#  Based-nonequilibrium-environment non-Markovianity, quantum Fisher
information and quantum coherence

Danping Lin  Hong-Mei Zou<EMAIL_ADDRESS>Jianhe Yang  Synergetic
Innovation Center for Quantum Effects and Application, Key Laboratory of Low-
dimensional Quantum Structures and Quantum Control of Ministry of Education,
School of Physics and Electronics, Hunan Normal University, Changsha, 410081,
P.R. China.

###### Abstract

In this work, we investigate the non-Markovianity, quantum Fisher information
(QFI) and quantum coherence of a qubit in a nonequilibrium environment and
have obtained the expressions of QFI and quantum coherence as well as their
relationship. We have also discussed in detail the influences of the different
noise parameters on these quantum sffects. The results show that the suitable
parameters of the nonequilibrium environment can retard the QFI and quantum
coherence in both Markovian and non-Markovian regions. In addition, the
smaller memory effects and the larger the jumping rate, the greater the QFI
and quantum coherence. And a larger QFI naturally corresponds to a larger
quantum coherence, which indicates that the quantum coherence can enlarge the
QFI and can effectively enhance the quantum metrology.

PACS numbers

03.65.Yz, 03.67.Lx, 42.50.-p, 42.50.Pq.

Keywords

Non-Markovianity; Quantum Fisher information; Quantum Coherence;
Nonequilibrium environment

##  I Introduction

The parameter estimation is a very basic and important content in quantum
information theory. Quantum metrology can attain a measurement precision that
surpasses the classically achievable limit by using quantum effects. As we
know, the metrology precision can be raised by increasing quantum Fisher
information (QFI)Fisher . And the Braunstein-Caves theorem proved that the QFI
basing on the symmetric logarithmic derivative(SLD) contains all possible
informationBraunstein . On the other hand, quantum coherence(QC) is

As we all knows, the memory effect of the environment is caused by the long
time correlation between the system and the environment and the environment
can be divided into Markovian and non-Markovian types according to the memory
effects. Many methods for quantifying non-Markovian features have been
proposed, such as Rivas-Huelga-Plenio (RHP) measureRivas , Breuer-Laine-Piilo
(BLP) measureBreuer  and Luo-Fu-Song (LFS) measureShunlong . Besides, Zhi He
et al.Zhi  has used the BLP measurement to calculate the non-Markovianity for
single-channel open systems, and they obtain a very tight lower bound of non-
Markovianity. Yu Liu et al.Yu  has calculated the non-Markovianity in the
dissipative cavity and used it to explain the dynamical behavior of quantum
coherence.

Recently, the influence of environments on the QFI and quantum coherence of
quantum systems has become an important research topic. …… one of our authors
have obtained the analytical solution of QFI through the SLD method and have
enhanced or retarded the QFI by the dissipation cavitylin . Very recently,
more and more attention has been paid to study the relationship between
different quantum effects. For examples, Zou and Fang have discussed that the
discord and entanglement in non-Markovian environments at finite
temperatureszou1 ; Zou has discussed the influence of the non-Markovian effect
and detuning on the lower bound of the quantum entropic uncertainty relation
and entanglement witness in detailzou2 ; Hou has studied the relationship
between quantum discord and coherenceHou . Feng has verified that the QFI can
be utilized to quantify the quantum coherenceFeng . Wang has shown that
quantum coherence also increases the precision of parameter estimationwang .

Although many important progresses have been acquired in experimental and
theoretical researches on the quantum Fisher information and quantum coherence
of open quantum systems, these investigations mentioned above are mainly
focused on the Lorentzian and Ohmic environments. Actually, the nonequilibrium
environment of dichotomic nature has been observed in some experimentsTaubert
; Nowack ; Granger , and the steady-state entanglement and coherence of two
coupled qubitsWang1  and the decoherence induced by non-Markovian noiseX  in a
nonequilibrium environment are investigated. In our work, we will study the
non-Markovianity, QFI and quantum coherence of a qubit in the nonequilibrium
environment. Our main purpose are to understand whether these quantum effects
of a qubit in a nonequilibrium environment are different from those in a
Lorentzian and an Ohmic environments, and to understand how nonequilibrium
environment parameters affect these quantum effects. In addition, we also hope
to get the relationship between the QFI and quantum coherence. Hence, we
believe that the study of the non-Markovianity, QFI and quantum coherence of a
qubit in the nonequilibrium environment is also important and meaningful

The outline of the paper is organized as follows. In Section II, we give the
model of a qubit in a nonequilibrium environment. In Section III, we calculate
the non-Markovianty, QFI and quantum coherence of the qubit in a
nonequilibrium environment. In Section IV, we analyze the influence of the
nonequilibrium parameter, the memory effects and the jumping rate on the non-
Markovianity, QFI and coherence. Finally, we end with a brief summary of
important results in Section V.

