Algebras with a negation map

0.9 MB·

by  Louis Halle Rowen

1
ALGEBRAS WITH A NEGATION MAP
LOUISHALLEROWEN
Abstract. Our objective in this is three-fold, the first two covered in this paper. In tropical mathe-
matics,aswellasothermathematicaltheoriesinvolvingsemirings,whentryingtoformulatethetropical
6 versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determi-
1 nants,Grassmannalgebras,Liealgebras,Liesuperalgebras,andPoissonalgebras,oneoftenischallenged
0 bythelackofnegation. FollowinganideaoriginatinginworkofGaubertandtheMax-Plusgroupand
2 brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation
maps, called systems, inthe context of universal algebra, showing how these encompass the morevi-
t
c able(super)tropicalversions,aswellashypergrouptheory. Specialattentionispaidtometa-tangible
O systems,whosealgebraictheoryisrichenoughtoprovideahostofstructuralresults. Basicresultsalso
areobtainedinlinearalgebra,linkingdeterminants tolinearindependence.

1 Formulatingthestructurecategoricallyenablesustoviewthetropicalizationfunctorasamorphism,
1 thereby further explaining the mysterious link between classical algebraic results and their tropical
analogs,aswellaswithhyperfields. Weusethetropicalizationfunctortoanalyzesometropicalstructures

] andproposetropicalanalogsofclassicalalgebraicnotions.
A In work currently in progress, having the basic category in place, we proceed to the third stage, a
R theoryofsheaves andschemesandderivedcategories withnegation.
.
h
t
a
m Contents
[
1. Introduction 3

3 1.1. A broad overview 4
v 1.2. Preliminaries 5
3
1.3. Introducing negation maps 7
5
1.4. The main tropical examples 8
3
0 1.5. Optional: Hypergroups 12
0 1.6. Introducing surpassing relations 13
2. 1.7. Introducing systems 15
0 1.8. Introducing symmetrization 17
6 2. Background from universal algebra 19
1
2.1. Algebraic structures 19
:
v 2.2. Varieties 20
i 2.3. Partial orders in universal algebra 21
X
2.4. (Optional) Application: -semirings† and their modules. 22
r T
a 2.5. Congruences 23
2.6. Free algebras 23
1
Thispaperoriginallywasentitled“Symmetriesintropicalalgebra”inarXiv:1602.00353[math.RA]
Date:October 12,2016.
2010 Mathematics Subject Classification. Primary16Y60,06F05, 13C10, 13C60,20N20, 18C05, 18E10,08A05, 12K10;
Secondary12K10,14T05, 08A72.
Key words and phrases. system, meta-tangible, negation map, symmetrization, congruence, tropical algebra, tropical
geometry, projective, hyperfield, fuzzy algebra, exploded algebra, ELT algebra, Lie algebra, Lie superalgebra, Grassmann
algebra,exterioralgebra,supertropicalalgebra,semigroup,module,semiring,semifield.
The author would like to thank the followingresearchers for helpful conversations: Oliver Lorscheid and Zur Izhakian
fordiscussionsontropicalization,MarianneAkianandGaubert,together withAdiNiv,fordiscussionsonsymmetrization
atBar-IlanUniversityandinParis,MaxKnebuschforinsightsonubsemirings,MattBakerforexplaininghyperfields,and
especiallytheroleofthe“signhyperfield,”andSergioLopezforhelpfulcommentsonthefirstarXivversion. Specialthanks
are due to Guy Shachar for a careful reading of the first arXivversion which led to several corrections, and to J. Jun for
findingseveralinaccuracies invariousversions,pointingouttheconnection tofuzzyrings,Henry’spaper,andmanyother
interestingpaperswithwhichhehasbeeninvolved.
TheauthorwouldalsoliketothanktheUniversityofVirginiaforitssupportduringtheinitialpreparationofthiswork.
Theauthor’sresearchwassupportedbyIsraelScienceFoundation grantNo. 1207/12.
