Hyperfields and dequantization

However, still, they have to find their way to the mainstream mathematics: still, it is easier to re-invent them than to find out that they have been invented.

When I realized that objects of this kind would be very useful in my efforts to find an appropriate base for tropical geometry, it was not difficult to build up a basic theory around my examples, but it took much longer to find the theory in literature.

I gave several talks about this matter (in particular, a talk [18] at MSRI workshop Tropical Structures in Geometry and Physics on November 30, 2009). The talks were attended by many mathematicians, but nobody told me about acquaintance with generalized fields having a multivalued addition. I am very grateful to Anatoly Vershik: when he listened to my talk in mid January, 2010, he told me that there were many papers devoted to multigroups and the likes. I found by Google a paper [10] of 2006 by Murray Marshall, where multiring and multifields were defined.

In the introduction to [10] Marshall wrote: "The idea of a multiring is very natural, although there seems to be no reference to it in the literature. Some basic properties of multigroups and multirings are established in Sections 1 and 2." In Section 2 Marshall defined also a multifield.

I have changed a draft of my paper, replacing my term "tropical field" by Marshall's "multifield" and uploaded it to arXiv as [20] . Of course, I contacted Marshall and expressed to him my excitement about multifields.

A couple of months later Marshall informed me that he learned from recent preprints [3] , [4] of Alain Connes and Caterina Consani that our multifields under the name of hyperfileds were introduced as early as in 1956 by Marc Krasner [6] . Krasner [6] introduced also hyperrings, but the notion of multiring introduced by Marshall [10] is more general than the notion of hyperring considered by Krasner, and the difference is essential for the applications that Marshall developed. Therefore both terms will be used. But Krasner's hyperfield and Marshall's multifield are absolutely the same, and the term hyperfield wins as it is older.

This paper is a new version of [20] . I insert the corresponding correction of references and terminology and make a few new remarks inspired by new information that I got from the papers [3] , [4] of Alain Connes and Caterina Consani.

Krasner, Marshall, Connes and Consani and the author came to hyperfields for different reasons, motivated by different mathematical problems, but we came to the same conclusion: the hyperrings and hyperfields are great, very useful and very underdeveloped in the mathematical literature.

Probably, the main obstacle for hyperfields to become a mainstream notion is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which forces to avoid multivalued maps at any cost.

I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Hyperfields, as well as multigroups, hyperrings and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to present new evidences for this.

I rediscovered hyperfields in an attempt [18] to find a true algebraic background of the tropical geometry. I believe hyperfields are to displace the tropical semifield in the tropical geometry. They suit the role better. In particular, with hyperfields the varieties are defined by equations, as in other branches of algebraic geometry.

These hyperfields are related to each other and the classical fields by hyperfield homomorphisms, and also via degenerations of the structures, similar to the Litvinov-Maslov dequantization [8] , which relates the semifield \((\R _+,+,\times )\) of non-negative real numbers with the usual arithmetic operations to the tropical semifield \(\T =(\R \cup \{0\},\max ,+)\). In particular, the fields \(\C \) and \(\R \) are dequantized. I call the results complex tropical hyperfield \(\tc \) and real tropical hyperfield \(\tr \).

A new hyperfield that does not appear via dequantizing a field, is a triangle hyperfield \(\mftr \). Its underlying set is \(\R _+\), and the addition is related to the triangle inequality: the sum of two non-negative numbers \(a\) and \(b\) is defined as the set of non-negative numbers \(c\) such that there exists an Euclidean triangle with sides of lengths \(a\), \(b\) and \(c\). This hyperfield dequantizes to a similar hyperfield \(\mfutr \) in which addition is related in the same way with the ultra-metric triangle inequality \(c\le \max (a,b)\).

Preliminary exposition of applications of the hyperfields introduced in this paper to the tropical geometry can be found in [19] .

In Section 3 , the notions related to multivalued generalizations of groups are discussed. This discussion is not complete, due to long history and a huge number of various level of the generalizations. We concentrate mainly on the notions needed to what follows.

In Section 4 we turn to multirings, hyperrings and hyperfields, their examples and general properties. Section 4 finishes with a discussion of multiring homomorphisms, their examples and first applications.

In Section 5 a few hyperfields related to triangle inequalities are introduced (triangle, ultra-triangle, tropical and amoeba hyperfields).

In Section 6 we introduce tropical addition of complex numbers and discuss its properties. In Section 7 subhyperfields of the complex tropical hyperfield are considered.

In Section 9 the dequantization are considered. We start with the Litvinov-Maslov dequantization, then study dequantization of the triangle hyperfield to the ultratriangle one, and dequantization of the field \(\C \) to the complex tropical hyperfield. All the dequantizations are related to each other at the end of Section 9 .

The reason for considering a multivalued map is usually a desire to emphasize an analogy to another situation, in which the corresponding map is univalued. In this paper we study a generalization of addition with the sum allowed to be multivalued. Usage of the modern set-theoretic terminology would make analogies with the usual addition more difficult to recognize. Cf., for example, [10] , where a multivalued binary operation is introduced, according to the standards of set theory, as a subset of the Cartesian cube of the underlying set, but a couple of pages after that the multivalued notation take over, anyway. Therefore we dare to use less conventional terminology of multivalued maps.

The term set-valued is used as a synonym for multivalued. A multivalued map \(f\) of \(X\) to \(Y\) is denoted by \(f:X\multimap Y\).

In the same spirit: the composition of multivalued maps \(f:X\multimap Y\) and \(g:Y\multimap Z\) is a multimap \(g\circ f:X\multimap Z\) that takes \(a\in X\) to \(g(f(a))=\cup _{y\in f(a)}g(y)\).

Other modifications may be quite confusing. For example, what is the preimage of a set \(B\subset Y\) under a multivalued map \(f:X\multimap Y\)? The set \(\{a\in X\mid f(a)\subset B\}\) or \(\{a\in X\mid f(a)\cap B\ne \varnothing \}\)? We see that the notion of the preimage of a set splits under the transition from univalued maps to multivalued ones. In cases of such ambiguity one needs to adjust the terminology. For example, the set \(\{a\in X\mid f(a)\subset B\}\) is called the upper preimage of \(B\) under \(f\) and denoted by \(f^+(B)\), while \(\{a\in X\mid f(a)\cap B\ne \varnothing \}\) is called the lower preimage of \(B\) under \(f\) and denoted by \(f^-(B)\). The names seems to be confusing because \(f^+(B)\subset f^-(B)\), the upper preimage is smaller than the lower one.

In order to take refuge in the standard set-theoretic terminology, we will pass from a multivalued map \(f:X\multimap Y\) to the corresponding univalued map \(X\to 2^Y\). The latter will be denoted by \(f^\uparrow \).

Sets, multimaps and their compositions form a category. Thus, although multivalued maps do not quite comply with the set-theoretic terminology, they fit comfortably to a more modern category-theoretic setup.

A binary multivalued operation \(f:X\times X\multimap X\) is said to be commutative if \(f(a,b)=f(b,a)\) for any \(a,b\in X\).

A binary multivalued operation \(f:X\times X\to 2^X\) is said to be associative if \(f(f(a,b),c)=f(a,f(b,c)\) for any \(a,b,c\in X\). Certainly, in the latter formula, by \(f\) we mean not only \(f\), but also its natural extension to all subsets of \(X\), that is

\[2^X\times 2^X\to 2^X:(A,B)\mapsto \bigcup _{a\in A,b\in B}f(a,b).\]

Let \(Y\subset X\) and \(f:X\times X\multimap X\) be a multivalued binary operation. A multivalued binary operation \(g:Y\times Y\multimap Y\) is said to be induced by \(f\), if \(g(a,b)=f(a,b)\cap Y\) for any \(a,b\in Y\). Of course, the induced operation is completely determined by the original one. It exists iff \(f(a,b)\cap Y\ne \varnothing \) for any \(a,b\in Y\) (recall that according to the definition of a multivalued operation the set \(g(a,b)\) is not allowed to be empty).

