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Categories with Negation 3 CATEGORIES WITH NEGATION JAIUNG JUN 1 AND LOUIS ROWEN 2 In honor of our friend and colleague, S.K. Jain. Abstract. We continue the theory of T -systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). Prime ground systems are introduced as a way of developing geometry. One basic result is that the polynomial system over a prime system is prime. For module systems, special attention also is paid to tensor products and Hom. Abelian categories are replaced by “semi-abelian” categories (where Hom(A, B) is not a group) with a negation morphism. The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Contents 1. Introduction 2 1.1. Objectives 3 1.2. Main results 3 2. Background 4 2.1. Basic structures 4 2.2. Symmetrization and the twist action 7 2.3. Ground systems and module systems 8 2.4. Morphisms of systems 10 2.5. Function triples 11 2.6. The role of universal algebra 13 3. The structure theory of ground triples via congruences 14 3.1. The role of congruences 14 3.2. Prime T -systems and prime T -congruences 15 3.3. Annihilators, maximal T -congruences, and simple T -systems 17 4. The geometry of prime systems 17 4.1. Primeness of the polynomial system A[λ] 17 4.2. Localization 19 4.3. Extensions of systems 21 4.4. Classical Krull dimension 23 arXiv:1709.03186v4 [math.RA] 31 Oct 2018 5. Tensor products 24 5.1. Tensor products of systems 24 6. The structure theory of module systems via congruences 25 6.1. Submodules versus subcongruences 26 6.2. The N-category of T -module systems 26 6.3. The category of congruences of T -module systems 27 7. Functors among semiring ground triples and systems 28 7.1. Valuations of semirings via systems 30 1Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA 2Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel E-mail addresses: [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary 08A05, 08A30, 14T05, 16Y60, 20N20. Secondary 06F05, 08A72, 12K10, 13C60. Key words and phrases. Category, pseudo-triple, matroid, pseudo-system, triple, system, semiring, semifield, congruence, module, negation, surpassing relation, symmetrization, congruence, tropical algebra, supertropical algebra, bipotent, meta- tangible, symmetrized, hypergroup, hyperfield, polynomial, prime, dimension, algebraic, tensor product. The authors would like to thank Oliver Lorscheid for helpful comments concerning the first draft of the paper. 1 2 J. JUN AND L. ROWEN 7.2. Functors among module triples and systems 31 8. Appendix A: Interface between systems and tropical mathematics 31 8.1. Tropical versus supertropical 31 9. Appendix B: The categorical approach 33 9.1. Categorical aspects of systemic theory 33 9.2. Functor categories 34 References 36 1. Introduction This paper, based on [39], is the continuation of a project started in [55, 56] as summarized in [57], in which a generalization of classical algebraic theory was presented to provide applications in related alge- braic theories. It is an attempt to understand why basic algebraic theorems are mirrored in supertropical algebra, spurred by the realization that some of the same results were obtained in parallel research on hypergroups and hyperfields [6, 24, 34, 35, 36, 61] and fuzzy rings [15, 17, 24], which lack negatives although each has an operation resembling negation 1 2. The underlying idea is to take a set that we want to study. In the situation considered here, also has a partial additive algebraic structureT which is not defined on all of ; this is resolved by havingT act on a set with a fuller algebraic structure. T T LorscheidA [44, 45] developed this idea when is a monoid which is a subset of a semiring . Since semirings may lack negatives, we introduce a formalT negation map ( ) in Definition 2.3, resemblingA negation, often requiring that generates additively. In tropical algebra,− ( ) can be taken to be the identity map. Or it can be suppliedT via “symmetrization”A ( 2.2), motivated by− [2, 21, 22, 53]. Together with the negation map, and comprise a pseudo-triple ( §, , ( )). This is rounded out to a pseudo- system ( , , ( ), ) withA a Tsurpassing relation , oftenA aT partial− order (PO), replacing equality in the algebraicA T theory.− Ironically, equality in classical mathematics is the only situation in which is an equivalence. We set forth a systemic foundation for affine geometry (based on prime systems) and representation theory, as well as to lay out the groundwork for further research, cf. [3] for linear algebra, [20] for exterior algebra, [37] for projective modules, and [38] for homology, and other work in progress. Familiar concepts from classical algebra were applied in [56] to produce new triples and systems, e.g., direct powers [56, Definition 2.6], matrices [56, 6.5], involutions [56, 6.6], polynomials [56, 6.7], localization [56, 6.8], and tensor products [56, 8.6].