1
2 LOUISROWEN
2.7. Homogeneous and multilinear operators 24
3. Tropical examples viewed in terms of universal algebra 25
3.1. Varieties arising naturally in tropical mathematics 25
3.2. Structures of tropical mathematics which do not comprise varieties 26
4. Negation maps and surpassing relation 27
4.1. Negation maps in universal algebra 27
4.2. Combining and comparing negation maps 28
5. Systems 29
5.1. A general overview of systems 29
5.2. Uniquely negated triples 29
5.3. The characteristic of a triple 31
5.4. Neutral elements 31
5.5. Polynomials and their roots 31
5.6. Triples with involution 32
5.7. Hyperstructures 33
6. Meta-tangible triples and their systems 34
6.1. The characteristic of a meta-tangible triple 36
6.2. Cancellative meta-tangible triples 37
6.3. Uniform elements and height in meta-tangible triples 38
6.4. Surpassing relations on meta-tangible triples 40
6.5. Most meta-tangible systems are matroidal and -reversible 41
T
6.6. -classical meta-tangible triples 43
T
6.7. Squares and sums of squares 44
6.8. Sign maps on 45
T
6.9. Classifying metasystems 46
6.10. Important examples of meta-tangible systems 46
6.11. Meta-tangible systems versus meta-tangible hypergroups 48
7. Symmetrization 49
7.1. Supermodules and super-semialgebras 49
7.2. Symmetrization of modules 49
7.3. Symmetrization of semirings and -semirings 50
T
7.4. Symmetrization in the language of universal algebra 50
7.5. Modification of symmetrization 51
7.6. The transfer principle 52
8. Categories of systems 52
8.1. Categories with negation 52
8.2. Morphisms of systems 52
8.3. Embedding hypergroups into systems 53
8.4. Congruences and -ideals on systems 53
T
8.5. Tensor products with a negation map, and their semialgebras 55
9. Linear algebra over a uniquely negated triple 56
9.1. Matrices over systems 56
9.2. Dependence relations of vectors 58
9.3. Ranks of matrices 59
10. Tropicalization 60
10.1. Tropicalization of Puiseux series 60
10.2. Tropicalization of classical systems defined over Puiseux series 61
11. Tropical structures arising from tropicalization 61
11.1. Exterior (Grassmann) semialgebras with a negation map 61
11.2. Nonassociative algebras with a negation map 63
11.3. Lie semialgebras and Lie super-semialgebras 63
11.4. Poisson algebras and their module congruences 64
12. Appendix A: Hyperfields as systems 65
12.1. Power sets of semigroups 65
ALGEBRAS WITH A NEGATION MAP 3
12.2. Major examples of hypergroups and hyperfields 67
13. Appendix B: Fuzzy rings 69
References 70
1. Introduction
Thispaperwasmotivatedbythe desiretounderstandamysteriousparallelbetweenstructuralresults
in what we will call the “classical algebraic theory” and theorems formulated directly in varied aspects
of tropical algebra, despite the former being taken over fields and the latter over the max-plus algebra
and related semifields. It is designed to lay the foundation for a unified algebraic theory, which also
encompasses diverse recent research,especially in the tropical setting and for hyperfields, and to a lesser
extent,“fuzzyrings.” Butourinitialpointofviewistropical. Oncetheoverallframework,theuniquely
negatedsystem,isestablished,itprovidesamechanismforobtainingeffectivedefinitionsofnewtropical
algebraic structures, and also provides a guide for applying classical algebraic techniques in these other
situations. For example, a useful definition of “characteristic” is given in Definition 5.10.
Tropicalizationoriginallywasviewedasalimitingprocesstakinglogarithmsandpassinginthelimiting
case to the max-plus algebra,whichis a semiring. Thus, tropicalalgebraalwayshas relied onthe theory
of semirings, which goes back to Costa [17] and Eilhauer [25], and for which we use [30] as our standard
reference. But lack of negatives obviously hampers the algebraic theory.
Over the years, various researchers, going back to Kuntzman [57] in 1972, have tackled the lack of
negation in the max-plus semiring, especially for matrices and the determinant. Some have used an
operation resembling negation. Gaubert [26] introduced such a structural approach in his dissertation,
motivated by [65, 71]; see for example [65, p.352, end of proof of (a)]. His work has been continued
together with the M. Plus group and Akian and Guterman and Henry, using a “symmetrized” theory
([64], [26], [6, 3.4], [27], [2], [35], [3], and [51, Appendix A]) leading to a general “transfer principle”
§
to generate semiring identities. More recently, Bertram and Easton [8] and Joo and Mincheva [52] have
utilized the “twist” of [2] to refine congruences on polynomial semirings.
As the field of Puiseux series came into play, the underlying semiring was viewed as the target of the
Puiseuxvaluation,whichdifferssomewhatfromthemax-plusalgebra. Towardsthisend,in[38,41,46,49]
a “supertropical”theory wasinitiated overa semiringR bymeans ofa“ghostmap”(where the negation
map actually is the identity map), with various applications to affine varieties, matrices, linear algebra,
and quadratic forms. In [2] and [47, 49] it was possible to transfer classical algebraic results to the
tropical theory by means of a somewhat mysterious “surpassing” relation on semirings, which satisfies
manypropertiesofequality,andreplacesequalityinmanygeneralizationsofclassicaltheorems,especially
for polynomials and matrix theory. This is given in Example 1.41(ii) for supertropical algebra, and in
Example1.41(iii)forGaubert’sdiodes. Thuswearemotivatedtoaskexactlyhowthissurpassingrelation
fits into the algebraic theory.
The same kind of relation (this time, as a subset), has also turned up in the theory of hypergroups.