1. the operation \((a,b)\mapsto a\cdot b\) is associative;
2. \(X\) contains an element \(1\) such that \(1\cdot a=a=a\cdot 1\) for any \(a\in X\);
3. for each \(a\in X\) there exists a unique \(b\in X\) such that \(1\in a\cdot b\) and there exists a unique \(c\in X\) such that \(1\in c\cdot a\). Furthermore, \(b=c\). This element is denoted by \(a^{-1}\).
4. \(c\in a\cdot b\) iff \(c^{-1}\in b^{-1}\cdot a^{-1}\) for any \(a,b,c\in X\).

The axioms of multigroup presented above are not minimal. I have chosen them, because they give a good idea what is the structure and why multigroups generalize groups. To my taste, they are convenient if you know already that you deal with a multigroup and want to deduce something from axioms. For checking if something is a multigroup, a more minimalistic set of axioms, like the one provided by following theorem, may serve better.

Proof.

Notice, that in the proof of Theorem 3.A we proved that in any multigroup \(1^{-1}=1\) and \((a^{-1})^{-1}=a\).

It is easy to verify that \(X\) with is a multigroup and \(-a=a\) for any \(a\in X\).

This construction gives \(Q_1\) if \(X=\{0,1\}\) and \(0\prec 1\).

In the same situation \(X\) can be turned into a different multigroup. For this, define a binary multivalued operation

It is easy to verify that \(X\) with is a multigroup and \(-a=a\) for any \(a\in X\). For \(X=\{0,1\}\) and \(0\prec 1\) this construction gives a group. If \(X\) consists of more than 2 elements, the operation is truly multivalued.

We will call a linear order multigroup, and a strict linear order multigroup.

Yet another multigroup of three elements can be defined as follows. In \(\{0,1,2\}\) define operation \(\tplus \) by formulas \(0\tplus x=x\tplus 0=x\) for any \(x\), \(1\tplus 1=2\), \(1\tplus 2=2\tplus 1=\{0,1\}\), \(2\tplus 2=\{1,2\}\). Denote this multigroup by \(M\).

A multigroup homomorphism \(f:X\to Y\) is said to be strong if \(f(a\cdot b)= f(a)\cdot f(b)\) for any \(a,b\in X\). If \(Y\) is a group, then any multigroup homomorphism \(f:X\to Y\) is strong.

Example. If \(X\) and \(Y\) are linearly ordered sets with the smallest elements \(0_X\) and \(0_Y\), respectively, then any monotone map \(X\to Y\) mapping \(0_X\) to \(0_Y\) is a multigroup homomorphism. Such a map is a strong homomorphism iff it is injective on the complement of the preimage of \(0_Y\).

A strong submultigroup \(Y\) of a multigroup \(X\) is said to be normal, if \(a^{-1}\cdot Y\cdot a=Y\) for any \(a\in X\). Observe, that a normal submultigroup of \(X\) contains the set \(a\cdot a^{-1}\) for any \(a\in X\).

For a multigroup homomorphism \(f:X\to Y\), the set \(\{a\in X\mid f(a)=e\}\) is called the kernel of \(f\) and denoted by \(\Ker f\). Obviously, this is a normal submultigroup of \(X\).

Proof.

Let \(\Ga ,\Gb \in X/\Ker f\) and their images \(\bar f(\Ga )\), \(\bar f(\Gb )\) under \(\bar f:X/\Ker f\to Y\) coincide. Take representative \(a,b\in X\) of \(\Ga \) and \(\Gb \), respectively. Then \(f(a)=\bar f(\Ga )=\bar f(\Gb )=f(b)\). Since \(f\) is strong, \(f(b^{-1}a)=f(b)^{-1}f(a)=f(a)^{-1}f(a)\ni 1\). Thus \(b^{-1}a\cap \Ker f\ne \varnothing \). Therefore there exists \(c\in b^{-1}a\cap \Ker f\). Then \(a\in bc\subset a\Ker f\) and \(\Ga =\Gb \).

The assumption that \(f\) is strong is necessary here. Without this assumption, a multigroup homomorphism with a trivial kernel may be non injective. On the other hand, most of interesting multigroup homomorphisms are not strong. This is a major new phenomenon distinguishing multigroups from groups.

Here is the simplest example: \(Q_2\to Q_1:1,-1\mapsto 1,0\mapsto 0\). It is easy to see that \(f\) is a multigroup homomorphism with \(\Ker f=\{0\}\), but \(f\) is not injective. In order to verify that \(f\) is not strong, consider \(f(1\cdot 1)=f(1)=0\), on the other hand, \(f(1)\cdot f(1)=1\cdot 1=\{0,1\}\).

Often the terms multigroup and hypergroup were used for objects of wider classes. For example, Dresher and Ore [5] used the word multigroup for much wider class of object, while what is called multigroup above, Dresher and Ore [5] would call a regular reversible in itself multigroup with an absolute unit.

The definition given in Section 3.1 seems to be the narrowest and closest multivalued generalization of the notion of group. In comparatively recent literature exactly the same notion was considered by S. D. Comer [2] (under the name of polygroup) and M. Marshall [10] . A. Connes and C. Consani [4] consider the same notion under the name of hypergroup, but restrict themselves to commutative hypergroups.

There is another breed of multigroups in which the value of the operation contains a fixed number of elements some of which may coincide to each other. Thus the operation takes values in the \(n\)th symmetric power of the set rather than in the set of all its subsets. This kind of multigroups was considered by Wall [24] and, more recently, by Buchstaber and Rees [1] . The author is not aware about any construction which would allow to relate multigroups of this kind with multigroups defined in Section 3.1 .

• \((X,\tplus )\) is a commutative multigroup,
• \((X,\cdot )\) is a monoid with unity \(1\) (i.e., multiplication \((a,b)\mapsto a\cdot b\) is associative and \(1\cdot a=a=a\cdot 1\) for any \(a\in X\)),
• \(0\cdot a=0\) for any \(a\in X\).
• the multiplication is distributive over \(\tplus \) in the sense that for every \(a\in X\) maps \(X\to X\) defined by formulas \(x\mapsto a\cdot x\) and \(x\mapsto x\cdot a\) are homomorphisms of multigroup \((X,\tplus )\) to itself.

The distributivity means that \(a\cdot (b\tplus c)\subset a\cdot b\tplus a\cdot c\) and \((b\tplus c)\cdot a\subset (b\cdot a)\tplus (c\cdot a)\) for any \(a,b,c\in X\).

A multiring is said to be commutative if the multiplication is commutative.

An equivalent description of this strong form of distributivity: for every \(a\in X\) maps \(X\to X\) defined by formulas \(x\mapsto a\cdot x\) and \(x\mapsto x\cdot a\) are strong homomorphisms of the multigroup \((X,\tplus )\) to itself.

Hyperrings were introduced by Krasner [6] , see also [7] and [3] . Multirings were introduced by Marshall [10] . In Marshall's work the extra generality of the notion of multirings is used: some of the multirings that he considers in [10] are not hyperrings.

For example, in a usual ring, distributivity implies that \((a+b)(x+y)=ax+ay+bx+by\). In a multiring and even in a hyperfield the proof fails. Moreover, the equality

\[(a\tplus b)(x\tplus y)=ax\tplus ay\tplus bx\tplus by\]

may be incorrect, see Sections 5.1 , 6.4 .

Let us analyze, why the arguments that deduce \((a+b)(x+y)=ax+ay+bx+by\) from distributivity for univalued addition do not work for multivalued addition. In the univalued case, \(x+y\) is just an element, and one can apply distributivity: \((a+b)(x+y)=a(x+y)+b(x+y)\). Then for each summand distributivity is applied again, giving the equality.