##  II Physical Model

We consider a qubit coupled to a nonequilibrium environment. The Hamiltonian
is written asXiang

$H=\frac{\hbar}{2}[\omega_{0}+\xi(t)]\sigma_{z},$ (1)

where $\sigma_{z}$ and $\omega_{0}$ are the Pauli matrix and the intrinsic
transition frequency for the qubit, respectively. $\xi(t)$ denotes the
environmental noise caused by environmental effects, which is originally
introduced by A. FulińskiFulinski . For simplify, we suppose that $\xi(t)$ is
subject to a nonstationary and non-Markovian stochastic process in this paper.

The time evolution of the system is described by the Liouville equation

$\frac{\partial}{\partial
t}\rho(t;\xi(t))=-\frac{i}{\hbar}[H(t),\rho(t;\xi(t))],$ (2)

where the notation $\rho(t;\xi(t))$ indicates the density operator under the
environmental noise $\xi(t)$. The reduced density operator of the qubit can be
derived by taking an average over the environmental noise as
$\rho(t)=\langle\rho(t;\xi(t))\rangle$. By solving Eq. (2), the density matrix
in the basis $\\{|e\rangle,|g\rangle\\}$ can be given in the following from

$\rho(t)=\left(\begin{array}[]{cccc}\rho_{ee}(0)&\rho_{eg}(0)e^{-i\omega_{0}t}G^{*}(t)\\\
\rho_{ge}(0)e^{i\omega_{0}t}G(t)&\rho_{gg}(0)\\\ \end{array}\right),$ (3)

where the $G(t)=\langle e^{i\int_{0}^{t}dt^{{}^{\prime}}\xi(t)}\rangle$ is a
complex time-dependent function.

According to Ref.X , the environmental noise $\xi(t)$ is subject to a
nonstationary random telegram process that randomly jumps back and forth
between $-\nu$ and $+\nu$ at an average rate of $\lambda$. The noise process
satisfies the conditional probability such as $\frac{\partial}{\partial
t}P(\pm\nu,t|\xi^{\prime},t^{\prime})=\mp\int^{t}_{t^{\prime}}dt^{\prime}K(t-\tau)[\lambda
P(\nu,\tau|\xi^{\prime},t^{\prime})-\lambda P(-\nu|\xi^{\prime},t^{\prime})]$,
where $K(t-\tau)$ represents a generalized memory kernel. By deriving the
calculation, the initial condition of this work is set to
$P(\xi_{0})=2^{-1}[(1\pm a)\delta_{\xi_{0,\pm\nu}}]$. And the memory kernel
$K$ satisfies $K(t-\tau)=\kappa e^{-\kappa(t-\tau)}$. The environment noise
can be characterized by the nonequilibrium parameter $a$ ($\mid a\mid\leq 1$)
and the memory decay rate $\kappa(\kappa>0)$ as well as the amplitude switches
randomly with the jumping rate $\lambda$ between the values $\pm\nu(\nu>0)$.
$a=0$ indicates that the environment is equilibrium (and $a\neq 0$ vice
versa). The smaller $\kappa$ and bigger $\nu$ represent the stronger non-
Markovianity. Under nonequilibrium environment, $G(t)$ in Eq. (3)can be
analytically solved asX

$G(t)=\mathcal{L}^{-}[G(s)],\\\
G(s)=\frac{s^{2}+(\kappa+ia\nu)s+\kappa(2\lambda+ia\nu)}{s^{3}+\kappa
s^{2}+(2\kappa\lambda+\nu^{2})s+\kappa\nu^{2}},$ (4)

where $\mathcal{L}^{-}$ indicates the inverse Laplace transform and the
initial condition is $G(0)=1$.

