4.1. The ring ${\mathcal{M}}_{0}$
First we consider a dense linear subspace ${c}_{00}$ of ${\mathit{\ell}}_{1}.$ Let ${c}_{00}$ be the vector space of all eventually zero sequences of complex numbers. Let ${c}_{00}\left({\mathbb{Z}}_{0}\right)$ be the subspace of ${\mathit{\ell}}_{1}\left({\mathbb{Z}}_{0}\right)$ consisting of all $\left(y\rightx)\in {\mathit{\ell}}_{1}\left({\mathbb{Z}}_{0}\right)$ such that $x,y\in {c}_{00}.$ To shorten the notation we will write elements of ${c}_{00}$ as $({x}_{1},\dots ,{x}_{n})$ instead of $({x}_{1},\dots ,{x}_{n},0,\dots ).$ Correspondingly, we will write elements of ${c}_{00}\left({\mathbb{Z}}_{0}\right)$ as $({y}_{1},\dots ,{y}_{m}{x}_{1},\dots ,{x}_{n}).$
Let us define the following equivalence relation on
${c}_{00}\left({\mathbb{Z}}_{0}\right).$ For
$a,b\in {c}_{00}\left({\mathbb{Z}}_{0}\right)$ let
$a\sim b$ if and only if
${T}_{n}\left(a\right)={T}_{n}\left(b\right)$ for every
$n\in \mathbb{N}.$ Let
${\mathcal{M}}_{0}={c}_{00}\left({\mathbb{Z}}_{0}\right)/\sim .$ Note that we have two types of equivalent elements:
where
$\tau $ and
$\sigma $ are permutations on sets
$\{1,\dots ,m\}$ and
$\{1,\dots ,n\}$ resp., and
Consequently, every element of
${\mathcal{M}}_{0}$ has the representative
$\left(y\rightx),$ where
$x,y\in {c}_{00},$ such that multisets of nonzero elements of
x and
y are disjoint. On the other hand, every pair of disjoint finite multisets of nonzero complex numbers define some element of
${\mathcal{M}}_{0}.$ So, we have the bijection between
${\mathcal{M}}_{0}$ and the set of all pairs of disjoint finite multisets of nonzero complex numbers. Let us define ring operations on
${\mathcal{M}}_{0}.$ First we define some auxiliary operations on
${c}_{00}.$ Let
and
for
$({x}_{1},\dots ,{x}_{n}),\left(\right)open="("\; close=")">{x}_{1}^{\prime},\dots ,{x}_{m}^{\prime}$ Let
and
for
$z=\left(y\rightx),{z}^{\prime}=\left(\right)open="("\; close=")">{y}^{\prime}{x}^{\prime}$ where
$x,y,{x}^{\prime},{y}^{\prime}\in {c}_{00}.$ By [
17]
${\mathcal{M}}_{0}$ with these operations is a ring, where
$\left[\right(y\leftx\right)]=[\left(x\righty\left)\right].$ Note that
${\mathcal{M}}_{0}$ is not a linear space, so it is not an algebra [
17].
Let
$a\sim b.$ Since
${T}_{n}\left(a\right)={T}_{n}\left(b\right)$ for every
$n\in \mathbb{N},$ it follows that
$f\left(a\right)=f\left(b\right)$ for every supersymmetric function
$f.$ That is, the value of a supersymmetric function does not depend on the choice of a representative of a class. So, we can set
for a supersymmetric function
f and for
$\left[a\right]\in {\mathcal{M}}_{0}.$
Let us consider how our ring operations interplay with the algebraic basis
${T}_{n}$ and the partition function
$\tilde{\mathcal{W}}\left(z\right)\left(t\right).$ By [
17],
for every
$n\in \mathbb{N}$ and
$\left[z\right],\left[{z}^{\prime}\right]\in {\mathcal{M}}_{0}.$ In other words, each
${T}_{n}$ is a ring homomorphism from
${\mathcal{M}}_{0}$ to
$\mathbb{C}.$ Also, it is easy to check (c.f. [
17]) that
and
The following example may be interesting for evaluating grand canonical partition functions “at infinity”.
Example 1.