In the case of multivalued addition \(\tplus \), \((x\tplus y)\) is not an element, but a set. Therefore the distributivity \((a\tplus b)c=ac\tplus bc\), in which \(c\) is a single element (that is an axiom in a multiring) cannot be applied in the situation when \(c\) is a set \(x\tplus y\).

Proof.

The opposite inclusion \((a\tplus b)(x\tplus y)\supset ax\tplus ay\tplus bx\tplus by\) in some multirings does not hold true (see Section 6.4 ). However, there are multirings in which it is true. Such multirings will be called doubly distributive.

In a doubly distributive multiring, \(\left ({\displaystyle \top }_{i=1}^na_i\right ) \left ({\displaystyle \top }_{j=1}^mb_j\right )= {\displaystyle \top }_{i,j}a_ib_j \). This can be easily proved by induction over \(m\) and \(n\).

Following Connes and Consani [3] , we will call the two-element hyperfield \(Q_1\) the Krasner hyperfield and denote it by \(\K \). It can be obtained from any field \(k\) with more than two elements by identifying all invertible elements. This is a multiplicative factorization (see Section 4.12 below) that was invented by Krasner [7] . To the best of my knowledge, \(\K \) did not appear in Krasner's papers.

The hyperfield \(Q_2\) is called the sign hyperfield and denoted by \(\S \).

These two hyperfields are doubly distributive.

Multigroup \(M\) defined also in Section 3.5 above cannot be turned into a hyperfield, unless a multivalued multiplication would be allowed. In this paper I prefer to stay with univalued multiplications only. If a multiplication in a hyperfield was allowed to be multivalued, one could define \(1\cdot x=x\) and \(0\cdot x=0\) for any \(x\) and \(2\cdot 2=\{1,2\}\). Then the multiplicative multigroup of \(M\) would be isomorphic to \(Q_1\).

Recall that the characteristic of a ring is the smallest positive integer \(n\) such that the sum \(1+\dots +1\) of \(n\) summands is 0, and zero if any sum \(1+\dots +1\) does not vanish.

This definition can be reformulated as follows: an integer \(n\) is the characteristic of a ring if \(n\) is the smallest positive number such that the sum \(1+\dots +1\) of \(n+1\) summands equals 1; if \(1+\dots +1\ne 1\) for any number \(k>1\) of summands, than the characteristic is zero.

For multirings, straightforward generalizations of these two definitions are not equivalent. I propose to preserve the old term of characteristic for the number defined by a generalization of the first definition, and to call the second one C-characteristic in honor of A. Connes and C. Consani, who discovered the opportunity of speaking about hyperrings of characteristic one, see [3] and [4] .

A natural number \(n\) is called the characteristic of a multiring if this is the smallest natural number such that \(0\in 1\tplus 1\tplus \dots \tplus 1\) where the number of summands on the right hand side is \(n\). A multiring which has no finite characteristic \(n\ge 2\) is said to be of characteristic 0. The characteristic of a multiring \(X\) is denoted by \(\chr X\).

A natural number \(n\) is called the C-characteristic of a multiring if \(n\) is the smallest positive number such that the sum \(1\tplus \dots \tplus 1\) of \(n+1\) summands contains 1; if \(1\not \in 1\tplus \dots \tplus 1\) for any number \(k>1\) of summands, than the C-characteristic is zero. The C-characteristic of a multiring \(X\) is denoted by \(\cchr X\).

Obviously, a multiring of characteristic \(p\ne 0\) has C-characteristic \(\le p\). On the other hand, \(\chr \S =0\) and \(\cchr \S =1\), while \(\chr \K =2\) and \(\cchr \K =1\).

A multiring is said to be idempotent, if \(a\tplus a=a\) for any \(a\) in it. A multiring is idempotent iff \(1\tplus 1=1\) in it.

An idempotent multiring has C-characteristic 1, but the converse is not true: in a multiring of C-characteristic 1 the set \(1\tplus 1\) may consist of more than one element. For example, \(\S \) is idempotent, \(\K \) is not idempotent (because \(1\tplus 1=\{0,1\}\) in \(\K \)), but both have C-characteristic 1.

The characteristic of an idempotent multiring is 0, because in it \(1\tplus \dots \tplus 1=1\) for any number of summands. In particular, \(\chr \S =0\).

A fundamental importance of the characteristic in the theory of rings comes from the fact that the characteristic determines the minimal subring of the ring. For multirings no structural theorem of this sort is known.

Moreover, there is no commonly accepted notion of submultiring or even subhyperfield. The point of disagreement is whether to require that the subset underlying a submultiring would be closed under the multivalued addition, or just require that the intersection of the subset with the sum of any of its two elements would be non-empty. In the univalued situation there is no difference between these two requirements.

If we accept the alternative in which the subset contains the whole sum of any two of its elements, then there is no hope for a reasonable list of simple hyperfields.

Under the other alternative, I am not aware about any conjectural list of simple hyperfields. However, the following two simple results in this direction sounds inspiring.

Theorem 4.C.

In any multiring \(R\) with \(\chr R=0\) and \(\cchr R=1\) the set \(\{-1,0,1\}\) inherits from \(R\) operations identical to the hyperfield operations in the sign hyperfield \(\S \).

Theorem 4.D.

In any multiring \(R\) with \(\chr R=2\) the set \(\{0,1\}\) inherits from \(R\) operations identical to the operations either in the Krasner hyperfield \(\K \), if \(\cchr R=1\), or in the field \(\F _2\), if \(\cchr R=2\).

The linear order \(\prec \) gives rise to two multigroup structures in \(Y\), with additions and , defined in Section 3.4 . Extend the multiplication in the group to by defining for each . It is easy to see that with any of the additions, either or , and this multiplication is a doubly distributive hyperfield.

The Krasner hyperfield can be obtained via this construction with applied to the trivial group. The construction with applied to the trivial group gives the field .

The sign hyperfield cannot be presented as a hyperfield of a linear order, because is idempotent, while any hyperfield of linear order is of characteristic 2: indeed, .

There are many well known commonly used maps which are multiring homomorphisms. Below we consider a few examples.

\[\R \to \{0,\pm 1\}:x\mapsto \begin {cases} \frac {x}{|x|}, &\text { if }x\ne 0\\ 0, &\text { if }x=0\end {cases} \]

is a multiring homomorphism of the field \(\R \) to the hyperfield \(\S \). For generalizations of this, see Marshall [10] .

As in the ring theory, for any ideal \(I\) of a multiring \(X\) one can construct the quotient \(X/I\), and a multiring structure in \(X/I\) such that the projection \(X\to X/I\) is a strong multiring homomorphism. Any multiring homomorphism \(f:X\to Y\) admits a natural factorization \(X\to X/\Ker f\to Y\). If \(f\) is surjective and strong, then the induced multiring homomorphism is an isomorphism.

The assumption that \(f\) is strong is necessary here. Without this assumption, a multiring homomorphism with a trivial kernel may be non injective. On the other hand, most of interesting multiring homomorphisms are not strong. This is a major new phenomenon distinguishing multirings from rings, cf 3.9 .

The example \(\S \to \K :1,-1\mapsto 1,0\mapsto 0\) of a non-injective multigroup homomorphism with trivial kernel considered in Section 3.9 above, is in fact a multiring homomorphism of the sign hyperfield \(\S \) to the Krasner hyperfield \(\K \) with the hyperfield structures defined in Section 4.5 .

The sign homomorphism \(\R \to \S \) defined in Section 4.9 above is also a non-injective multiring homomorphism with trivial kernel.

In a hyperfield \(X\), the only ideals are \(\{0\}\) and \(X\).