Using $G(0)=1$, one can obtain easily
$G(s)=\frac{(s-u_{+})(s-u_{-})}{(s-s_{1})(s-s_{2})(s-s_{3})}$ where $u_{\pm}$
are roots of $s^{2}+(\kappa+ia\nu)s+\kappa(2\lambda+ia\nu)=0$, and
$s_{j}(j=1,2,3)$ are roots of $s^{3}+\kappa
s^{2}+(2\kappa\lambda+\nu^{2})s+\kappa\nu^{2}=0$

We can acquire the exact solution of $G(t)$ asPiessens , i.e.

$G(t)=\sum_{j=1}^{3}G(s)e^{s_{j}t}.$ (5)

Let an arbitrary initial state is
$|\psi\rangle=\cos\frac{\theta}{2}|e\rangle+e^{i\phi}\sin\frac{\theta}{2}|g\rangle$.
The reduced density matrix $\rho(t)$ in Eq. (3) can write as

$\rho(t)=\left(\begin{array}[]{cccc}\rho_{ee}(t)&\rho_{eg}(t)\\\
\rho_{ge}(t)&\rho_{gg}(t)\\\ \end{array}\right),$ (6)

with the elements

$\begin{split}\rho_{ee}(t)=&\frac{1}{2}(1+\cos\theta),\\\
\rho_{eg}(t)=&\frac{1}{2}\sin\theta e^{-i(\phi+\omega_{0}t)}G^{*}(t),\\\
\rho_{ge}(t)=&\rho^{\ast}_{eg}(t),\\\
\rho_{gg}(t)=&1-\rho_{ee}(t).\end{split}$ (7)

##  III Non-Markovianity, QFI and Quantum Coherence

In the three subsections, we briefly review the measurements of the non-
Markovianity, QFI and quantum coherence.

###  III.1  non-Markovianity

In this part, we introduce the non-Markovianity of the qubit by using the BLP
measure, which based on the trace distance between two quantum states
$\rho_{1}$ and $\rho_{2}$Breuer . The non-Markovianity $\mathcal{N}$ is
defined as

$\displaystyle\begin{array}[]{cccc}\mathcal{N}&=&\max\limits_{\rho_{1}(0),\rho_{2}(0)}\int_{\sigma>0}\sigma[t,\rho_{1}(0),\rho_{2}(0)]dt\end{array},$
(9)

where $\sigma[t,\rho_{1}(0),\rho_{2}(0)]$ is the change rate of the trace
distance defined as
$\sigma[t,\rho_{1}(0),\rho_{2}(0)]=\frac{d}{dt}D[\rho_{1}(t),\rho_{2}(t)]$,
and the trace distance is defined as
$D[\rho_{1}(t),\rho_{2}(t)]=\frac{1}{2}Tr|\rho_{1}(t)-\rho_{2}(t)|$ with
$|A|=\sqrt{A^{{\dagger}}A}$ and $0\leq D\leq 1$. If $D=0$, the two state are
the same, and if $D=1$, the two state are totaly distinguishable. That is,
when $\rho_{1}(0)=|e\rangle\langle e|$ and $\rho_{2}(0)=|g\rangle\langle g|$,
the trace distance is maximum.

###  III.2 Quantum Fisher Information

Based on thee classical Fisher information (CFI)C ; A ; Pairs , the quantum
Cramér-Rao (QCR) theorem of the QFI has also be proposed, which is
$\langle(\bigtriangleup\hat{\lambda})^{2}\rangle_{\lambda}\geq\frac{1}{N\mathcal{F}_{\lambda}}$,
where $\hat{\lambda}$ is the unbiased estimator of parameter $\lambda$. The
QFI theorem gives the highest accuracy of parameter estimation in the case of
quantum mechanics. The smaller the
$\langle(\bigtriangleup\hat{\lambda})^{2}\rangle_{\lambda}$, the larger the
$\mathcal{F}_{\lambda}$. That is, we can get the higher the quantum metrology
precision.