Let λ and μ be positive numbers. Set
Taking into account [17] and relations between $\mathcal{W}$ and $\tilde{\mathcal{W}},$ we can see that if $n\to \infty ,$ then both $\mathcal{W}\left({z}^{\left(n\right)}\right)\left(t\right)$ and $\tilde{\mathcal{W}}\left({z}^{\left(n\right)}\right)\left(t\right)$ approach the function ${e}^{(\lambda +\mu )t}.$ Moreover, at the “limit point” ${Z}_{1}\left(\beta \right)=\lambda +\mu ,$ and ${Z}_{N}\left(\beta \right)=0$ for every $N>1.$
Consider the case when sequences
$\tilde{x}$ and
$\tilde{y},$ defined by (
26) and (
25) resp., have only finite number of nonzero elements, i.e.,
$\tilde{x},\tilde{y}\in {c}_{00}.$ Then
$\left(\tilde{y}\right\tilde{x})\in {c}_{00}\left({\mathbb{Z}}_{0}\right).$ So,
$\left[\left(\tilde{y}\right\tilde{x})\right]\in {\mathcal{M}}_{0}.$ Since functions
${\tilde{W}}_{n},$ used in the representations (
24) and (
27) of partition functions are supersymmetric, it follows that values
${\tilde{W}}_{n}\left(u\right)$ do not depend on the choice of the representative
$u\in \left[\left(\tilde{y}\right\tilde{x})\right].$ So, it is natural to consider partition functions as functions on such equivalence classes. Note that all elements of the sequence
$\tilde{x}$ are nonnegative and all elements of the sequence
$\tilde{y}$ are nonpositive. So, the equivalence class
$\left[\left(\tilde{y}\right\tilde{x})\right]$ belongs to the subset
${\mathcal{M}}_{0}^{\pm}$ of
${\mathcal{M}}_{0}$ defined in the following way. Let us denote by
${\mathcal{M}}_{0}^{\pm}$ the set of elements
$\left[u\right],$ where
u is of the form
Note that
${\mathcal{M}}_{0}^{\pm}$ can be completed with respect to a ring norm on
${\mathcal{M}}_{0}$ (see [
15,
17]). In Sub
Section 4.2 we consider such completions more detailed.
For every
$\left[u\right]\in {\mathcal{M}}_{0}^{\pm}$ and odd number
$k,$
where
$x=({x}_{1},\dots ,{x}_{n})$ and
$y=({y}_{1},\dots ,{y}_{m}),$ and it is equal to zero if and only if
$u=0.$
It is known that
${\mathcal{M}}_{0}$ contains divisors of zero. For example,
Proposition 1. The set ${\mathcal{M}}_{0}^{\pm}$ is a commutative semiring with respect to the ring operations in ${\mathcal{M}}_{0},$ without divisors of zero.
Proof. It is easy to check that if
$\left[u\right]$ and
$\left[v\right]$ are in
${\mathcal{M}}_{0}^{\pm},$ then both
$\left[u\right]+\left[v\right]$ and
$\left[u\right]\left[v\right]$ are in
${\mathcal{M}}_{0}^{\pm}.$ But for a given
$\left[u\right]=\left[({y}_{1},\dots ,{y}_{m}{x}_{1},\dots ,{x}_{n})\right]\in {\mathcal{M}}_{0}^{\pm},$$u\ne 0,$ the element
$\left[u\right]=\left[({x}_{1},\dots ,{x}_{n}{y}_{1},\dots ,{y}_{m})\right]$ does not belong to
${\mathcal{M}}_{0}^{\pm}.$ Thus,
${\mathcal{M}}_{0}^{\pm}$ is a semiring but not a ring. If
$\left[u\right]\left[v\right]=0,$ then, by (
31),
${T}_{1}\left(u\right){T}_{1}\left(v\right)=0.$ So either
${T}_{1}\left(u\right)=0$ or
${T}_{1}\left(v\right)=0.$ Thus, either
$u=0$ or
$v=0.$ □
The semiring
${\mathcal{M}}_{0}^{\pm}$ has the following important property that
if and only if
$m={m}^{\prime},$$n={n}^{\prime}$ and there are permutations
$\sigma $ and
$\tau $ such that
Let $\u266f\left[u\right]$ be a pair $(m,n)$ such that in the representation $\left[u\right]=\left[({y}_{1},\dots ,{y}_{m}{x}_{1},\dots ,{x}_{n})\right]$ in ${\mathcal{M}}_{0}^{\pm}$ the number of nonzero elements ${y}_{j}$ is equal to m and the number of nonzero elements ${x}_{i}$ is equal to $n.$ From the definition of the ring operations in ${\mathcal{M}}_{0},$ we have that if $\u266f\left[u\right]=(m,n)$ and $\u266f\left[v\right]=(k,s),$ then $\u266f\left(\right[u]+[v\left]\right)=(m+k,n+s)$ and $\u266f\left(\right[u\left]\right[v\left]\right)=(ms+nk,mk+ns).$ In particular, $\u266f{\left[u\right]}^{2}=(2mn,{m}^{2}+{n}^{2}).$