If \(f:X\to Y\) is a multiring homomorphism, and \(X\) is hyperfields, then either \(\Ker f=0\) or \(f=0\). Indeed, any ideal of a hyperfield \(X\) is either \(0\) or \(X\) exactly for the same reasons as if \(X\) was a field.

A hyperfield belongs to the traditional algebra at least in its multiplicative structure. In a hyperfield the complement of the zero is a commutative group. A non-trivial multiring homomorphism between hyperfields is a group homomorphism of the multiplicative groups. As such, it has a kernel, the preimage of unity.

In the univalued algebra, preimages of any two elements under a ring homomorphism are cosets related by translations which map bijectively one of them onto another. In multivalued algebra this phenomenon has no analogue. The formula \(x\mapsto x+a\) defining a translation by \(a\) in a ring, in a multiring turns into \(x\mapsto x\tplus a\) which defines a multivalued map. This map restricted to a preimage \(f^{-1}(b)\) of an element \(b\) under a multiring homomorphism \(f:X\to Y\) does not send it to the preimage of an element, but to the preimage of a set \(b\tplus f(a)\), and this restriction is not invertible. So, everything is broken.

If \(X\) and \(Y\) are hyperfields and \(f:X\to Y\) is a multiring homomorphism, then nonempty preimages of any non-zero element \(b\in Y\) is related via natural bijections, which are multiplicative translations, with \(f^{-1}(1)\). For any \(\Gb \in f^{-1}(b)\) formula \(x\mapsto \Gb ^{-1}x\) maps \(f^{-1}(b)\) onto \(f{-1}(1)\), and this map has inverse \(x\mapsto \Gb x\). The set \(f^{-1}(1)\) is the kernel of the group homomorphism \(X\sminus 0\to Y\sminus 0\) induced by \(f\). Denote this kernel by \(\Ker _mf\) and call it the multiplicative kernel of \(f\). Obviously, \(\Ker _mf\) is a subgroup of the multiplicative group of \(X\).

Some fragments of this nice picture take place in a more general setup, when \(X\) and \(Y\) are multirings and \(f:X\to Y\) is a multiring homomorphism. Still the multiplicative kernel \(\Ker _mf\) is defined as \(f^{-1}(1)\). This set is obviously closed under multiplication, but may be not a subgroup. Let \(b\in f(X)\) and \(\Gb \in X\) such that \(f(\Gb )=b\). Then multiplication by \(\Gb \) maps \(\Ker _mf\) to \(f^{-1}(b)\). However, as \(\Gb \) may be non-invertible, the construction for the inverse map \(f^{-1}(b)\to \Ker _mf\) is not available. Moreover, simple examples show that the map \(x\mapsto \Gb x:\Ker _mf\to f^{-1}(b)\) may be neither injective nor surjective.

Elements \(\Gb ,\Gg \in X\) have the same image under a map \(f\) with given \(\Ker _mf\) if there exist \(s,t\in \Ker _mf\) such that \(s\Gb =t\Gg \). This is the weakest sufficient conditions, which can be formulated solely in terms of \(\Ker _mf\). However, this is not a necessary condition.

The resulting hyperfield is denoted by \(X/_mS\). As a set, this is \((X^{\times }/S)\cup \{0\}\), a disjoint union the zero and the quotient of the multiplicative group \(X^{\times }\) by the subgroup \(S\). The multiplication in \(X/_mS\) is defined by the multiplication in the quotient group and the identity \(x0=0\). The addition in \(X/_mS\) induced by the addition in \(X\). For cosets \(aS,bS\in X^{\times }/S\) the sum is \(\{cS\mid c\in aS\tplus bS\}\), where \(\tplus \) denotes the addition of subsets of \(X\) induced by the addition in \(X\).

The natural map \(X\to X/_mS\) is a multiring homomorphism with multiplicative kernel \(S\).

Examples. 1. \(X/_m(X\sminus \{0\})=\K \) for any hyperfield \(X\ne \F _2\).
2. \(\R /_m\R _{>0}=\S \).

Marshall [10] , Example 2.6 introduced the multiplicative factorization for more general situation in which \(X\) is an arbitrary multiring and \(S\) any subset of \(X\) closed under multiplication. Then \(X/_mS\) is a multiring obtained as the set of equivalence classes for the following equivalence relation: \(a\sim b\) if there exist \(s,t\in S\) such that \(sa=tb\). If \(0\in S\), then \(X/_mS=0\).

Marshall's papers [10] , [11] contain numerous interesting applications of this construction. We restrict here to a simple elementary example that was not considered in these papers.

In a ring \(\Z \) of integers, let \(S\) be the set of all odd numbers. Then \(\Z /_mS\) can be identified with the set \(\{2^n\mid n=0,1,2,\dots \}\) of powers of 2. The multiplication in this multiring is the usual multiplication of powers of \(2\) (i.e., addition of the exponents). The multivalued addition is the strict linear order operation from Section 3.4 for the order opposite to the ordering \(<\) (i.e., \(2^p\prec 2^q\) iff \(p>q\)). This operation addresses to the following question: given two powers of 2, what is the highest power of 2 that can divide a sum of two integers \(m\) and \(n\) for which the highest powers of 2 that divide \(m\) and \(n\) are the given powers of 2. Clearly, \(\chr \Z /_mS=2\) and \(\cchr \Z /_mS=2\).

If in this example we would replace 2 by an odd prime number \(p\), then the strict linear order operation would be replaced by a non-strict one, the characteristic of the multiplicative quotient would be still 2, and the C-characteristic would change to 1.

Notice that the kernel of any multiring homomorphism \(f:X\to \K \) is a prime ideal in \(X\). Vice versa, any prime ideal can be presented in this way. Indeed, for any prime ideal \(I\) of a multiring \(X\), define

\[f_I:X\to \K :x\mapsto \begin {cases} 0, &\text { if }x\in I,\\ 1, &\text { if }x\not \in I. \end {cases} \]

This gives a multiring interpretation of prime ideals in usual rings. Thus, the prime ideal spectrum \(Spec K\) of a multiring \(K\) can be identified with the set of multiring homomorphisms \(K\to \K \).

In other words, is the set of all real numbers such that there exists an Euclidean triangle with sides of lengths .

Theorem 5.A.

The set \(\R _+\) with the multivalued addition and usual multiplication is a hyperfield.

Proof.

This addition is obviously commutative. It is also associative. In order to prove this, just observe that both and coincide with the set of real numbers \(x\) such that there exists a Euclidean quadrilateral with sides of lengths \(a,b,c,x\).

The usual multiplication is distributive over .The role of zero is played by \(0\). The negation \(a\mapsto -a\) for is identity, as for any \(a\in \R _+\) the only real number \(x\) such that is \(a\).

This hyperfield is called the triangle hyperfield and denoted by \(\mftr \).

Proof.

Indeed, .‘:Therefore .

On the other hand,

contains 0, because there exists an isosceles trapezoid with sides 4, 2, 1, and 2. In fact, .

The operation appears in the representation theory. Denote by \(V^{(a)}\) the \(a\)th irreducible representation of \(sl_2\C \) (i.e., the symmetric power \(\Sym ^aV\) of the standard 2-dimensional representation \(V\)). Then the set parametrizes the set of irreducible representations of \(sl_2\C \) which are the summands in \(V^{(a)}\otimes V^{(b)}\):

the multiplication is the usual multiplication of real numbers. As any hyperfield of a linear order, this one is doubly distributive, see Section 4.7 .

There is another way to construct the same hyperfield. It is completely similar to the construction of the triangle hyperfield of Section 5.1 , but with the triangle inequality replaced by the non-archimedian (or ultra) triangle inequality \(|c|\le \max (|a|,|b|)\). This hyperfield is called the ultratriangle hyperfield and denoted by \(\mfutr \).

The hyperfield structure of \(\mftrop \) can be obtained by the construction of Section 4.7 applied to the additive group of all real numbers with the usual order \(<\). The hyperfield addition here differs from the semifield addition \((a,b)\mapsto \max (a,b)\) in \(\T \) only on the diagonal: , although .