Using positive-operator valued measurement (POVM), $\mathcal{F}_{\lambda}$ can
be written as

$\displaystyle\mathcal{F}_{\lambda}$ $\displaystyle=$ $\displaystyle
Tr(\rho_{\lambda}L_{\lambda}^{2})=Tr(\partial_{\lambda}\rho_{\lambda}L_{\lambda}),$
(10)

this is quantum Fisher information (QFI), where $L_{\lambda}$ is symmetric
logarithmic derivatives(SLD) for the parameter $\lambda$, which is a Hermitian
operator determined by
$\partial_{\lambda}\rho_{\lambda}=\frac{1}{2}\\{\rho_{\lambda},L_{\lambda}\\}$
where $\partial_{\lambda}\equiv\frac{\partial}{\partial\lambda}$ and
$\\{\cdot,\cdot\\}$ denotes the anticommutator. If the density operator
$\rho_{\lambda}$ satisfies the spectral decomposition
$\rho_{\lambda}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ and without
considering the existence of zero eigenvalues, the $\mathcal{F}_{\lambda}$ can
be expressed as follows

$\displaystyle\begin{array}[]{cccc}\mathcal{F}_{\lambda}&=&\sum_{i^{{}^{\prime}}}\frac{(\partial_{\lambda}p_{i^{{}^{\prime}}})^{2}}{p_{i^{{}^{\prime}}}}+2\sum_{i\neq
j}\frac{(p_{i}-p_{j})^{2}}{p_{i}+p_{j}}|\langle\psi_{i}|\partial_{\lambda}\psi_{j}\rangle|^{2}\\\
\end{array},$ (12)

Using Eq. (10) to calculate the QFIs of parameter $\theta$ and $\phi$.
Inserting Eq. (6) into Eq. (10), the QFIs are obtained as

$\displaystyle\begin{array}[]{cccc}\mathcal{F_{\theta}}&=&1\\\
\mathcal{F_{\phi}}&=&|G(t)|^{2}\sin^{2}\theta\\\ \end{array},$ (15)

From Eq. (15), it is known that $\mathcal{F_{\theta}}$ is always 1 and the
$\mathcal{F_{\phi}}$ is determined by two factors, $G(t)$ and $\theta$.

###  III.3  Quantum Coherence

A reasonable measure to quantify quantum coherence should fulfill the
following condition: non-egativity, monotonicity, strong monotonicity,
convexity, uniqueness for pure states and additivityStreltsov . According to a
set of properties of quantum coherence measure, some coherence measures are
proposed. Here, we focus on the $l_{1}$ norm of coherenceBaumgratz . From Eq.
(6) it can be expressed as

$\displaystyle\mathcal{C}_{l}(t)$ $\displaystyle=$
$\displaystyle\sum_{i,j(i\neq j)}|\rho_{ij}(t)|=|G(t)||\sin\theta|,$ (16)

where $\mid\rho_{ij}(t)\mid(i\neq j)$ is the absolute value of the non-
diagonal elements $\rho_{ij}(t)$ of the density matrix $\rho(t)$.

Comparing Eq. (15) and Eq. (16), we can obtain the analytical relationship
between the $\mathcal{F_{\phi}}$ and the $\mathcal{C}_{l}(t)$, which is

$\displaystyle\mathcal{F_{\phi}}$ $\displaystyle=$
$\displaystyle\mathcal{C}^{2}_{l}(t),$ (17)

From Eq. (16), we know that the QFI will enlarge with the quantum coherence
increasing, which indicates that the quantum coherence can augment the QFI and
can effectively enhance the quantum metrology.

##  IV Discussion and results

In this section, we illustrate the influence of environmental parameters (the
nonequilibrium parameter $a$, the memory decay rate $\kappa$ and the jumping
rate $\lambda$) on non-Markovianity, QFI and quantum coherence. The quantum
system exhibits a Markovian behavior when $(\nu/\lambda<1)$, the evolution of
the quantum system is non-Markovian only when $(\nu/\lambda>1)$Xiang .