Proposition 2. Every invertible element in ${\mathcal{M}}_{0}^{\pm}$ is of the form $\left(0\rightx)$ for some $x>0$ or $(y0)$ for some $y>0.$ Every idempotent $\left[u\right]$ in ${\mathcal{M}}_{0}^{\pm}$ is of the form $\left[u\right]=\left[\right(0\left1\right)]$ or $\left[u\right]=\left[\right(1\left0\right)].$
Proof. Let $\left[u\right]\left[v\right]=\left[\right(0\left1\right)],$ then $\u266f\left(\right[u\left]\right[v\left]\right)=(0,1)$ and, so, $\u266f\left[u\right]=(0,1)$ and $\u266f\left[v\right]=(0,1)$ or $\u266f\left[u\right]=(1,0)$ and $\u266f\left[v\right]=(1,0).$ Consequently, $u=\left(0\rightx)$ and $v=\left(0\right1/x)$ for some $x>0$ or $u=(y0)$ and $v=(1/y0)$ for some $y>0.$
Let $\left[u\right]$ be an idempotent in ${\mathcal{M}}_{0}^{\pm},$ that is, ${\left[u\right]}^{r}=\left[u\right]$ for some positive integer $r>1.$ Then $\u266f{\left[u\right]}^{r}=\u266f\left[u\right]$ only if $\u266f\left[u\right]=(1,0)$ or $\u266f\left[u\right]=(0,1).$ Elements of the form $\left[\right(a\left0\right)$ and $\left[\right(0\lefta\right)],$$a>0,$ are idempotents only if $a=1.$ □
Proposition 3.
Elements of the form $\left[({x}_{1},\dots ,{x}_{n}{x}_{1},\dots ,{x}_{n})\right],$${x}_{i}>0$ can be represented as
for every integer $k,$$0\le k\le n.$
Proof. The straightforward computation. □
From the proposition it follows that we have no multiplicative cancelation in ${\mathcal{M}}_{0}^{\pm},$ that is, the equalities $\left[u\right]\left[v\right]=\left[w\right]$ and $\left[u\right]\left[{v}^{\prime}\right]=\left[w\right]$ do not imply $\left[v\right]=\left[{v}^{\prime}\right].$
4.2. A tropical semiring structure
We introduce another semiring structure on
${\mathcal{M}}_{0}^{\pm}$ which is related to Tropical Mathematics. Some applications of tropical semirings to Quantum Mechanics can be found in [
34]. Let us recall that the
min tropical semiring is the semiring
$\left(\right)open="("\; close=")">\mathbb{R}\cup \{+\infty \},\underline{\oplus},\odot $ where the operations
$\underline{\oplus}$ and ⊙ are defined by
The operations
$\underline{\oplus}$ and ⊙ are called the
tropical addition and the
tropical multiplication respectively. The unit for
$\underline{\oplus}$ is
$+\infty ,$ and the unit for ⊙ is
$0.$
Similarly, the
max tropical semiring is the semiring
$\left(\right)$ such that
In this semiring, the unit for
$\overline{\oplus}$ is
$\infty $ and the unit for ⊙ is
$0.$ The semirings are isomorphic with respect to the mapping
$x\mapsto x.$ The usual metric
$\rho (a,b)=ab$ on
$\mathbb{R}$ can be extended to
$\mathbb{R}\cup \{\infty \}$ by setting
$\rho (a,\infty )=1$ for every
$a\in \mathbb{R}.$ Similarly,
$\rho (a,+\infty )=1,$$a\in \mathbb{R},$ for the case
$\mathbb{R}\cup \{+\infty \}.$
Let
$({y}_{m},\dots ,{y}_{1}{x}_{1},\dots ,{x}_{n})$ be a representation of
$\left[u\right]\in {\mathcal{M}}_{0}^{\pm}.$ We say that this representation is
ordered if
${x}_{1}\ge {x}_{2}\ge \cdots \ge {x}_{n}$ and
${y}_{1}\le {y}_{2}\le \cdots \le {y}_{m}.$ The ordered representation of
$\left[u\right]$ is unique and we denote it by
${({y}_{m},\dots ,{y}_{1}{x}_{1},\dots ,{x}_{n})}_{o}.$ Let us denote by
$\mathfrak{e}$ the formal element
Definition 2.
Let us define a tropical semiring ${\mathcal{M}}_{0}^{\oplus}$ as the set ${\mathcal{M}}_{0}^{\pm}\cup \left\{\mathfrak{e}\right\}$ with operations ⊕ and ⊙ such that
and
Proposition 4.
