > 0, taking L = Z n , identify each level modulo n . (This has height n and characteristic n .) (The truncated algebra) A weird example, which leads to counterexamples in linear algebra and must be confronted. Fixing n > 1, we say that L = { 1 , . . . , n } is truncated at n if addition and multiplication are given by identifying every number greater than n with n . In other words, k 1 + k 2 = n in L if k 1 + k 2 ≥ n in N ; k 1 k 2 = n in L if k 1 k 2 ≥ n in N . The negation map is the identity. This T -triple has characteristic 0, since m � = 1 for all m > 1, but it has height n . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 43 / 75
Overview A somewhat more esoteric example from the tropical standpoint, but quite significant algebraically. Fixing n > 0, taking L = Z n , identify each level modulo n . (This has height n and characteristic n .) (The truncated algebra) A weird example, which leads to counterexamples in linear algebra and must be confronted. Fixing n > 1, we say that L = { 1 , . . . , n } is truncated at n if addition and multiplication are given by identifying every number greater than n with n . In other words, k 1 + k 2 = n in L if k 1 + k 2 ≥ n in N ; k 1 k 2 = n in L if k 1 k 2 ≥ n in N . The negation map is the identity. This T -triple has characteristic 0, since m � = 1 for all m > 1, but it has height n . L is some classical algebraic structure, such as a ring, or an exterior algebra, or a Lie algebra. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 43 / 75
Overview The “exploded” T -system, where A = L × G with L the set of lowest coefficients of Puiseux series, T = ( L \ 0) × G , and ( − )( ℓ, a ) = ( − ℓ, a ) , is ( − )-bipotent of the second kind, provided L is not of characteristic 2. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 44 / 75
Overview Hypergroups Recent interest has arisen in the study of hypergroups and hyperfields. It turns out that the hypergroups can be injected naturally into their power sets, which have a negation map, whereby the hyperfield is identified with the subset of singletons. The idea is to formulate all of our extra structure in terms of addition (and possibly other operations such as multiplication) on P ( T ), the set of subsets of T , viewed as an additive semigroup, identifying T with the singletons in P ( T ). But this is not so easy since T 0 itself need not be closed under addition. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 45 / 75
Overview The “intuitive” definition: A hyper-semigroup should be a structure ( T , ⊞ , 0) where ⊞ : T × T → P ( T ), for which the analog of associativity holds: ( a 1 ⊞ a 2 ) ⊞ a 3 = a 1 ⊞ ( a 2 ⊞ a 3 ) , ∀ a ∈ T . There is a fundamental difficulty in this definition: a 1 ⊞ a 2 need not be a singleton, so technically ( a 1 ⊞ a 2 ) ⊞ a 3 is not defined. This difficulty is exacerbated when considering generalized associativity; for example, what does ( a 1 ⊞ a 2 ) ⊞ ( a 3 ⊞ a 4 ) mean in general? Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 46 / 75
Overview A hyper-semigroup is ( T 0 , ⊞ , 0), where ⊞ is a commutative binary operation T 0 × T 0 → P ( T 0 ) , which also is associative in the sense that if we define a ⊞ S = ∪ s ∈ S a ⊞ s , then ( a 1 ⊞ a 2 ) ⊞ a 3 = a 1 ⊞ ( a 2 ⊞ a 3 ) for all a i in T 0 . 0 is the neutral element. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 47 / 75
Overview A hyper-semigroup is ( T 0 , ⊞ , 0), where ⊞ is a commutative binary operation T 0 × T 0 → P ( T 0 ) , which also is associative in the sense that if we define a ⊞ S = ∪ s ∈ S a ⊞ s , then ( a 1 ⊞ a 2 ) ⊞ a 3 = a 1 ⊞ ( a 2 ⊞ a 3 ) for all a i in T 0 . 0 is the neutral element. We always think of ⊞ in terms of addition. Note that repeated addition in the hyper-semigroup need not be defined until one passes to the power set, which makes it difficult to check basic universal relations such as associativity. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 47 / 75
Overview A hypernegative of an element a in a hyper-semigroup ( T , ⊞ , 0) is an element − a for which 0 ∈ a ⊞ ( − a ) . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 48 / 75
Overview A hypernegative of an element a in a hyper-semigroup ( T , ⊞ , 0) is an element − a for which 0 ∈ a ⊞ ( − a ) . A hypergroup is a hyper-semigroup ( T , ⊞ , 0) for which every element a has a unique hypernegative − a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 48 / 75
Overview A hypernegative of an element a in a hyper-semigroup ( T , ⊞ , 0) is an element − a for which 0 ∈ a ⊞ ( − a ) . A hypergroup is a hyper-semigroup ( T , ⊞ , 0) for which every element a has a unique hypernegative − a . The hypernegation is a negation map, and induces a negation map on P ( T 0 ), via ( − ) S = {− s : s ∈ S } . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 48 / 75