Since \(\mftrop \) will play an important role in what follows, let me describe it explicitly and independently of constructions above. The underlying set of \(\mftrop \) is \(\R \cup \{-\infty \}\), the addition is

the multiplication is the usual addition of real numbers extended in the obvious way to \(-\infty \), the hyperfield zero is \(-\infty \), the hyperfield unity is \(0\in \R \).

The addition in \(\mfamb \) is defined by formula

while the multiplication in \(\mfamb \) is the usual addition.

• If \(|a|>|b|\), then \(a\smallsmile b=a\).
• If \(|a|<|b|\), then \(a\cplus b=b\).
• If \(|a|=|b|\) and \(a+b\ne 0\), then \(a\cplus b\) is the set of all complex numbers which belong to the shortest arc connecting \(a\) with \(b\) on the circle of complex numbers with the same absolute value.
  In formulas: if \(a=re^{\Ga i}\), \(b=re^{\Gb i}\) with \(|\Gb -\Ga |<\pi \), then \(a\cplus b=\{re^{\Gf i} \mid |\Ga -\Gf |+|\Gf -\Gb |=|\Ga -\Gb |\}\).
• If \(a+b=0\), then \(a\cplus b\) is the whole closed disk \(\{c\in \C \mid |c|\le |a|\}\).

The zero plays the same role of the neutral element as it plays for the usual addition: \(a\cplus 0=a\) for any \(a\in \C \).

Furthermore, for any complex number \(a\) there is a unique \(b\) such that \(0\in a\cplus b\). This \(b\) is \(-a\).

A straightforward proof is elementary, but quite cumbersome. It is postponed till Appendix 1.

Indeed, all the constructions and characteristics of summands involved in the definition of tropical addition are invariant under multiplication by a complex number: the ratio of absolute values of two complex numbers is preserved, an arc of a circle centered at \(0\) is mapped to an arc of a circle centered at \(0\), a disk centered at \(0\) is mapped to a disk centered at \(0\).

Proof.

The proof of Theorem 6.D is elementary and straightforward. See Appendix 2.

Corollary 6.E.

The tropical sum of any finite set of complex numbers equals the tropical sum of a subset consisting at most of three summands. If the tropical sum does not contain the zero, then the number of summands can be reduced to two.

Corollary 6.F.

The tropical sum of a finite set of complex numbers contains the zero iff the zero is contained in the convex hull of the summands having the greatest absolute value.

In particular, any subfield of \(\C \) inherits structure of hyperfield from \(\tc \).

\[ a\cplus _{\R } b = \begin {cases} \{a\}, &\text { if }\quad |a|>|b|,\\ \{b\}, &\text { if }\quad |a|<|b|,\\ \{a\}, &\text { if }\quad a=b,\\ [-|a|,|a|], &\text { if }\quad a=-b. \end {cases} \]

The operation \((a,b)\mapsto a\cplus _{\R }b\) is called the tropical real addition or even just tropical addition, when there is no danger of confusion. The set \(\R \) with the tropical real addition and usual multiplication is called the tropical real hyperfield and denoted by \(\tr \).

Proof.

Recall that there is a multiring homomorphism

\[\R \to \{0,\pm 1\}:x\mapsto \begin {cases} \frac {x}{|x|}, &\text { if }x\ne 0\\ 0, &\text { if }x=0\end {cases} \]

of the field \(\R \) to the hyperfield \(\S \).

This map is also a multiring homomorphism \(\tr \to \S \).

The map

\[\C \to \Phi : z\mapsto \begin {cases} \dfrac {z}{|z|}, &\text { if }z\ne 0\\ 0, &\text { if }z=0 \end {cases} \]

is called the phase map. This is a multiring homomorphism in two senses: \(\C \to \Phi \) and \(\tc \to \Phi \).

A classical example of a semifield is the set \(\R _+\) of non-negative real numbers with the usual addition and multiplication. Another semifield structure in the same set is defined by replacing the usual addition with the operation of taking the greatest of two numbers: \((a,b)\mapsto \max (a,b)\).

There is an isomorphism of the tropical semifield \(\T \) onto the semifield \(\R _{\ge 0, \max ,\times }\) mapping \(x\mapsto \exp x\) for \(x>0\), and \(-\infty \mapsto 0\).

Observe that the semifield addition \((a,b)\mapsto \max (a,b)\) in \(\R _+\) is induced from the addition in \(\tc \) (or \(\tr \), does not matter). Indeed, \(a\cplus b=\max (a,b)\) for any \(a,b\in \R _+\).

Thus, the semifield \(\R _{\ge 0,\max ,\times }\) is a subset of the hyperfield \(\tc \) closed with respect to both binary operations of \(\tc \), and the binary operations coincide with the operations of the semifield \(\R _{\ge 0,\max ,\times }\). In particular, the inclusion \(\R _{\ge 0,\max ,\times }\to \tc \) and its composition \(\T \to \tc \) with the isomorphism \(\T \to \R _{\ge 0,\max ,\times }\) are homomorphisms.

Warning. There is a natural map in the opposite direction \(\tc \to \R _+:z\mapsto |z|\). It is a right inverse for the inclusion. However, this is not a homomorphism for the tropical addition \(\cplus \). Indeed, \(x\cplus (-x)\cap \R _+=[0,|x|]\) for any \(x\in \R \), but \(|x|\cplus |-x|=|x|\), which does not contain \([0,|x|]\) for \(x\ne 0\).

In order to make the map \(\tc \to \R _+:z\mapsto |z|\) a homomorphism, one should consider a hyperfield structure in \(\R _+\).

• a multiring homomorphism \(\C \to \mftr \) from the field of complex numbers to the triangle hyperfield (see Section 5.1 );
• a multiring homomorphism \(\tc \to \mftr \) from the complex tropical hyperfield \(\tc \) to the triangle hyperfield;
• a multiring homomorphism \(\tc \to \mfutr \) from \(\tc \) to the ultratriangle hyperfield (see Section 5.2 );

• \(\C \to \mfamb \);
• \(\tc \to \mfamb \);
• \(\tc \to \mftrop \).

Let \(p(X)\in \C [X]\) be a polynomial in one variable \(X\) with complex coefficients, \(p(X)=\sum _{k=0}^na_kX^k\), where \(a_k\in \C \), \(a_n\ne 0\). Let \(w(p)=\frac {a_n}{|a_n|}e^n\). Further, let \(w(0)=0\). This defines a map \(\C [X]\to \C :p\mapsto w(p)\).

Proof..

Let us replace \(\C [X]\) by the group algebra \(\C [\R ]\) of the additive group \(\R \). Elements of \(\C [\R ]\) can be thought of as \(\sum _ka_kX^{r_k}\), where \(a_k\in \C \), \(r_k\in \R \). The formal variable \(X\) symbolizes here the transition from additive notation for addition in \(\R \) to multiplicative notation in \(\C [\R ]\), where additive notation is reserved for the formal sum.

Elements of \(\C [\R ]\) may be interpreted as functions \(\C \to \C \). For this, let us turn \(\sum _ka_kX^{r_k}\) into an exponential sum \(\sum _ka_ke^{r_kT}\) by replacing \(X\) with \(e^T\).

The map \(w:\C [X]\to \C \) extends to \(\C [\R ]\) as follows: choose from the sum \(\sum _ka_kX^{r_k}\) the summand with the greatest exponent, say, \(a_nX^{r_n}\) and apply the same formula to it \(\frac {a_n}{|a_n|}e^{r_n}\). The map is a multiring homomorphism of the ring \(\C [\R ]\) onto the hyperfield \(\tc \). The proof that this is a multiring homomorphism is literally the same as the proof of Theorem 7.C above.

