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An Algebraic Structure Theory Applicable to Tropical Mathematics Shirshov 100 Conference

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An Algebraic Structure Theory Applicable to Tropical Mathematics Shirshov 100 Conference Semialgebra systems: An algebraic structure theory applicable to tropical mathematics Shirshov 100 Conference. Louis Rowen, Bar-Ilan University 16 August, 2021 Ode to Shirshov I first heard of Shirshov from Amitsur (also born the same year!) who discovered the amazing Shirshov Height Theorem, which revolutionized PI-theory. I did not have the opportunity to meet Shirshov, but have the privilege of knowing several of his students, including Bokut (who has been a frequent guest at Bar-Ilan University and introduced me to many members of the important algebra group in Novosibirsk), Zelmanov, Kemer, (both joint students of Bokut and Shirshov), and Shestakov. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to 1. Tropical background in 3 slides Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. Tropical background in 3 slides Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to 1. Tropicalization For any complex affine variety W = f(z1;:::; zn): zi 2 Cg; and any t > 0, its amoeba A(W ) is defined as f(logt jz1j;:::; logt jznj) :(z1;:::; zn) 2 W g (n) ⊂ (R [ {−∞}) ; graphed according to the (rescaled) coordinates logt jz1j;:::; logt jznj. Thus, respectively, multiplication has been replaced by addition, whereas addition has been replaced by the maximum. The degeneration t ! 1 is called tropicalization, honoring Imre Simon, who worked in Brazil. a b a b Note that logt (t t ) = logt (t ) + logt (t ): Also, for a = b + c a b b c c logt (t + t ) = logt (t (t + 1)) = b + logt (t + 1) a ≈ b + c = a = logt (t ); as t ! 1. a b a b Note that logt (t t ) = logt (t ) + logt (t ): Also, for a = b + c a b b c c logt (t + t ) = logt (t (t + 1)) = b + logt (t + 1) a ≈ b + c = a = logt (t ); as t ! 1. Thus, respectively, multiplication has been replaced by addition, whereas addition has been replaced by the maximum. The degeneration t ! 1 is called tropicalization, honoring Imre Simon, who worked in Brazil. Any ordered group (G; ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation. For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5: We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 : Semirings The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be maxfa; bg: This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A n f0g; ·) is a group. For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5: We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 : Semirings The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be maxfa; bg: This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A n f0g; ·) is a group. Any ordered group (G; ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation. We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 : Semirings The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be maxfa; bg: This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A n f0g; ·) is a group. Any ordered group (G; ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation. For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5: Semirings The underlying algebraic structure reverted from R to the max-plus semiring Rmax, an ordered multiplicative monoid in which one defines a + b to be maxfa; bg: This is a semiring, which by definition satisfies all the axioms of ring except the existence of negatives), and is clearly idempotent in the sense that a + a = a. A semifield is a semiring A for which (A n f0g; ·) is a group. Any ordered group (G; ·) gives rise to a max-plus semifield in the same way. We append the subscript max to indicate the corresponding max-plus algebra, e.g., Nmax or Qmax, but we use the usual algebraic notation. For example, in Nmax, 1 + 3 = 3 and 2 · 3 = 5: We always assume that our semirings have a unit element1, and we that its additive identity0 is an absorbing element, in the sense that0 a = a0=0 : For any set S, Fun(S; A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A; by the relation yx = xy + 1: One also can define modules over semirings in the usual way, sometimes called \semimodules." The free A-module of rank n is just the Cartesian product A(n); with operations componentwise. Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). Likewise one can define the Weyl semialgebra over a commutative semiring A; by the relation yx = xy + 1: One also can define modules over semirings in the usual way, sometimes called \semimodules." The free A-module of rank n is just the Cartesian product A(n); with operations componentwise. Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S; A) is the semiring of functions from S to A, with operations defined elementwise. One also can define modules over semirings in the usual way, sometimes called \semimodules." The free A-module of rank n is just the Cartesian product A(n); with operations componentwise. Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S; A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A; by the relation yx = xy + 1: Semirings admit the usual algebraic constructions: If A is a semiring then so are the polynomial semiring A[Λ] and the matrix semiring Mn(A). For any set S, Fun(S; A) is the semiring of functions from S to A, with operations defined elementwise. Likewise one can define the Weyl semialgebra over a commutative semiring A; by the relation yx = xy + 1: One also can define modules over semirings in the usual way, sometimes called \semimodules." The free A-module of rank n is just the Cartesian product A(n); with operations componentwise. The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0: Nmax has characteristic 1 since1+1=1 : (But in this definition one could have1+1 6=1 with2+1=2 :) Characteristic 1 geometry has been quite popular in recent years. Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N !A given by n 7! n for each n: Nmax has characteristic 1 since1+1=1 : (But in this definition one could have1+1 6=1 with2+1=2 :) Characteristic 1 geometry has been quite popular in recent years. Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N !A given by n 7! n for each n: The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0: Every semiring A has a sub-semiring additively generated by1. We write n := n1. Note that there is always a semiring homomorphism N !A given by n 7! n for each n: The characteristic char(A) of A is the minimal n such that k + n = k for some k; if there is none such, then char(A) = 0: Nmax has characteristic 1 since1+1=1 : (But in this definition one could have1+1 6=1 with2+1=2 :) Characteristic 1 geometry has been quite popular in recent years.
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