Viro [74] has viewed tropical theory in terms of hyperfields, and it turns out that the hyperfield theory
can be embedded into the theory, as spelled out in Appendix A. Indeed, the recent spur in research
in hypergroups provides grounds for the study of “uniquely negated systems.” Recently ties have been
found in [29] between hyperfields and fuzzy rings, and these also are reflected in the theory.
Inalloftheseinstances,thereisaset ofmaininterest(themax-plusalgebra,thesymmetrizedmax-
T
plus algebra, and the underlying hypergroup), together with a special operation resembling negation
(the switch map or the hypernegative in the latter two cases), but its intrinsic algebraic structure is not
sufficient for satisfactory investigation, causing many algebraic results to be formulated and proved on
an ad hoc basis. The situation is significantly clarified by embedding into a much richer structure
T
(such as supertropical, symmetrized, or power set) which can be studied via well-known techniques
A
from universal algebra. The interplay between and is intriguing. In brief, a “system” is comprised
A T
of an algebraic structure , (often a semiring), a designated subset , a negation map ( ), and the
A T −
surpassing relation .
(cid:22)
Our major goal with these “systems” is to build an algebraic foundation that unifies all of these
approaches in a way that also includes the classical algebraic theory, and in which the “surpassing
4 LOUISROWEN
relation”is anintrinsic component that encompassesequality. But we wantthe axiomsto be sufficiently
restrictive to specialize naturally to our main examples from tropical mathematics and the theory of
hypergroups,therebyprovidinganaxiomaticset-upthatwilldrivethetheory,showingthewaytonatural
new definitions, and eventually yielding intrinsic theorems.
The appropriate setting for the investigation seems to be that of universal algebra, to be reviewed
below in 2.5, where we start with addition as the basic operation (of a semigroup), treated differently
from all others, and bring in other operations as seen fit. This provides a vehicle (the “system”) for
linking more sophisticated theorems from classical algebra and algebraic geometry to tropical algebra
(and also to hyperfields). This can be viewed in context of Lorscheid’s “blueprint,” but also involves
specific extra information to permit us to hone in on the applications, as described in Sections 4,5,6,7.
Systems behave well categorically,especially under tensor products, as considered in 8. In particular
§
one can turn to modules and linear algebra ( 9). They meld well with tropicalization ( 10), inspiring
§ §
tropical versions of classical algebraic structures in 11.
§
Since the tropical structures are distributive, but the applications to hyperfields and fuzzy rings are
notnecessarilydistributive,andsometimesrathertechnical,wedefersomedetailstoAppendicesAandB
respectively. The applications to hyperfields are direct and yield immediate results in hyperfield theory,
via the pointof view of 12.2.1. The applicationsto fuzzy ringsaremoretenuous, butin factgiverise to
§
a slightly more generalversion(Definition 13.2) which “explains” the role of invertible elements through
the structural results in Proposition 13.7 and Proposition 13.8.
1.1. A broad overview.
We start with some semigroup ( ,+), perhaps with extra structure (often multiplication), in which
A
we are interested in a certain subset called the tangible elements. could be all of in classical
T T A
algebra, or an ordered subgroup identified with the max-plus algebra (or related structures) in tropical
algebra, or a hyperfield. Usually, is taken to be a semiring. However, to accommodate application to
A
hyperrings we might relax distributivity, cf. 2.4.
§
Inspired by [2], we couple addition with a negation map (Definition 1.9), which is a formal map
a ( )a that satisfies allof the properties ofnegationexcept a+(( )a)=0. This comes automatically
7→ − −
for classical algebra and for hyperfields. Initially, negation is notably absent in the tropical theory, but
is circumvented in two main ways: The identity itself is a negation map, leading to the “supertropical
theory,” or else one can introduce a negation map, called a “symmetry” in [2], through the process of
“symmetrization” ( 1.8), passing to .
§ A×A
To simplify notation, we write a( )b for a+(( )b). To avoidambiguity,we then write the product of
− −
a and ( )b as a(( )b), which occurs much more rarely. Also we write ( )a for “a or ( )a,′′ and a( )b
− − ± − ±
for “a+b or a( )b.′′
−
Asto be expected,the flavorofthe theorydiffers accordingto whetherornot( )is the identity map,
−
called respectively the first and second kind (Definition 1.10). This enables us to distinguish between
“supertropical”and“symmetrized”tropicalalgebra,andhelpstoexplainwhytheoremsforsupertropical
algebras might fail for symmetrized algebras, as illustrated in [5].
We define a◦ := a( )a, called a quasi-zero. When considering structures with a multiplicative unit
−
element 1 (defined as satisfying 1a=a=a1, a), we define e:=1◦ =1( )1.
∀ −
Sofarwehavethedata( , ,( )),whichwecallatriple. Wesaythatthetripleisuniquelynegated
A T −
when for any a in , ( )a is the only element in for which a( )a is a quasi-zero. This already is a
T − T −
rather powerful condition, with some crucial consequences given in Proposition 5.2 and Corollary 5.3.