A ring can be replaced here by an algebraically closed field real-power Puiseux series \(\sum _{r\in I}a_rt^r\), where \(I\subset \R \) is a well-ordered set. Cf. Mikhalkin [13] , Section 6.

This construction demonstrates how one can obtain the tropical addition of complex numbers from the usual addition of polynomials. It is clear why it should be multivalued. For complex numbers \(a\) and \(b\) with \(|a|=|b|\), but \(a\ne -b\) any \(c\) for the open arc \((a\tplus b)\sminus \{a,b\}\), one can find \(A,B,C\in \C [\R ]\) such that \(w(A)=a\), \(w(B)=b\) and \(w(C)=c\), see Figure 3 . Complex numbers \(a,b\in \C \) with \(a+b=0\) are represented as the images under \(w\) of polynomials \(A,B\in \C [\R ]\) with highest degree terms opposite to each other and annihilating under addition of the polynomials. The highest degree term of \(A+B\) is not controlled by the highest degree terms of the summands \(A\) and \(B\), but its degree does not exceed the degree of the summands.

This topology is quite odd. For example, it is far from being Hausdorff: sets with non-empty intersection cannot have disjoint neighborhoods in it. Therefore usually a limit in the upper Vietoris topology is not unique. By adding new points to a limit we would get a limit. Probably this is what motivates the word upper in the name of the topology.

The lower Vietoris topology in the set \(2^X\) of all subsets of a topological space \(X\) is the topology generated by the sets \(2^X \sminus 2^{C}\), where \(C\) is a closed subset of \(X\). In other words, the lower Vietoris topology is generated by sets \(\{Y\subset X\mid Y\cap U\ne \varnothing \}\), where \(U\) is an open set of \(X\). In the lower Vietoris topology, closed sets are generated by closed sets of \(X\) in the most direct way: a closed set \(C\subset X\) gives rise to the set \(2^C\subset 2^X\) closed in the lower Vietoris topology. Recall that in the upper Vietoris topology open sets are generated similarly by open subsets of \(X\). A neighborhood of a set \(A\in 2^X\) in the lower Vietoris topology should contain all sets intersecting with open sets \(U_1,\dots ,U_n\subset X\) which meet \(A\). A limit in the lower Vietoris topology also usually is not unique, but for the opposite reason: it would stay a limit under removing of its points.

The topology generated by the upper and lower Vietoris topologies is called just the Vietoris topology.

• upper semi-continuous if the corresponding map \(f^\uparrow :X\to 2^Y\) is continuous with respect to the upper Vietoris topology in \(2^Y\);
• lower semi-continuous if \(f^\uparrow :X\to 2^Y\) is continuous with respect to the lower Vietoris topology in \(2^Y\);
• continuous if \(f^\uparrow :X\to 2^Y\) is continuous with respect to the Vietoris topology in \(2^Y\) (i.e., \(X\multimap Y\) is both upper and lower semi-continuous).

Recall that the set \(\{a\in X\mid f(a)\subset B\}\) is called the upper preimage of \(B\) under \(f\), and the set \(\{a\in X\mid f(a)\cap B\ne \varnothing \}\) is called the lower preimage of \(B\) under \(f\).

It is easy to see that \(f:X\multimap Y\) is upper (respectively, lower) semi-continuous if and only if the upper (respectively, lower) preimage of any set open in \(Y\) is open in \(X\).

Proof.

It would suffice to find a set such that it is open in the lower Vietoris topology and its preimage is not open in the classical topology of \(\C ^2\). Take, for instance, the set \(H\) consisting of sets \(A\) meeting the open disk of radius 1 and center 0. Its preimage is the set of pairs \((a,b)\) of complex numbers such that \(a\cplus b\) meets the disk. The preimage of \(H\) consists of pairs \((a,b)\) which satisfy one of the following two conditions: either \(|a|<1\) and \(|b|<1\), or \(a=-b\). Obviously, this set is not open.

Similarly one can prove that the additions in the ultratriangle hyperfield \(\mfutr \) (see Section 5.2 ), the tropical hyperfield \(\mftrop \) (Section 5.3 ) and the real tropical hyperfield \(\tr \) (Section 7.2 ) are not lower semi-continuous.

Proof.

Similarly one can prove that the tropical addition of real numbers is upper semi-continuous.

Lemma 8.C.

A multimap \(f:X\multimap \R \) is continuous if there exist continuous functions \(f_\pm :X\to \R \) with \(f(x)=[f_-(x),f_+(x)]\) for any \(x\in X\) and \(f_+(x)>f_-(x)\) on everywhere dense subset of \(X\).

The triangle addition satisfies the hypothesis of Lemma, hence it is continuous, (i.e., the corresponding map \(\R _+\times \R _+\to 2^{\R _+}\) is continuous with respect to the classical topology in \(\R _+\times \R _+\) the the Vietoris topology in \(2^{\R _+}\)).

Similarly, from Lemma 8.C it follows that the addition in the amoeba hyperfield \(\mfamb \) is continuous.

First, notice, that for univalued maps upper semi-continuity is equivalent to continuity.

Second, obviously, a composition of upper semicontinuous maps is upper semi-continuous.

From these two statements it follows immediately that a multimap defined by a polynomial over \(K\) is upper semi-continuous.

Proof.

The set \(\{B\in 2^Y\mid B\subset X\sminus C\}\) is open in the upper Vietoris topology of \(2^Y\). Therefore, due to upper semi-continuity of multimap \(f\), the preimage \(f^{+}=\{a\in X\mid f(a)\subset X\sminus C\}\) of this set under the \(f^\uparrow :X\to 2^Y\) is open. Consequently, the set \(\{a\in X\mid f(a)\cap C\ne \varnothing \}=X\sminus \{a\in X\mid f(a)\subset X\sminus C\}\) is closed.

Corollary 8.E.

For any polynomial \(p\) over \(K\), the set defined by condition \(0\in p(x_1,\dots ,x_n)\) is closed in \(K^n\).

Finally, recall two well-known theorems about upper semi-continuous multimaps.

Theorem 8.F.

The image of a connected set under an upper semi-continuous multimap is connected, if the image of each point is connected.

Theorem 8.G.

The image of a compact set under an upper semi-continuous multimap is compact, if the image of each point is compact.

Corollary 8.H.

A multimap defined by a polynomial \(p(x_1,\dots ,x_n)\) over \(K\) maps connected sets to connected and compact sets to compact. In particular, the graph of \(p\) is connected.

\begin{align} a+_h b&=\begin{cases} h\ln (e^{a/h}+e^{b/h}),& \text { if }h>0 \\ \max \{a,b\}, & \text { if } h=0 \end {cases}\label {oplus}\\ a\times _h b&= a+b\label {odot} \end{align} These operations depend continuously on \(h\). For each \(h>0\) the map

\[D_h:\R _{>0} \to S_h: x\mapsto h\ln x\]

is a semiring isomorphism of \(\left \{\R _{>0},+,\cdot \right \}\) onto \(\left \{S_h,+_h,\times _h\right \}\), that is

\[D_h(a+b)=D_h(a)+_hD_h(b),\qquad D_h(ab)=D_h(a)\times _hD_h(b). \]

Thus \(S_h\) with \(h>0\) can be considered as a copy of \(\R _{>0}\) with the usual operations of addition and multiplication. On the other hand, \(S_0\) is a copy \(\R _{\max ,+}\) of \(\R \), where the operation of taking maximum is considered as an addition, and the usual addition, as a multiplication.

Applying the terminology of quantization to this deformation, we must call \(S_0\) a classical object, and \(S_h\) with \(h\ne 0\), quantum ones. The whole deformation is called the Litvinov-Maslov dequantization of positive real numbers. The addition in the resulting semiring \(\R _{\max ,+}\), is idempotent in the sense that \(\max (a,a)=a\) for any \(a\).