In Fig. 1, we plot the non-Markovianity $\mathcal{N}$ as function of the
dimensionless quantity $\nu/\lambda$, which shows that $\mathcal{N}$ is
dependent on $\nu/\lambda$. However, due to the difference between the
selected parameters and the calculation method, the curve change of
$\mathcal{N}$ is different from that of the Ref.Xiang . In short,
$\mathcal{N}$ increases as $\nu/\lambda$ increases. For the same equilibrium
parameter $a$, the larger the value of $\nu$ is, the larger $\mathcal{N}$ is.
For the same the value of $\nu/\lambda$ , the smaller the equilibrium
parameter $a$, the larger $\mathcal{N}$. As is plotted in Fig. 2,
$\mathcal{N}$ associated to the memory decay rate $\kappa/\lambda$.
$\mathcal{N}$ decreases as $\kappa/\lambda$ increases, and $\mathcal{N}$ will
eventually disappear when $\kappa\rightarrow\infty$. When $\nu=0.8\lambda$ and
$\kappa/\lambda=0$, the maximum value of $\mathcal{N}$ can reach 4.0,
indicating that the non-Markovian effect is very obvious. When
$\nu=0.8\lambda$ and $\kappa/\lambda=10$, the non-Markovianity is zero. The
dynamic behavior of the qubit presents Markovian characteristics. For the same
$\nu$, the smaller $\kappa$, the larger $\mathcal{N}$. And, the non-
Markovianity has the similar dynamic behaviors in the cases of
$\nu=0.8\lambda$ and $\nu=4\lambda$, while the $\mathcal{N}$ is larger in the
former than in the latter.

Figure 1: (Color online) Non-Markovianity $\mathcal{N}$ as a function of the
dimensionless quantity $\nu/\lambda$. Here $\theta=\frac{\pi}{2},\phi=0$. The
memory decay rate $\kappa=8\lambda.$ Figure 2: (Color online) Non-Markovianity
$\mathcal{N}$ as a function of the dimensionless quantity $\kappa/\lambda$.
Here $\theta=\frac{\pi}{2},\phi=0$, $a=0.5$.

Figure 3: (Color online) QFI $\mathcal{F}_{\phi}$ and quantum coherence
$\mathcal{C}_{l}$ as a function of $\lambda t$ for $a=0.5$ and different
values of the memory decay rate $\kappa$: $\kappa=0$ (red dot dash line),
$\kappa=\lambda$ (blue dotted line), $\kappa=10\lambda$ (green line). (a) and
(c)$\nu=0.8\lambda$; (b) and (d)$\nu=4\lambda$. The other parameter are
$\theta=\frac{\pi}{2}$ and $\phi=0$.

Fig. 3 is plotted $\mathcal{F}_{\phi}$ and $\mathcal{C}_{l}$ as a function of
$\lambda t$ under different values of $\kappa$ and $\nu$ for $a=0.5$. Fig. 3
(a) shows that the $\mathcal{F}_{\phi}$ has the periodic oscillation when
$\kappa=0$ and $\nu=0.8\lambda$. However, when $\kappa=10\lambda$ and
$\nu=0.8\lambda$, the $\mathcal{F}_{\phi}$ decays monotonically, which implies
that the $\mathcal{F}_{\phi}$ dynamics of the qubit becomes Markovian.
Therefore, with $\kappa$ decreasing, the evolution of the qubit will change
from Markovian to non-Markovian, corresponding to the tendency of
$\mathcal{F}_{\phi}$ from monotonical decay to oscillation. As shown in Fig. 3
(b), when $\kappa=10\lambda$ and $\nu=4\lambda$, the $\mathcal{F}_{\phi}$
could also take the behavior of oscillations. In addition, the maximum value
of oscillation recovery of $\mathcal{F}_{\phi}$ increases with the decreasing
of $\kappa$, which implies that the smaller the value of $\kappa/\lambda$, the
more obvious the non-Markovian effects. In Fig. 3 (c) and (d),
$\mathcal{C}_{l}$ is plotted as a function of $\lambda t$ for $a=0.5$ and
different values of $\kappa$ and $\nu$. Fig. 3 (c) displays an oscillatory
behavior of $\mathcal{C}_{l}$ when $\kappa$ is very small and
$\nu=0.8\lambda$. However, $\mathcal{C}_{l}$ decays monotonically for larger
$\kappa$, which shows that the dynamical evolution becomes Markovian. Similar
to the case of Fig. 3 (c), as $\kappa$ increase, the $\mathcal{C}_{l}$ rapidly
oscillations to zero when $\nu=4\lambda$, as shown in the Fig. 3 (d).

Figure 4: (Color online) QFI $\mathcal{F}_{\phi}$ as a function of the
dimensionless quantity $\lambda t$ and the memory decay rate $\kappa$ for
$a=0.5$. Here $\theta=\frac{\pi}{2}$ and $\phi=0$. (a)$\nu=0.8\lambda$;
(b)$\nu=4\lambda$.

