Algebraic Aspects in Tropical Mathematics Tal Perri Mathematics Department, Bar-Ilan University Under the supervision of Professor Louis Rowen November 1, 2018 arXiv:1305.2764v3 [math.AG] 22 Jun 2013 Abstract Much like in the theory of algebraic geometry, we develop a correspondence be- tween certain types of algebraic and geometric objects. The basic algebraic envi- ronment we work in is the a semifield of fractions H(x1, ..., xn) of the polynomial semidomain H[x1, ..., xn], where H is taken to be a bipotent semifield, while for the geometric environment we have the space Hn (where addition and scalar mul- tiplication are defined coordinate-wise). We show that taking H to be bipotent n makes both H(x1, ..., xn) and H idempotent which turn out to satisfy many de- sired properties that we utilize for our construction. The fundamental algebraic and geometric objects having interrelations are called kernels, encapsulating congruences over semifields (analogous to ideals in alge- braic geometry) and skeletons which serve as the analogue for zero-sets of alge- braic geometry. As an analogue for the celebrated Nullstellenzats theorem which provides a correspondence between radical ideals and zero sets, we develop a correspondence between skeletons and a family of kernels called polars originally developed in the theory of lattice-ordered groups. For a special kind of skeletons, called principal skeletons, we have simplified the correspondence by restricting our algebraic environment to a very special semifield which is also a kernel of H(x1, ..., xn). After establishing the linkage between kernels and skeletons we proceed to con- struct a second linkage, this time between a family of skeletons and what we call ‘corner-loci’. Essentially a corner locus is what is called a tropical variety in the theory of tropical geometry, which is a set of corner roots of some set of tropical polynomials. The relation between a skeleton and a corner-locus is that they define the exact same subset of Hn, though in different ways: while a corner locus is defined by corner roots of tropical polynomials, the skeleton is defined by an equality, namely, equating fractions from H(x1, ..., xn) to 1. All the con- nections presented above form a path connecting a tropical variety to a kernel. In this paper we also develop a correspondence between supertropical varieties (generalizing tropical varieties) introduced by Izhakian, Knebusch and Rowen to the lattice of principal kernels. Restricting this correspondence to a sublattice of kernels, whom we call regular kernels, yields the correspondence described above. The research we conduct shows that the theory of supertropical geometry is in fact a natural generalization of tropical geometry. We conclude by developing some algebraic structure notions such as composi- tion series and convexity degree, along with some notions holding a geometric interpretation, like reducibility and hyperdimension. 4 Contents 1 Overview................................ 7 2 Background .............................. 9 2.1 Basic notions in lattice theory . 9 2.2 Semifields ........................... 10 2.3 Semifield with a generator and generation of kernels . 21 2.4 Simple semifields . 27 2.5 Irreducible kernels, maximal kernels and the Stone topology 28 2.6 Distributive semifields . 32 2.7 Idempotentsemifields:Part1. 33 2.8 Affine extensions of idempotent archimedean semifields . 39 3 Lattice-ordered groups, idempotent semifields and the semifield of fractions H(x1, ..., xn). ........................ 41 3.1 Lattice-orderedgroups . 42 3.2 Idempotent semifields versus lattice-ordered groups . 45 3.3 Idempotentsemifields:Part2 . 46 3.4 Archimedean idempotent semifields . 52 4 Skeletonsandkernelsofskeletons . 56 4.1 Skeletons ........................... 56 4.2 Kernelsofskeletons...................... 58 5 Thestructureofthesemifieldoffractions. 61 5.1 Bounded rational functions in the semifield of fractions . 61 5.2 The structure of hHi ..................... 65 6 Thepolar-skeletoncorrespondence . 69 6.1 Polars ............................. 69 6.2 Thepolar-skeletoncorrespondence . 75 6.3 Appendix: The Boolean algebra of polars and the Stonerepresentation . 81 7 The principal bounded kernel - principal skeleton correspondence 84 5 8 Thecoordinatesemifieldofaskeleton. 93 9 Basic notions : Essentiality, reducibility, regularity andcorner-integrality. 95 9.1 Essentiality of elements in the semifield of fractions . 95 9.2 Reducibility of principal kernels and skeletons . 98 9.3 Decompositions ........................ 106 9.4 Regularity........................... 111 9.5 Corner-Integrality. .. .. 115 9.6 Appendix : Limits of skeletons . 117 10 The corner loci - principal skeletons correspondence . 120 10.1 Cornerloci .......................... 120 10.2 Cornerlociandprincipalskeletons . 124 10.3 Example: The tropical line . 135 11 Corner-integralityrevisited. 137 12 Composition series of kernels of an idempotent semifield . 146 13 The Hyperspace-Region decomposition and the Hyperdimension . 150 13.1 Hyperspace-kernels and region-kernels. 150 13.2 Geometric interpretation of HS-kernels and region kernels andtheuseoflogarithmicscale . 