The analogy with Quantum Mechanics motivated the following

Correspondence principle (Litvinov and Maslov [8] ). “There exists a (heuristic) correspondence, in the spirit of the correspondence principle in Quantum Mechanics, between important, useful and interesting constructions and results over the field of real (or complex) numbers (or the semiring of all nonnegative numbers) and similar constructions and results over idempotent semirings.”

This principle proved to be very fruitful in a number of situations, see [8] , [9] . The Litvinov-Maslov dequantization helps to relate the corresponding things.

Indeed, any valid formula involving only positive real numbers and only arithmetic operations survives under the limit and turns into a valid formula in \(\R _{\max ,+}\).

The correspondence principle is formulated much wider, than this transition to limit allows: not only for the semirings of all positive real numbers and \(\R _{\max ,+}\), but for any idempotent semiring, on one hand, and the fields \(\R \) and \(\C \), on the other hand.

One may expect that there are extra mathematical reasons for this heuristic correspondence. Below similar dequantization deformations are presented. However, the dequantized objects are not semifields, but rather mutlifields.

Obviously, \(R_h\) is an isomorphism with respect to multiplication, and does not commute with the triangular addition

In order to make \(R_h\) a hyperfield isomorphism, pull back the triangular addition, that is define

Observe that if , then

and if \(a=b\), then \(|a^{1/h}-b^{1/h}|^h=0\), while \(\lim _{h\to 0}(a^{1/h}+b^{1/h})^h=a\). Thus the endpoints of the segment tend to the endpoints of the segment as \(h\to 0\). Define to be .

For \(h\ge 0\), denote by \(\mftr _h\) the hyperfield with the underlying set \(\R _+\), addition and multiplication coinciding with the usual multiplication of real numbers. For \(h>0\), the map \(R_h\) is an isomorphism \(\mftr _h\to \mftr \). The hyperfield \(\mftr _0\) coincides with \(\mfutr \).

Thus, \(\mftr _h\) is a dequantization (degeneration) of \(\mftr \) to \(\mfutr \). The map \(\log :\R _+\to \R \cup \{-\infty \}\) converts \(\mftr _h\) into a dequantization of the amoeba hyperfield \(\mfamb \) to the tropical hyperfield \(\mftrop \).

\[z\mapsto \begin {cases} |z|^\frac 1h\frac {z}{|z|} &\text { for }z\ne 0;\\ 0 &\text { for }z=0. \end {cases}\]

This map is invertible. Its inverse is defined by the formula

\[S_h^{-1}:z\mapsto \begin {cases} |z|^h\frac {z}{|z|} &\text { for }z\ne 0;\\ 0 &\text { for }z=0. \end {cases}\]

Obviously, \(S_h\) is an isomorphism with respect to multiplication, that is \(S_h(ab)=S_h(a)S_h(b)\). However, it does not commute with addition.

In order to make \(S_h\) an isomorphism with respect to addition, let us redefine the addition on the source of the map. In other words, induce a binary operation on the set of complex numbers:

\[a+_hb=S_h^{-1}(S_h(a)+S_h(b)).\]

In this way we get a field \(\C _h=(\C ,+_h,\times )\) (which is nothing but a copy of \(\C \)) and an isomorphism \(S_h:\C _h\to \C \).

It is easy to see that converges as \(h\) tends to zero. Namely:

• if \(|a|>|b|\), then ;
• if \(|a|=|b|\) and \(a+b\ne 0\), then ;
• if \(a+b=0\), then .

Denote \(\lim _{h\to 0}(a+_hb)\) by \(a+_0b\). See figure 4 .

Some properties of the operation \((a,b)\mapsto a+_0b\) are nice. It is commutative, distributive for the usual multiplication of complex numbers, the zero behaves appropriately: \(a+_00=a\) for any \(a\in \C \). Furthermore, for any \(a\in \C \) there exists a unique complex number \(b\) such that \(a+_0b=0\), and this \(b\) is nothing but \(-a\).

However, the operation \((a,b)\mapsto a+_0b\) is far from being perfect. First, \(a+_0b\) is not continuous as a function of \(a\) and \(b\). Certainly, this happens because the convergence \(a+_hb\to a+_0b\) is not uniform. Second, it is not associative.

In order to see the latter, compare \((-1+_0i)+_01\) and \(-1+_0(i+_01)\):

\begin{multline*} (-1+_0 i)+_0 1= \left (\exp \left (\pi i\right )+_0\, \exp \left (\frac {\pi i}2\right )\right )+_0\, 1\\ =\exp \left (\frac {3\pi i}4\right )+_0\, \exp (0)=\exp \left (\frac {3\pi i}8\right ) \end{multline*} On the other hand,

The tropical addition \((a,b)\mapsto a\cplus b\) introduced in Section 6.1 above does not have this defect. It is associative, see Appendix 1. Though, it is multivalued.

Another advantage of the tropical addition is that it is upper semicontinuous, see Section 8.3 . The tropical addition is also a limit of as \(h\to 0\), but not in the sense of pointwise convergence.

Thus the tropical addition of complex numbers is a dequantization of the usual addition of complex numbers in the same way as taking maximum is a dequantization of the usual addition of positive real numbers.

Proof of Theorem 9.A .

In order to prove the equality

\[\GG _{\cplus }\times \{0\}=\Cl (\GG )\cap \C ^3\times \{0\},\]

that is the content of Theorem 9.A , we will prove the corresponding two inclusions:

\(\seteqnumber{0}{}{2}\)

\begin{equation} \label {incl1} \GG _{\cplus }\times \{0\}\subset \Cl (\GG )\cap \C ^3\times \{0\}, \end{equation}

\(\seteqnumber{0}{}{3}\)

\begin{equation} \label {incl2} \GG _{\cplus }\times \{0\}\supset \Cl (\GG )\cap \C ^3\times \{0\}. \end{equation}

Proof of ( 3 ). There are three types of points in \(\GG _{\cplus }\subset \C ^3\):

1. \((a,b,a)\) with \(|a|>|b|\), or \((a,b,b)\) with \(|a|<|b|\);
2. \((a,b,c)\) with \(|a|=|b|=|c|\) and \(a+b\ne 0\);
3. \((a,-a,b)\) with \(|b|\le |a|\).

In the first case, if \(|a|>|b|\), then \(a=a+_0b\), hence \((a,b,a)\) belongs to the graph of \(+_0\), and therefore \((a,b,a,0)\) belongs to the closure of \(\GG \). If \(|a|<|b|\), then \(b=a+_0b\), hence \((a,b,b)\) belongs to the graph of \(+_0\), and therefore \((a,b,b,0)\) belongs to the closure of \(\GG \).

In the second case, let \(|a|=|b|=|c|=r\). Recall that \(c\) belongs to the shortest arc connecting \(a\) and \(b\) on the circle \(|z|=r\). Therefore \(c=\Gl a+\Gm b\) with \(\Gl ,\Gm \in [0,1]\).

Assume that \(c\ne a,b\). From this assumption, it follows that \(0<\Gl ,\Gm <1\). Let us prove, first, that \(c=(\Gl ^ha)+_h(\Gm ^hb)\).

Indeed, \(|S_h(\Gl ^ha)|=|\Gl ^ha|^{\frac 1h}=\Gl |a|^{\frac 1h}\) and \(S_h(\Gl ^ha)= \Gl \frac {a}{|a|}|a|^{\frac 1h}=\Gl ar^{\frac {1-h}h}\). Similarly, \(S_h(\Gm ^hb)=\Gm \frac {b}{|b|}|b|^{\frac 1h}=\Gm br^{\frac {1-h}h}\) and \((\Gl ^ha)+_h(\Gm ^hb)=S_h^{-1}(S_h(\Gl ^ha)+S_h(\Gm ^hb))= S_h^{-1}(r^{\frac {1-h}h}(c))=c\).