§ Recently tracts, a special§ case of pseudo-systems,§ were introduced§ in [7] in order to investigate matroids.§ At the end of this paper we also view systems categorically, to make them formally applicable to varied situations. Since the main motivation comes from tropical and supertropical algebra, in Appendix A ( 8), we coordinate the systemic theory with the main approaches to tropical mathematics, demonstrating§ the parallels of some tropical notions such as bend congruences (introduced by J. Giansiracusa and N. Gi- ansiracusa in [23], and tropical ideals (introduced by Maclagan and Rincon in [46]). For example, [57, Proposition 7.5] says that the bend relation implies -equivalence, cf. Definition 8.8. Category theory involving explicit algebraic structure◦ can be described in universal algebra in terms of operations and identities, reviewed briefly in 2.6. But the “surpassing relation” also is required. A crucial issue here is the “correct” definition§ of morphism. One’s initial instinct would be to take “homomorphisms,” preserving all the structure from universal algebra. However, this does not tie into hyperfields, for which a more encompassing definition pertains in [34, 36]. Accordingly, we define the more general -morphism in Definition 2.37, satisfying f(a + a′) f(a)+ f(a′). Another key point is that in the theory of semirings and their modules, homomorphisms are described in terms of congruences. The trivial congruences contain the diagonal, not just zero. Mor(A, B) denotes the semigroup of -morphisms from A to B, and has the sub-semigroup Hom(A, B) of homomorphisms. Since (Mor(A, B), +) and (Hom(A, B), +) no longer are groups, one needs to weaken the notion of additive and abelian categories, respectively in 9.2.2 to semi-additive categories and semi-abelian categories, [26, 1.2.7]. § § 1 In a hypergroup, for each element a there is a unique element called −a, such that 0 ∈ a ⊞ (−a). 2 A fuzzy ring A has an element ε such 1 + ε is in a distinguished ideal A0, and we define (−)a = εa. CATEGORIES WITH NEGATION 3 The tensor product and its abstraction to monoidal categories, an important aspect of algebra, is exposed in [18] for monoidal abelian categories. But, to our dismay, the functoriality of the tensor product runs into stumbling blocks because of the asymmetry involved in . So we have a give and play between -morphisms and homomorphisms, which is treated in 5. § 1.1. Objectives. Our objectives in this paper are as follows: (1) Lay out the notions in 2 of -module, negation map, “ground” triples and -systems, which should parallel the classical§ structureT theory of algebras. In the process, we considerT convolutions and the ensuing construction of polynomials, Laurent series, etc. (2) Study affine geometry in in 3.2 in terms of polynomials, with special attention to the theory of prime ground triples, to lay§ out the groundwork for the spectrum of prime congruences. (3) Elevate congruences to their proper role in the theory, in 3 and 6, since the process of modding out ideals suffers from the lack of negation. Investigate which§ class§ ical module-theoretic concepts (such as Hom and direct sums) have analogs for module triples over ground triples, viewed in terms of their congruences. Special attention is given to the tensor product of triples (Definition 5.5). In general -morphisms do not permit us to build tensor categories, as seen in Example 5.6, but we do have the usual theory using homomorphisms, in view of [43]. (4) Express these notions in categorical terms. This should parallel the theory of modules over semirings, which has been developed in the last few years by Katsov [40, 41, 42], Katsov and Nam [43], Patchkoria [51], Macpherson [48], Takahashi [59]. (5) Provide the functorial context for the main categories of this paper, as indicated in the diagram given in 7. (6) Relate this§ theory in Appendix A to other approaches in tropical algebra. (7) Define negation morphisms and negation functors, together with a surpassing relation, in the context of N-categories, in Appendix B. In the process, we generalize abelian categories to semi- abelian categories with negation, and lay out the role of functor categories. 1.2. Main results. Our results in geometry require prime congruences (Definition 3.7). Proposition A (Proposition 3.17). For every -congruence Φ on a commutative -semiring system, T T √Φ is an intersection of prime -congruences. T Theorem B (Theorem 4.6). Over a commutative prime triple , any nonzero polynomial f [λ] of degree n cannot have n +1 distinct -roots in .
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