Nextwedefineasurpassing relation ,the majorexamplebeing definedbya biffb=a+c◦
◦ ◦
(cid:22) (cid:22) (cid:22)
for some c . We write a b when b a. In order to obtain that is a surpassing relation requires
◦
∈A (cid:23) (cid:22) (cid:22)
a basic assumption on the triple ( , ,( )), such as meta-tangibility (Definition 1.51), characterized
A T −
by the property that the sum of tangible elements that are not quasi-negativesof each other is tangible.
(Another example of surpassing relation is arising in the theory of hypergroups.) Ironically, instead
⊆
of being symmetric (and thus an equivalence), in the non-classical cases the surpassing relation often is
antisymmetric, although it restricts to equality on .
T
Altogether,ourstructureofchoice,asystem(Definition5.1),isaquadruple( , ,( ), ),where is
A T − (cid:22) T
thesetoftangibleelements,generating additively,( )isanegationmap,and isasurpassingrelation.
A − (cid:22)
Thiscoverstheclassicalcase,the“standard”supertropicalsemiring,the“symmetrized”semiringof[2,3],
ALGEBRAS WITH A NEGATION MAP 5
the “exploded” algebra [62], the “layered” semiring of [40], and some hyperfields (but not all, cf. the
examples in [7], reviewed in Remark 12.4).
Expressed in these terms, one of the obstacles to tropical structure theories has been to describe a◦
accurately. Theanswerseemstobetotreatthisdifferentlyfroma+bwhereb=( )a,suchasinuniquely
6 −
negated systems (Definition 1.50). Another example — we define ( )-bipotence by a+b a,b when
− ∈{ }
b = ( )a (Definition 1.51); this turns out to classify precisely the variants of the max-plus algebra that
6 −
have arisen in the tropical literature.
In 5.5 we discuss polynomials and their roots, to pave the way for affine geometry (but not in this
§
paper).
There are two ways of approaching systems — one is as the basic algebraic structure, such as a ring,
andthe otherisasasecondarystructure(suchasamoduleorhyper-module)whichprovidesinsightinto
the former. Ouremphasisinthispaperisonthe former,sinceonehastopausesomewhere,and72pages
seems enough. This covers the basic tropical algebraic structures, hypergroups, and analogs of classical
constructions.
Meta-tangible triples and their systems, lying at the center of this study, are treated in considerable
detail in 6, where we show in Theorem 6.18 that the cancellative meta-tangible triples are either ( )-
§ −
bipotent (Definition 1.51) or satisfy e+1=1. Their elements all have the “uniform” presentationgiven
in Theorem 6.25. Cancellative meta-tangible systems are classified in Theorem 6.57, often reducing to
the familiar examples from tropical theory.
One important application of systems (which actually motivated this paper) is the symmetrization
process, studied in these terms in 7.
§
Seeingthatuniquelynegatedsystems,especiallymeta-tangiblesystems,havearobustalgebraictheory,
we proceedto view them categoricallyin 8,utilizing the surpassingrelation asanessentialingredient
§ (cid:22)
inthedefinitionofmorphisminDefinition8.4. Tensorproductsaredefinedinthis context,althoughone
could lose meta-tangibility. This approachleads to an intrinsic version of tropicalization, as a morphism
of systems, given in 10.
§
Linear algebra over systems is particularly intriguing, since some of the supertropical results go over,
butothershavecounterexamples,asdiscussedin 9. Thedifferenceslargelycomefromwhether negation
§
is of the first or second kind. This delicate issue is treated explicitly in [5].
Forthoseresearchersalsointerestedinhypergroups,weincludeasemiring-likestructure( -semirings)
T
from universalalgebra, satisfying distributivity only over , encompassing hypergroups and hyperfields,
T
cf. 2.4. This specialized notion actually helps our intuition, since its assortment of examples, given
§
in Appendix A, casts a strong light on the axiomatic theory.
1.2. Preliminaries.
Inordertoelaboratefurthertheresultsinthispaper,weneedsomepreliminaryterminologyandfacts.
As customary, N denotes the positive natural numbers, N denotes N 0 , Q the rationalnumbers, and
0
∪{ }
R the real numbers, all ordered monoids under addition.
1.2.1. Ongoing hypothesis.
Fromnowon,wecarrytheongoinghypothesisthat( ,+)isasemigroup,withadistinguishedsubset
A T
of that additively generates . When contains an element 0, we write 0 for 0 0 and assume
A A A T T ∪{ }
that 0 additivelygenerates . (Presumingthattheextraoperatorspreserve intheappropriatesense,
T A T
this situation is realized when one replaces by the additive sub-semigroup spanned by 0, so this
A T
requirement is rather mild.)