Thus \((\Gl ^ha,\Gm ^hb,c,h)\in \GG \). Since \(\lim _{h\to 0}x^h=1\) for any \(x\in (0,1)\),

\[(a,b,c,0)=\lim _{h\to 0}(\Gl ^ha,\Gm ^hb,c,h)\in \Cl (\GG ).\]

Thus each interior point of the arc \((a\cplus b)\times 0\) belongs to the closure of \(\GG \). Therefore, its boundary points belong to the closure of \(\GG \), too.

Consider finally the last case, \((a,-a,b)\) with \(|b|\le |a|\). It would suffice to prove that \((a,-a,b,0)\) belongs to the closure of \(\GG \) for \(b\) with \(|b|<|a|\). Obviously, \((a+_hb,-a,b,h)\) belongs to \(\GG \). Indeed, \((a+_hb)+_h(-a)=a+_h(-a)+_hb=b\). Further, \(\lim _{h\to 0}(a+_hb)=a+_0b=a\), since \(|b|<|a|\).

Proof of ( 4 ). The Inclusion ( 4 ) follows from the following lemmas.

Lemma 9.B.

If \((a,b,c,0)\in \Cl \GG \), then \(|c|\le \max (|a|,|b|)\).

Lemma 9.C.

If \((a,b,c,0)\in \Cl \GG \) with \(|a|>|b|\), then \(c=a\).

Lemma 9.D.

If \((a,b,c,0)\in \Cl \GG \) and \(|a|=|b|\), but \(a+b\ne 0\), then \(|c|\ge |a|\).

Lemma 9.E.

If \((a,b,c,0)\in \Cl \GG \) and \(|a|=|b|\), but \(a+b\ne 0\), then \(c\in a\R _++b\R _+\).

Proof of Lemma 9.B .

\(\seteqnumber{0}{}{4}\)

\begin{multline*} |a+_hb|=|S_h^{-1}(S_h(a)+S_h(b))|\\ =|S_h(a)+S_h(b)|^h\\ \le (|S_h(a)|+|S_h(b)|)^h\\ \le (2\max (S_h(a),S_h(b))^h\\ =2^h\left (\max (|a|^{\frac 1h},|b|^{\frac 1h})\right )^h\\ =2^h\max (|a|,|b|) \end{multline*} Since \(2^h\underset {h\to 0}{\to }1\), it follows that for any \(C>\max (|a|,|b|)\) there exists neighborhoods \(U\) and \(V\) of \(a\) and \(b\), respectively, and a real number \(\Ge >0\) such that \(\sup \{|x|\mid x\in U+_hV\}\) is not greater than \(C\) for any \(h\in (0,\Ge )\).

Proof of Lemma 9.C .

For any complex numbers \(x,y\) with \(|x|>|y|\),

\(\seteqnumber{0}{}{4}\)

\begin{multline*} x+_hy=S_h^{-1}(S_h(x)+S_h(y))=\\ S_h^{-1}\left (S_h(x)\left (1+\frac {S_h(y)}{S_h(x)}\right )\right )= xS_h^{-1}\left (1+\frac {S_h(y)}{S_h(x)}\right ) \end{multline*} Further, \(|\frac {S_h(y)}{S_h(x)}|=\left |\frac {y}x\right |^{\frac 1h}\). Hence \(\left |1+\frac {S_h(y)}{S_h(x)}\right |\le 1+\left |\frac {y}x\right |^{\frac 1h}\) and

\[\left |S_h^{-1}\left (1+\frac {S_h(y)}{S_h(x)}\right )\right |= \left |1+\frac {S_h(y)}{S_h(x)}\right |^h \le \left |1+\left |\frac {y}{x}\right |^{\frac 1h}\right |^h.\]

The family \(|1+a^{\frac 1h}|h\) converges to 1 as \(h\to 0\) uniformly for \(a\in (0,r)\) if \(r<1\). Therefore \(x+_hy\) converges to \(x\) as \(h\to 0\) uniformly on the set \(|x|\le R\) and \(|y|\le r\) if \(R\) and \(r\) are positive real numbers with \(0<r<R\).

If \((a,b,c,0)\in \Cl \GG \) and \(|a|>|b|\), then for any neighborhoods \(U\), \(V\) and \(W\) of \(a\), \(b\) and \(c\), respectively, and any \(\Ge >0\) there exist \(h\in (0,\Ge )\) and \((x,y)\in U\times V\) such that \(x+_hy\in W\). Let \(R\) and \(r\) be real numbers with \(|a|>R>r>|b|\). We may take neighborhoods \(U\) and \(V\) such that \(|y|<r\) ad \(R<|x|\) for any \(x\in U\) and \(y\in V\). When \((x,y)\in U\times V\), \(x+_hy\) uniformly converges to \(x\) as \(h\to 0\). On the other hand we see that by shrinking \(W\) towards \(c\) and pushing \(\Ge \) to 0, we force \(x+_hy\) converge to \(c\), while by shrinking \(U\) towards \(a\), we force \(x\) converge to \(a\). Hence \(c=a\).

Proof of Lemma 9.D .

The numbers \(a\) and \(b\) can be related by formula \(b=ae^{i\Gf }\) with \(|\Gf |<\pi \). Then \(S_h(b)=S_h(ae^{i\Gf })=e^{i\Gf }S_h(a)\) and \(|a+_hb| =|S_h^{-1}(S_h(a)+S_h(b))| =|S_h^{-1}(S_h(a)(1+e^{i\Gf }))| =|a||1+e^{i\Gf }|^h\)

Proof of Lemma 9.E .

Fix \(h>0\). The numbers \(S_h(a)\) and \(S_h(b)\) have the same arguments as \(a\) and \(b\). Therefore their sum \(S_h(a)+S_h(b)\) belongs to \(a\R _++b\R _+\). The number \(a+_hb=S_h^{-1}(S_h(a)+S_h(b))\) has the same argument as \(S_h(a)+S_h(b)\). Hence, it also belongs to \(a\R _++b\R _+\).

• \(\mftr _h\) degenerating the triangle hyperfield \(\mftr \) to the ultratriangle hyperfield \(\mfutr \);
• \(\mfamb _h\) degenerating the amoeba hyperfield \(\mfamb \) to the tropical hyperfield \(\mftrop \);
• \(\C _h\) degenerating the field \(\C \) of complex numbers to the complex tropical hyperfield \(\tc \).

\[ \begin {CD}\C \cong \C _h@>{h\to 0}>>\C _0=\tc \\ @V{x\mapsto |x|}VV @VV{x\mapsto |x|}V\\ \mftr \cong \mftr _h@>>{h\to 0}>\mftr _0=\mfutr \\ @V{x\mapsto \log x}VV @VV{x\mapsto \log x}V\\ \mfamb \cong \mfamb _h@>>{h\to 0}>\mftrop \end {CD} \]

All vertical arrows in this diagram are multiring homomorphisms discussed above. The horizontal arrows denote passing to limits.

Double distributivity of a hyperfield is not preserved under dequantization. In the first line of the diagram the original hyperfield is a field \(\C \). It is doubly distributive. The complex tropical hyperfield is not (see Section 6.4 ). In the second and third lines the original hyperfields are not doubly distributive (see Section 5.1 ), while the dequantized hyperfields are (cf. Sections 4.7 , 5.2 and 5.3 ).

Each of the hyperfields gives rise to its own algebraic geometry. The classical complex algebraic geometry corresponds to the left upper corner of the diagram. The left vertical arrows correspond to construction of amoeba for a complex algebraic variety. The bottom right corner of the diagram corresponds to the tropical geometry.

The least studied of these algebraic geometries is the one corresponding to the right upper corner of the diagram. This is the complex tropical geometry. It occupies an intermediate position between the complex algebraic geometry and tropical geometry, cf. [22] .