${\mathcal{M}}_{0}^{\pm}\cup \left\{\mathfrak{e}\right\}$ is a semiring and the unit for ⊕ is $\mathfrak{e}$ and the unit for ⊙ is $0.$
Proof. Let us check the distributive law. From the distributive laws in the min tropical semiring and in the max tropical semiring,
□
Let
X be a Banach space with an unconditional Schauder basis
$\left({e}_{n}\right),$$n\in \mathbb{N}.$ Then any vector
$x\in X$ can be represented as
Denote by
${\mathcal{M}}_{X}$ the ring of elements
such that
$x=({x}_{1},\dots ,{x}_{n},\dots )$ and
$y=({y}_{1},\dots ,{y}_{m},\dots )$ are in
X endowed with the following ring norm
where the infimum is taken over all representations
$u=(\dots ,{y}_{m},\dots ,{y}_{1}{x}_{1},\dots ,{x}_{n},\dots ).$ It is known that this norm generates a metric
$d\left(\right[u],[v\left]\right)=\parallel \left[u\right]\left[v\right]\parallel $ and
${\mathcal{M}}_{X}$ is a complete metric space with respect to the metric. Moreover, the ring operations in
${\mathcal{M}}_{X}$ are continuous and
${\mathcal{M}}_{0}$ is a dense subring in
${\mathcal{M}}_{X}$ [
15,
17].
Let us denote by
${\mathcal{M}}_{X}^{\pm}$ the closed subset in
${\mathcal{M}}_{X}$ consisting of elements
Thus
${\mathcal{M}}_{X}^{\pm}$ is a complete metric space and a topological semiring.
We can extend the metric to
${\mathcal{M}}_{X}^{\oplus}={\mathcal{M}}_{X}^{\pm}\cup \left\{\mathfrak{e}\right\}$ by setting
$d(u,\mathfrak{e})=1$ for every
$u\in {\mathcal{M}}_{X}^{\oplus}.$ Note that
${\mathcal{M}}_{X}^{\oplus}\setminus \left\{\mathfrak{e}\right\}$ is a commutative group with respect to “⊙” and
Theorem 1. For any Banach space X with an unconditional basis the following statements are true:
 1
The tropical operations are continuous in ${\mathcal{M}}_{X}^{\oplus}$;
 2

are continuous semiring homomorphisms from ${\mathcal{M}}_{X}^{\oplus}$ to the max tropical semiring $\left(\right)$ and to the min tropical semiring $\left(\right)$ respectively.
Proof. 1. If
$\left[u\right]$ and
$\left[v\right]$ are not equal to
$\mathfrak{e},$ then
and we know that the operation “•” is continuous.
2. Clearly,
${\mathrm{\Phi}}^{+}\left(\left[u\right]\right)={x}_{1}$ and
${\mathrm{\Phi}}^{}\left(\left[u\right]\right)={y}_{1},$ in particular,
${\mathrm{\Phi}}^{+}\left(\mathfrak{e}\right)=+\infty ,$ and
${\mathrm{\Phi}}^{}\left(\mathfrak{e}\right)=\infty .$
Also,
and
Thus
${\mathrm{\Phi}}^{+}$ is a semiring isomorphism.
To show the continuity, we observe that the function
$x=({x}_{1},{x}_{2},\dots )\mapsto {max}_{n}\left{x}_{n}\right$ is bounded (on bounded subsets) on every Banach space
X with a Schauder basis
$\left({e}_{n}\right).$ Indeed, if
$\left({\pi}_{n}\right),$$n\in \mathbb{N}$ be the sequence of projections,
then
(see [
35], pp. 12) and so
${x}_{n}={\pi}_{n+1}\left(x\right){\pi}_{n}{\left(x\right)\le 2K\parallel x\parallel}_{X}.$ Hence
${\mathrm{\Phi}}^{+}\left(\left[u\right]\right)\le 2K\parallel \left[u\right]\parallel .$ The continuity of
${\mathrm{\Phi}}^{+}$ follows from ([
36], Theorem 11.22) taking into account that
${\mathrm{\Phi}}^{+}$ is a bounded homomorphism of the multiplicative normed group
${\mathcal{M}}_{X}^{\oplus}\setminus \mathfrak{e}$ such that
$\parallel {\left[u\right]}^{\odot k}\parallel =k\parallel u\parallel ,$$k\in \mathbb{N}.$ The same works for
${\mathrm{\Phi}}^{}.$ □