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A General Algebraic Structure Theory for Tropical Mathematics Algebra Conference in Spa A general algebraic structure theory for tropical mathematics Algebra Conference in Spa Louis Rowen, Bar-Ilan University Tuesday 20 June, 2017 Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 1 / 75 Overview PART I: Overview Many algebraic theories involve the study of a set T with incomplete structure that can be understood better by embedding T in a larger set A endowed with more structure. We start with the following set-up: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 2 / 75 Overview A T -module over a set T is an additive monoid (A; +; 0A) together with scalar multiplication T × A ! A satisfying distributivity over T in the sense that a(b1 + b2) = ab1 + ab2 for a 2 T ; bi 2 A, also with the stipulation that a0A = 0A for all a in T . Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 3 / 75 A semiringy is idempotent if a + a = a; it is bipotent if a + b 2 fa; bg: We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , 8b 2 M: Overview Semiringsy Usually A is a semiring. We delete the zero element in the definition of a y semiring, because it just gets in the way: A semiring (R; +; ·; 1R ) is a set R equipped with binary operations + and · such that: (R; +) is an Abelian semigroup; (R; · ; 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition. There exist a; b 2 R such that a + b = 1R : (The last axiom is much weaker than requiring the existence of a zero element.) Thus, we have all the ring axioms except negation. y y A semifield is a semiring for which (R; · ; 1R ) is an Abelian group. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , 8b 2 M: Overview Semiringsy Usually A is a semiring. We delete the zero element in the definition of a y semiring, because it just gets in the way: A semiring (R; +; ·; 1R ) is a set R equipped with binary operations + and · such that: (R; +) is an Abelian semigroup; (R; · ; 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition. There exist a; b 2 R such that a + b = 1R : (The last axiom is much weaker than requiring the existence of a zero element.) Thus, we have all the ring axioms except negation. y y A semifield is a semiring for which (R; · ; 1R ) is an Abelian group. A semiringy is idempotent if a + a = a; it is bipotent if a + b 2 fa; bg: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 Overview Semiringsy Usually A is a semiring. We delete the zero element in the definition of a y semiring, because it just gets in the way: A semiring (R; +; ·; 1R ) is a set R equipped with binary operations + and · such that: (R; +) is an Abelian semigroup; (R; · ; 1R ) is a monoid with identity element 1R ; Multiplication distributes over addition. There exist a; b 2 R such that a + b = 1R : (The last axiom is much weaker than requiring the existence of a zero element.) Thus, we have all the ring axioms except negation. y y A semifield is a semiring for which (R; · ; 1R ) is an Abelian group. A semiringy is idempotent if a + a = a; it is bipotent if a + b 2 fa; bg: We also want to consider modules M over a semiring R with zero 0R , often called semimodules in the literature; one must stipulate that 0R b = 0M , 8b 2 M: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 4 / 75 Overview Some universal algebra Universal algebra is a natural general structure theory described in terms of \congruences" which, although more complicated than the usual algebraic structure theory because of the lack of an intrinsic negative, has a wide range of applications. Not every structure involved in tropical mathematics can be put in the framework of signatures in universal algebra. Nevertheless, the language of universal algebra unifies many algebraic theories, including the common tropical theories. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 5 / 75 Overview Definition: A carrier is a collection of sets A1; A2;:::; At . A set of operators is a set Ω := [m2NΩ(m) where each Ω(m) := f!m;j : j 2 Jmg in turn is a set of formal symbols !m;j = !m;j (x1;j ;:::; xm;j ) interpreted as a map !m;j : Aj1 × · · · × Ajm !Aim;j . Each operator !m;j , called an (m-ary) operator, has a target index im;j , indicating where the operator takes its values. We define a targeted Ω-formula inductively: Each formal letter xu;i is an Ω-formula with target i, and if φ1; : : : ; φm are Ω-formulas with respective targets iu;j , 1 ≤ u ≤ m; and if !m;j (x1;j ;:::; xm;j ) 2 Ω is compatible with φ in the sense that iu;j is the subscript for xu;j for each u, then !m;j (φ1; : : : ; φm) also is an Ω-formula. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 6 / 75 One can formulate T -modules in terms of universal algebra, often with extra structure on T passed on to A: For example T could be a monoid, in which case we also require associativity ((a1a2)b = a1(a2b) for all ai 2 T and b 2 A). Or T could have a Lie structure, and A could be a Lie module. Overview A universal relation is a pair (φ, ) of Ω-formulas (having the same target). It is the way we identify two expressions. (Associativity is a good example of a universal relation.) A signature is a pair (Ω; I); where Ω is a set of operators and I is a set of universal relations. Writing I for the set of universal relations, we also call fA1; A2;:::; At g an (Ω; I)-algebra. In our theory, the carrier will include both A and T , and the signature will take their structure into account. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 7 / 75 Overview A universal relation is a pair (φ, ) of Ω-formulas (having the same target). It is the way we identify two expressions. (Associativity is a good example of a universal relation.) A signature is a pair (Ω; I); where Ω is a set of operators and I is a set of universal relations. Writing I for the set of universal relations, we also call fA1; A2;:::; At g an (Ω; I)-algebra. In our theory, the carrier will include both A and T , and the signature will take their structure into account. One can formulate T -modules in terms of universal algebra, often with extra structure on T passed on to A: For example T could be a monoid, in which case we also require associativity ((a1a2)b = a1(a2b) for all ai 2 T and b 2 A). Or T could have a Lie structure, and A could be a Lie module. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 7 / 75 Overview Universal algebra and categories A signature in universal algebra gives rise to a category C whose objects are the carriers and whose morphisms are functions that preserve the operators, i.e., f (!(a1;:::; am)) = !(f (a1);:::; f (am)): But later we will weaken the definition of morphism, to make it more appropriate to \systems." Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 8 / 75 When S is a set, we usually write I instead of S. The support of f is fs 2 S : f (s) 6= 0g: In this situation, we can take T to be the morphisms having support of order 1. Overview Function systems Given a category C from universal algebra and a small category S, we define CS to be the category of morphisms from S to C. CS can be seen to have the same signature as C, where operations are defined componentwise, i.e., if fi : S !C; then !(f1;:::; am)(s) := !(f1(s);:::; fm(s)); 8s 2 S: Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 9 / 75 The support of f is fs 2 S : f (s) 6= 0g: In this situation, we can take T to be the morphisms having support of order 1. Overview Function systems Given a category C from universal algebra and a small category S, we define CS to be the category of morphisms from S to C. CS can be seen to have the same signature as C, where operations are defined componentwise, i.e., if fi : S !C; then !(f1;:::; am)(s) := !(f1(s);:::; fm(s)); 8s 2 S: When S is a set, we usually write I instead of S. Louis Rowen, Bar-Ilan University A general algebraic structure theory for tropical mathematicsTuesday 20 June, 2017 9 / 75 Overview Function systems Given a category C from universal algebra and a small category S, we define CS to be the category of morphisms from S to C.
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