Thishypothesis enablesusto define multilinearoperators( 2.7) on viatheir actiononthe elements
§ A
of , and to lift various properties from to . We define the height of an element c as the
minTimal t such that c = t a with eacTh a A . (We say that 0 has height 0.) The hei∈ghAt of is
i=1 i i ∈ T A
the maximal height of its elements. Thus has height 1 iff = or 0, and height 2 also will play an
P A A T T
important role. Height 3 involves extra considerations, as indicated for example in Definition 6.24 and
Theorem 6.25.
1.2.2. General preliminaries.
Recallthatamonoidisasemigroupwithatwo-sidedidentityelement,denotedas0foraddition,and
1 for multiplication. For any multiplicative semigroup :=( , ) we can formally adjoin the identity
M M ·
6 LOUISROWEN
element 1 by declaring that 1 a=a 1 =a for all a , and when dealing with multiplication
M M M
· · ∈M
we always work with monoids. We customarily write ab for a b.
·
In additive notation, when (M,+) is a semigroup, we write M for the monoid M 0 where 0
+0
∪{ } { }
is formally adjoined satisfying 0+a=a+0=a for all a M.
∈
A semiring† is a semiring (R,+, ,1 ) without 0, i.e., an additive Abelian semigroup (R,+) and
R
·
multiplicative monoid (R, ,1 ) satisfying the usual distributive laws. The reasonthat we do not always
R
·
require R necessarily to have the element 0 is that 0 just distracts from our true goal, which is negation
maps and quasi-zeros, not negatives.1 One may assume that R has an undesignated element 0 if one so
wishes. The theory of semirings† is essentially the same as that of semirings.
Definition 1.1. A semiring† (R,+, ,1 ) is a semifield† if (R, ) is an Abelian group. A semiring† R
R
· ·
is idempotent if a+a=a for all a R, R is bipotent if a+b a,b for all a,b.
∈ ∈{ }
Themax-plusalgebraisbipotent,butbipotence(barely)failsintheotherexamples,therebymotivating
( )-bipotence (Definition 1.51).
−
Definition 1.2. An R-module (often called a semimodule in the literature) over a semiring† R is
an additive monoid (M,+,0 ) together with scalar multiplication R M M paralleling the module
M
× →
axioms of classical algebra, although now one must stipulate that r0 =0 for all r in R.2
M M
For example, R := R 0 is naturally an R-module. Likewise, one defines submodules and
+0 R
∪ { }
homomorphic images,andthe semigroup(Hom(M,N),+,0), for any R-modules M andN. We saythat
a module can be written in the form M M when every element can be written uniquely as the sum
1 2
⊕
ofanelementofM andM . Thenwecandefine directsums,andM(I) := M where eachM =M.
1 2 i∈I i i
⊕
Definition 1.3. The free module over a semiring† R is R(I), denoted R(I) for short, where we identify
+0
the base element e with 1 in the i component.
i R
Wemayconsidersemiringmodules ratherthantheunderlyingsemirings,asinclassicalrepresentation
theory. Manyconceptsdonotinvolvemodulemultiplication,andareformulatedforadditivesemigroups.
In the otherdirectionthe following simple observationenables us to apply module theory to semigroups:
Remark 1.4. Any semigroup is an N-module in the obvious way.
Definition 1.5. As in classical algebra, a semialgebra over a commutative (associative) semiring† C
is a module which also has a multiplication with respect to which it becomes a semiring satisfying the
A
usual law
c(a a )=a (ca )=(ca )a , c C, a . (1.1)
1 2 1 2 1 2 i
∀ ∈ ∈A
Attheoutset,weworkwithassociativesemialgebrasbutlaterontheycanbenonassociative,according
to the context.
Werecalltheusualdefinitionofthemonoid semialgebraC[ ]ofamonoid overacommutative,
M M
associative semiring† C, by taking the free module over C whose base is the elements of , with
M
multiplication induced by the given multiplication in C and in , extended via distributivity.
M
Definition 1.6. A partial pre-order is a transitive relation satisfying a a for all a.
≤
A partially pre-ordered monoid is a monoid with a partial pre-order satisfying
M
a b implies ca cb, ac bc (1.2)
≤ ≤ ≤
forallelementsa,b,c . Amonoid isordered(resp.partiallyordered)ifitspre-order(resp.par-
∈M M
tial pre-order) is antisymmetric. We write PO for partial order.
Remark 1.7. (Q×, ) is not an ordered monoid under this definition, since taking inverses reverses the
·
order.
1Tobefair,oneoftenwantsazeroelementinordertobeabletodefinesuchfamiliaralgebraicvarietiesasxy=0,but
alsothiscouldbeviewedastheasymptoteofthehyperbolasxy=cascdecreases.