Overview A hypernegative of an element a in a hyper-semigroup ( T , ⊞ , 0) is an element − a for which 0 ∈ a ⊞ ( − a ) . A hypergroup is a hyper-semigroup ( T , ⊞ , 0) for which every element a has a unique hypernegative − a . The hypernegation is a negation map, and induces a negation map on P ( T 0 ), via ( − ) S = {− s : s ∈ S } . A T - hyperzero of a hypergroup ( T , ⊞ , 0) is a set of the form a ⊞ ( − a ) ∈ P ( T ) . (This is not the usual definition, which is any subset of T containing 0 , but serves just as well since, by definition, if 0 ∈ a ⊞ b for a , b ∈ T then b = − a , implying a ⊞ b is a hyperzero in our sense.) Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 48 / 75
Overview A hypernegative of an element a in a hyper-semigroup ( T , ⊞ , 0) is an element − a for which 0 ∈ a ⊞ ( − a ) . A hypergroup is a hyper-semigroup ( T , ⊞ , 0) for which every element a has a unique hypernegative − a . The hypernegation is a negation map, and induces a negation map on P ( T 0 ), via ( − ) S = {− s : s ∈ S } . A T - hyperzero of a hypergroup ( T , ⊞ , 0) is a set of the form a ⊞ ( − a ) ∈ P ( T ) . (This is not the usual definition, which is any subset of T containing 0 , but serves just as well since, by definition, if 0 ∈ a ⊞ b for a , b ∈ T then b = − a , implying a ⊞ b is a hyperzero in our sense.) ( T , ⊞ , · , 1) is a hyperfield if ( T , ⊞ , 0) is also a group ( T , · , 1). Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 48 / 75
Overview Here are some main examples of hyperfields. The tropical hyperfield. Define R ∞ = R ∪ {−∞} and define the product a � b := a + b and � max ( a , b ) if a � = b , a ⊞ b = { c : c ≤ a } if a = b . Thus 0 is the multiplicative identity, −∞ is the additive identity, and we have a hyperfield (satisfying Property P), easily seen to be isomorphic (as semirings) to Izhakian’s extended tropical arithmetic , where we identify ( −∞ , a ] := { c : c ≤ a } with a ν , and have a natural hyperfield isomorphism of this tropical hyperfield with the sub-semiring � R ∞ of P ( R ∞ ), because b if b > a ; ( −∞ , a ] + b = ( −∞ , a ] if b = a ( −∞ , b ] ∪ ( b , a ] = ( −∞ , a ] if b < a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 49 / 75
Overview The Krasner hyperfield. Let K = { 0; 1 } with the usual operations of Boolean algebra, except that now 1 ⊞ 1 = { 0; 1 } . Again, this generates a sub-semiring of P ( K ) having three elements, and is just the supertropical algebra of the monoid K , where we identify { 0; 1 } with 1 ν . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 50 / 75
Overview The Krasner hyperfield. Let K = { 0; 1 } with the usual operations of Boolean algebra, except that now 1 ⊞ 1 = { 0; 1 } . Again, this generates a sub-semiring of P ( K ) having three elements, and is just the supertropical algebra of the monoid K , where we identify { 0; 1 } with 1 ν . (Hyperfield of signs) Let S := { 0 , 1 , − 1 } with the usual multiplication law and hyperaddition defined by 1 ⊞ 1 = { 1 } , − 1 ⊞ − 1 = {− 1 } , x ⊞ 0 = 0 ⊞ x = { x } , and 1 ⊞ − 1 = − 1 ⊞ 1 = { 0 , 1 , − 1 } = S . Then S is a hyperfield (satisfying Property P), called the hyperfield of signs. The four elements {{ 0 } , {− 1 } , { 1 } , S } constitute the sub-semiring † � S of P ( S ), and comprises a meta-tangible system. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 50 / 75
Overview The phase hyperfield. Take T = S 1 , the complex unit circle, together with the center { 0 } , and Points a and b are antipodes if a = − b . Multiplication is defined as usual (so corresponds on S 1 to addition of angles). We call an arc of less than 180 degrees short . all points in the short arc from a to b if a � = b ; a ⊞ b = {− a , 0 , a } if a = − b � = 0; { a } if b = 0 . T is a hyperfield, called the phase hyperfield . At the power set level, given W 1 , W 2 ⊆ S 1 , we define W 1 ⊞ W 2 to be the union of all (short) arcs from a point of W 1 to a non-antipodal point in W 2 (which together makes a connected arc), together with { 0 } if W 2 contains an antipode of W 1 . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 51 / 75
Overview Thus the system spanned by T is not meta-tangible, and its elements can be described as follows: { 0 } , which has height 0, T , the points on S 1 , which has height 1, Short arcs (the sum of non-antipodal distinct points), which have height 2, The sets { a , 0 , − a } = a − a , which we write as a ◦ , which have height 2, Semicircles with 0, having the form a ◦ + b where b � = ± a , which have height 3, S 1 ∪ { 0 } = a ◦ + b ◦ where b � = ± a . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 52 / 75
Overview Viro’s “triangle” hyperfield T defined over R + by the formula a ⊞ b = { c ∈ R + : | a − b | ≤ c ≤ a + b } . In other words, c ∈ a ⊞ b iff there exists a Euclidean triangle with sides of lengths a , b , and c . Here T + T = { [ a 1 , a 2 ] : a 1 ≤ a 2 } , although not meta-tangible, has height 2, since [ a 1 , a 2 ] = a 1 + a 2 + a 2 − a 1 ∈ ˆ A whereas [ a 1 , a 2 ] + [ a ′ 1 , a ′ 2 ] is some interval 2 2 going up to a 2 + a ′ 2 . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 53 / 75
Overview Hypergroup systems ( P ( T ) , T , ( − ) , ⊆ ) is a system, which we call a hypergroup system . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematics Tuesday 20 June, 2017 54 / 75
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