Algebras with a negation map

Abstract

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, one often is challenged by the lack of negation when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called systems, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in the various theories. Special attention is paid to meta-tangible T -systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. The systems studied here are "ground" systems, insofar as they are the underlying structure which can be studied via other "module" systems.Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby explaining the mysterious link between classical algebraic results and their tropical and hyperfield analogs. The tropicalization functor indicates analogs of classical algebraic notions, with applications to determinants, linear algebra, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras.

Commutative $ν$-algebra and supertropical algebraic geometry

“…However, the use of congruences raises the issue of defining primeness that exhibits the desired attributes. This approach is taken by Joó-Mincheva [41] and Rowen [58], who use twisted products, by Bertram-Easton [4], and by Lescot [45,46,47]. Primeness in [45,46,47] depends only on equivalence to zero, ignoring other relations that are determined by a congruence.…”

Hypergroups and hyperfields in universal algebra

“…This note, a companion to [19], grew out of a conversation with Matt Baker, in which we realized that the "tropical hyperfield" of [3] and [22, §5.2] is isomorphic to the "extended" tropical arithmetic in Izhakian's Ph.D. dissertation (Tel-Aviv University) in 2005, also cf. [10,11].…”

“…Thus, the theme is to embed the category of hyperfields (and their modules) into the category of semirings with negation (and their modules), as studied in [19], defined on power sets. The tricky part in passing to the power set is distributivity, which must be weakened at times to a notion that we call "weak distributivity," and we thereby weaken "semiring" to "T -semiring," where distributivity holds only with respect to a special subset T .…”

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Hypergroups are lifted to power semigroups with negation, yielding a method of transferring results from semigroup theory. This applies to analogous structures such as hypergroups, hyperfields, and hypermodules, and permits us to transfer the general theory espoused in [19] to the hypertheory.Definition 1.1. A monoid (A, •, 1) acts on a set S if there is a multiplication A × S → S satisfying 1s = s and (a 1 a 2 )s = a 1 (a 2 s) for all a i ∈ A and s ∈ S.A pre-semiring (A, •, +, 1) is a multiplicative monoid (A, •, 1 R ) also possessing the structure of an additive Abelian semigroup (A, +), on which (A, •) acts.A pre-semifield is a pre-semiring (A, •, +, 1) for which (A, •, 1) is a group.

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“…Hypergroups are lifted to power semigroups with negation, yielding a method of transferring results from semigroup theory. This applies to analogous structures such as hypergroups, hyperfields, and hypermodules, and permits us to transfer the general theory espoused in [19] to the hypertheory.…”