2Ifinsteadwestudymodulesoversemiringswithzero0R thenwealsostipulatethat0Ra=0M,∀a∈M.Thisleadsto
ambiguity in defining modules M over semirings† containing a zero element 0R that has not been designated as such; to
resolvethisambiguity,onecouldmodM outbytheequivalence givenby0Ra1≡0Ra2 forallai∈M.
ALGEBRAS WITH A NEGATION MAP 7
Recallthata(nonarchimedean) valuationfromaringRtoanorderedmonoid( ,+,0)isamonoid
G
homomorphism v :(R, ) satisfying
· →G
v(a+b) min v(a),v(b) , a,b R.
≥ { } ∀ ∈
It is well known that v( 1)=0, and if v(a)>v(b) then v(a+b)=v(b).
±
1.2.3. Digression: Modules over monoids.
For hypergroups, as we shall see shortly, addition passes outside the original set, which is why the
following more general version of modules could come in handy.
Definition 1.8. A monoid ( , ,1) acts on a set if there is a multiplication satisfying
T · S T ×S → S
1s=s and (a a )s=a (a s) for all a and s S.
1 2 1 2 i
∈T ∈
Amodule overamonoid( , ,1)isanAbeliansemigroup( ,+,0 )onwhich acts,alsosatisfying
A
A T · A T
the condition:
a0 =0 , a .
A A
∀ ∈T
1.3. Introducing negation maps.
Definition 1.9. A negation map on an additive semigroup ( ,+) is a semigroup homomorphism
A
( ): of order 2, written a ( )a. (Thus ( )(a+b)=( )a+( )b.)
− A→A ≤ 7→ − − − −
A negation map on a module M over a semiring† (R, ,+,1 ) is simultaneously a negation map on
R
·
the additive semigroup M(,+), as well as satisfying
( )(ra)=r(( )a), r R, a M. (1.3)
− − ∀ ∈ ∈
A negation map on a semiring† (R, ,+,1 ) is simultaneously a negation map on the additive semi-
R
·
group (R,+), as well as satisfying
( )(a a )=(( )a )a =a (( )a ), a R. (1.4)
1 2 1 2 1 2 i
− − − ∀ ∈
Negation maps are best understood as 1-ary operators in universal algebra3, as developed in 4, but
§
here is the specific case of main interest.
The two obvious examples are the identity map, and (for modules over rings) the usual negation map
( )a= a. This gives rise to two kinds of negation maps.
− −
Definition 1.10. A negation map ( ) is of the first kind if ( )a=a for all a . The negation map
− − ∈A
is of the second kind if ( )a=a for some a .
− 6 ∈T
Recall that ( , ( ),( )) is called a triple.
A T A −
Definition 1.11. A triple ( , ( ),( )) is -cancellative over a monoid ( , ,1) acting on , if
A T A − T T · A
a b=a b implies a =a for a ,a and b ( ). A -cancellative triple is -invertible if is a
1 2 1 2 1 2
∈T ∈T A T T T
multiplicative group.
In this paper, we always have = ( ), with ( , ) acting on in the natural way. In this context,
T T A T · A
we write cancellative for “ - cancellative.”
T
Lemma 1.12. When the triple ( , ,( )) is cancellative, the negation map ( ) is of the first kind iff
A T − −
( )1=1; ( ) is of the second kind iff ( )1=1.
− − − 6
Proof. ( )1=1 iff ( )a=a for all a, since we can cancel a. (cid:3)
− −
As in [3], one has:
Definition 1.13. When R already has a negation map, we say that the negation maps on R and M are
compatible if
( )(ra)=(( )r)a =r(( )a), r R, a M.
− − − ∀ ∈ ∈
3It is convenient to cast our considerations in terms of universal algebra. Although more complicated than the usual
algebraic structure theory because an intrinsic negative is not available, universal algebra enhances the tropical and su-
pertropicalstructures,andhasawiderangeofapplicationsgivenin§3.1.
8 LOUISROWEN
Example 1.14. Suppose R already has a negation map ( ), with 1 R. Then any R-module M has a
− ∈
compatible negation map given by ( )a=(( )1 )a. Thus, we can view the negation map on M in terms
R
− −
of the single element ( )1.
−
Also, any module homomorphism ϕ satisfies ϕ(( )a)=ϕ(( )1 a)=(( )1 )ϕ(a)=( )ϕ(a).
R R
− − − −
Thisraisesthe questionofhowtocopesimultaneouslywithdifferentnegationmapsatonce,whichwe
discuss briefly in 4.2. But our applications are for a single given negation map on M.
§
The following notion takes the role customarily assigned to the zero element. Recall that a◦ denotes
a( )a.
−
Remark 1.15. a◦ =(( )a)◦.
−
Lemma 1.16. If 0 ( ,+), then ( )0=0.
∈ A −
Proof. ( )0=( )0+0=( )0+(( )( )0)=( )(0( )0)=( )(( )0)=0. (cid:3)
− − − − − − − − −
Definition 1.17. Given a semigroup ( ,+) and a subset , we denote
A T ⊆A
◦ = a◦ :a , + := 0 ◦.