156 13.3 A preliminary discussion . 159 13.4 TheHO-decomposition. 160 13.5 The lattice generated by regular corner-integral principal kernels............................. 169 13.6 Convexity degree and hyperdimension . 170 Bibliography 186 6 1 Overview In the following overview, we will establish a linkage between our construction and the widely known theory of algebraic geometry, in order to give the reader some additional insights. As noted in the abstract, we develop a geometric structure corresponding to a semifield of fractions H(x1, ..., xn) over a bipotent semifield H. We concentrate on the case where H is a divisible archimedean semifield, motivated by our interest to link the theory to tropical geometry. We also consider a divisible bipotent archimedean semifield R, which is also complete (and thus isomorphic to the positive reals with max-plus operations). The role of ideals is played by an algebraic structures called kernels. While ideals encapsulate the structure of the preimages of zero of rings homomor- phisms, kernels encapsulate the preimages of {1} with respect to semifield ho- momorphisms with the substantial difference that considering homomorphisms one gets a sublattice of kernels with respect to intersection and multiplication, (Con(R(x1, ..., xn)), ·, ∩). For each kernel K of Con(R(x1, ..., xn)), we define an analogue for zero sets corresponding to ideals, namely, a skeleton Skel(K) ⊆ Rn, which is defined to be the set of all points in Rn over which all the elements of the kernel K are evaluated to be 1. While in the classical theory all ideals are finitely generated, in our case we explicitly consider the finitely generated kernels in Con(R(x1, ..., xn)). This family of finitely generated kernels in Con(R(x1, ..., xn)) forms a sublattice of (Con(R(x1, ..., xn)), ·, ∩), which we denote as PCon(R(x1, ..., xn)). At this point, the theory is considerably simplified, since a well-known result in the theory of semifields states that every finitely generated kernel is principal, i.e., generated as a kernel by a single element. The generator of a principal kernel is not unique, though there is a designated set of generators which provides us with many tools for implementing the theory. First we establish a Zariski-like correspondence between a special kind of ker- 7 nels called polars and general skeletons. Then we find some special semifield of R(x1, ..., xn) over which the above correspondence gives rise to a simplified corre- spondence between principal (finitely generated) kernels of and principal (finitely generated) skeletons which hold most of our interest. We show that a skeleton is uniquely defined by any of its corresponding kernel generators, define a maximal kernel, and show that the maximal principal kernels correspond to points in Rn. Along the way, we provide some theory concerning general (not necessarily prin- cipal) kernels and skeleton. The geometric interpretation established for the theory of semifields provides us with a topology, called the Stone Topology, defined on the family of the so- called irreducible kernels (analogous to prime ideals) and the family of maximal kernels. This topology is in essence the semifields version of the famous Zariski topology. After establishing this geometric framework, we introduce a map linking su- pertropical varieties (cf. [4]), which we call ‘Corner loci’, to a subfamily of princi- pal skeletons. We use the latter to construct a correspondence between principal supertropical varieties (the analogue for hypersurfaces of algebraic geometry) and a lattice of kernels generated by special principal kernels which we call corner- integral. In addition, we characterize a family of skeletons that coincide with the family of ‘regular’ tropical varieties via which we obtain a correspondence with a sublattice of principal kernels called regular kernels. Finally, we establish a notion of reducibility and some other notions and prop- erties of kernels and their corresponding skeletons, such as convex-dependence, dimensionality etc. We begin our thesis by introducing the relevant results in the theory of semifields. 8 2 Background 2.1 Basic notions in lattice theory Definition 2.1.1. A poset (X, ≤) is directed if for any pair of elements a, b ∈ X there exists c ∈ X such that a ≤ c and b ≤ c, i.e., c is an upper bound for a and b. A poset (X, ≤) is a lattice or lattice-ordered set if for a, b ∈ X, the set {a, b} has a join a ∨ b (also known as the least upper bound, or the supremum) and a meet a ∧ b (also known as the greatest lower bound, or the infimum). Equivalently, a lattice can be defined as a directed poset (X, ∨, ∧), consisting of a set X and two associative and commutative binary operations ∨ and ∧ defined on X, such that for all elements a, b ∈ X. a ∨ (a ∧ b)= a ∧ (a ∨ b)= a. The first definition is derived from the second by defining a partial order on X by a ≤ b ⇔ a = a ∧ b or equivalently, a ≤ b ⇔ b = a ∨ b. Definition 2.1.2. A lattice (X, ∨, ∧) is said to be distributive if the following condition holds for any a, b, c ∈ X: a ∧ (b ∨ c)=(a ∧ b) ∨ (a ∧ c).