T { ∈T} T T ∪T
◦ is the analog of the “balanced elements” of [2].
T
Sometimes we write M instead of when we want to stress the module structure.
A
Lemma 1.18 ([2, Remark 4.5]). M◦ is a submodule of M, for any module M with negation map.
Proof. 0◦ =0( )0=0+0=0 M◦, and r(a◦)=r(a( )a)=(ra)( )(ra) =(ra)◦. (cid:3)
− ∈ − −
The introduction of the negation map to replace negatives enables us to develop the tropical analogs
of some of the most basic structures of algebra, applicable to Parker’s exploded algebra [62], Sheiner’s
ELT algebra[70], Grassmannalgebras[28], Blachar’sELT Lie algebras[11], Lie super-semialgebras,and
Poisson algebras,and unifies research coming from different directions as well.
Occasionally we want the following notion.
Definition 1.19. We say that a module M is -ordered if M◦ is ordered, and we write a > a if
1 ◦ 2
◦
a◦ >a◦.
1 2
In classical algebra,the only -order is trivial, since a◦ =0 for all a.
◦
Definition 1.20. Often we are given a multiplication . We say that contains 1 if there
T ×A→A A
is an element 1 such that 1a = a for all a . For example, this holds in any semiring† with a
∈ T ∈ A
negation map, and we designate several important elements, for future reference:
e=1( )1, e′ =e+1, e◦ =e+e. (1.5)
−
Also we define 1 = 1, and inductively n+1 = n+1, and N( ) to be the semiring† n : n N .
A { ∈ } ⊆ A
When is understood we write N for N( ).
A A
The mostimportant quasi-zeroin a semiring† is e, whichacts similarly to 0. But e need notabsorbin
multiplication! Rather:
Remark 1.21. e2 =((1( )1)2 =(1( )1)+(1( )1)=e◦ =2e.
− − −
1.4. The main tropical examples.
To prepare for the general algebraic theory, let us review some of the structures that have played a
major role so far in tropical algebra.
1.4.1. The max-plus algebra.
The parent structure in tropical algebra is the well-known max-plus algebra. We append the sub-
script to indicate the corresponding max-plus algebra, e.g., N or Q . But to emphasize the
max max max
algebraicstructure westill use the usualalgebraicnotationof and+ throughout,evenfor the max-plus
·
algebra. Themax-plusalgebrareallyconcernsorderedgroups,suchas(Q,+)or(R,+),whichareviewed
at once as max-plus semifields†, generalizing to the following elegant observation of Green:
ALGEBRAS WITH A NEGATION MAP 9
Remark 1.22. (i) Any ordered monoid ( , ) gives rise to a bipotent semiring†, where we define a+b
M ·
to be max a,b . Indeed, associativity is clear, and distributivity follows from the inequalities (1.2).
{ }
(ii) Conversely, any semigroup has a naturalpartial pre-order given by a a in if a =a +b
1 2 1 2
M ≥ M
for some b . It is a pre-order when is bipotent.
∈M M
(One could tighten this correspondence by considering lattice-ordered monoids as in [10, 53, 69], but
this would take us too far afield here.)
Remark 1.23. The max-plus algebra can be viewed as the triple ( , ,( )) of the first kind where
A T −
= , ( ) is the identity, and is equality, but then a=( )a=a◦, which is too crude for us, and we
T A − (cid:22) −
search for alternatives.
1.4.2. Supertropical semirings† and supertropical domains†.
To remedy Remark 1.23, we recall briefly some basics of supertropical algebra.
Definition 1.24. A ν-semiring† is a quadruple R := (R, , ,ν) where R is a semiring†, is a
T G T
submonoid, and R is a semiring† ideal, with a multiplicative monoid homomorphism ν : R ,
G ⊂ → G
satisfying ν2 =ν as well as the condition:
a+b=ν(a) whenever ν(a)=ν(b).
R is called a supertropical semiring† when ν is onto, is ordered, and
G
a+b=a whenever ν(a)>ν(b).
(In this paper we focus on supertropical semirings†, but the more general definition of ν-semiring†
enables one to work with polynomials and matrices.)
The elements of are called ghost elements and ν : R is called the ghost map. is the
G → G T
monoidof tangible elements,andencapsulatesthe tropicalaspect. Here we take( )a=a, a negation
−
map of the first kind.
Definition1.25. Asupertropicalsemiring† Riscalledasupertropicaldomain† whenthemultiplicative
monoid (R, ) is commutative, ν is 1:1, and R is -cancellative.
T
· | T
In this case ν : is a monoid isomorphism, and inherits the order from . In this case,
T
| T → G T G
the standard supertropical semifield† is ν (where customarily =Q orR ). Addition is
max max
T ∪T T
now given by
ν(a) whenever a=b,
a+b= a whenever a>b,

b whenever a<b.
R is called a supertropical semiring†when ν is onto, is ordered, and
G
a+b=a whenever ν(a)>ν(b).
The standard supertropical semifield is the standard supertropical semifield† with 0 adjoined.
We can write Rν in place of , which may be more suggestive. When dealing with supertropical
G
domains we expand both and to include the same 0. In other words, 0ν =0.
T G
We define e=1 +1 =1ν. Thus, e is the multiplicative unit of , and Rν =eR.
R R R G
Conversely, if e = 1 +1 is an additive and multiplicative idempotent of a semiring† R, then one
R R
can define = Re and the projection ν : R given by r re, thereby recovering the ν-semiring†
G → G 7→
structure.
Remark 1.26. As observed by Knebusch, any module M defined over a supertropical domain† itself
inherits a map ν :M M given by aν =ea.
→
Module homomorphisms send ghosts to ghosts, since
f(aν)=f(ea)=ef(a)=f(a)ν.
10 LOUISROWEN
1.4.3. The standard ( )-supertropical semifield†.
−
One can modify Definition 1.25 in the presence of a negation map ( ).
−
Definition 1.27. The standard ( )-supertropical semifield is a generalization of the standard su-
−
pertropical semifield, defined the same way but with addition now given as:
a whenever a>( )b,
a+b= − (1.6)
(a◦ for b=( )a.
−
The supertropical case can be viewed as a special case, when one takes ( )a=a (i.e., ( ) of the first
− −
kind) and a◦ =a+a=aν. One might expect the standard ( )-supertropicalsemifield to have the same
−
theory as the standard supertropical semifield, but as we shall see, there are significant differences for
negation maps of the second kind.
1.4.4. Layered semirings†.
“Layered semirings” are described in [40], also cf. [3, Proposition-Definition 2.12]. They are of the
formL , where L is the “layeringsemiring”and( , ) is an orderedmonoid. The motivating example
×G G ·
arises from a valuation L , so this contains extra information for tropicalization, to be discussed
→ G
later. We will also consider more generally when is a -ordered monoid with a negation map.
G ◦
Example 1.28. We assume that the “layering semiring” L has a negation map that we designate as .
−
We can define the layered semiring† as follows:
=L . Multiplication is defined componentwise. Addition is given by:
A ×G
(ℓ ,a ) if a >a ;
1 1 1 2
(ℓ ,a )+(ℓ ,a )= (ℓ ,a ) if a <a ; .
1 1 2 2  2 2 1 2
(ℓ1+ℓ2, a1) if a1 =a2.
Define eℓ =(ℓ,1). Thus e1 =1A =(1,1)∈T, and by induction, for ℓ∈N,
e =e +e =1+ +1,
ℓ ℓ−1 1
···
taken ℓ times. Then clearly the e generate a sub-semiring with negation map, and = e .
ℓ ℓ ℓ
A ∪ T
Example 1.29. Here are some natural explicit examples of layered semirings:
(i) L = N with = (ℓ,a) L : ℓ = 1 , and ( ) is the identity (thus of the first kind). ◦ is
T { ∈ ×G } − T
the layer 2. (The higher levels, if they exist, are neither tangible nor in ◦ when e′ =e. In fact
T 6
e′ =1+1+1 has layer 3.)
(ii) Take L=N in (i), and formally adjoin 0 at level 0, to be tangible.
0
{ }
(iii) L = Z with the usual negation, = (ℓ,a) L : ℓ = 1 , and ( )(ℓ,a) = ( ℓ,a), of the
T { ∈ ×G ± } − −
second kind.
(iv) L is the residue ring of a valuation, where now = (ℓ,a) L : ℓ = 0 , and ( )(ℓ,a) =
T { ∈ ×G 6 } −
( ℓ,a).
−
(v) L is a finite field of characteristic 2, where = (ℓ,a) L : ℓ = 0 , and ( ) is the
T { ∈ × G 6 } −
identity (thus of the first kind). This has several interesting theoretical properties, to be specified
in Example 6.19, in the context of meta-tangible systems.
(vi) Asomewhatmoreesotericexamplefromthetropicalstandpoint,butquitesignificantalgebraically.
Fixing n > 0, taking L = Z , identify each level modulo n. (This has height n and characteris-
n
tic n, cf. Definition 5.10.)
(vii) (Cycling) A weird example, which must be confronted. Fixing n > 0, take L = 0,...,n , but
{ }
with addition and multiplication given by identifying every number greater than n with n. In
other words,
k +k =n in L if k +k n in L;
1 2 1 2
≥
k k =n in L if k k n in L;
1 2 1 2
≥
There are two candidates for the negation map:
(First kind) The negation map is the identity.
•
(Second kind, for n>2) The negation map sends level ℓ to layer n ℓ.
• −