In order to observe comprehensively the dependence of $\mathcal{F}_{\phi}$ on
$\kappa$ and $\lambda t$, we can plot a three-dimensional diagram. Fig. 4 is
plotted $\mathcal{F}_{\phi}$ as a function of the dimensionless quantity
$\lambda t$ and the memory decay rate $\kappa$ for $a=0.5$. Fig. 4(a) shows
that $\mathcal{F}_{\phi}$ appears the phenomenon of rapidly revivals when
$\kappa/\lambda=10\lambda$ and $\nu=0.8\lambda$. However, when
$\kappa/\lambda\gg 1$ and $\nu=0.8\lambda$, $\mathcal{F}_{\phi}$ appears decay
phenomenon. Fig. 4(b) shows that no matter what $\kappa$ is,
$\mathcal{F}_{\phi}$ appears the phenomenon of oscillations. The memory effect
is the reason for the revival of $\mathcal{F}_{\phi}$. Namely, the memory
effect can enhance the non-Markovian behavior and the value of QFI. But, the
oscillatory behavior of the $\mathcal{F}_{\phi}$ does not disappear in the
memoryless limit. The three-dimensional picture of $\mathcal{C}_{l}(t)$ is
similar to the $\mathcal{F}_{\phi}$, we omit it in order to reduce the space.

Figure 5: (Color online) $\mathcal{F}_{\phi}$ as a functions of the
dimensionless quantity $\lambda t$ and the memory decay rate $\kappa$ for
$a=0.5$. Here $\theta=\frac{\pi}{2}$ and $\phi=0$. (a)$\nu=0.8\lambda$;
(b)$\nu=4\lambda$.

In Fig.5, we plot the $\mathcal{F}_{\phi}$ as a functions of the dimensionless
quantity $\lambda t$ and the memory decay rate $\kappa/\lambda$ for $a=0.5$.
From this figure, it can be seen that the smaller the value of
$\kappa/\lambda$ and $\lambda t$, the larger the value of
$\mathcal{F}_{\phi}$. Meanwhile, the evolution behavior of
$\mathcal{C}_{l}(t)$ is similar to the $\mathcal{F}_{\phi}$, we omit it in
order to reduce the space. Furthermore, as is shown above, the evolution
behavior of $\mathcal{C}_{l}$ is similar qualitatively to the
$\mathcal{F}_{\phi}$ in both Markovian and non-Markovian regions. That is, the
smaller $\kappa$ is beneficial to the $\mathcal{F}_{\phi}$ and
$\mathcal{C}_{l}$. Comparing Fig. 3 (a) and (c) (or Fig. 3 (b) and (d)), we
can see the tendency of $\mathcal{F}_{\phi}$ and $\mathcal{C}_{l}$ with
$\nu=0.8\lambda$ and $\nu=4\lambda$ is coincident.

Figure 6: (Color online) QFI $\mathcal{F}_{\phi}$ and coherence
$\mathcal{C}_{l}$ as a function of dimensionless quantity $\lambda t$ for the
initial state $|\psi\rangle$. (a), (c)$\nu=0.8\lambda$; (b),
(d)$\nu=4\lambda$. The memory decay rate $\kappa=8\lambda.$ The insets show
the evolution of a short time. The other parameter are $\theta=\frac{\pi}{2}$
and $\phi=0$.

In Fig. 6(a), $\mathcal{F}_{\phi}$ is plotted as a function of the
dimensionless quantity $\lambda t$ for the different nonequilibrium parameter
$a$ in the Markovian dynamic region ($\nu=0.8\lambda$). As is shown above,
there is a little difference among the five lines and $\mathcal{F}_{\phi}$
decreases monotonically with time $t$ and vanishes in the asymptotic limit
$t\rightarrow\infty$, which implies the accuracy of the parameter estimation
is smaller. If the environment is out of equilibrium ($0<|a|\leq 1$), the
greater the value of $\mid a\mid$, the greater the initial value of
$\mathcal{F}_{\phi}$. But they all close to zero when time $t$ increases. And
the curves of $\mathcal{F}_{\phi}$ are coincident for $a=0.5$ and $a=-0.5$,
the curves of $\mathcal{F}_{\phi}$ are also coincident for $a=1$ and $a=-1$.
The inset of Fig. 6(a) shows that the initial value of $\mathcal{F}_{\phi}$
will be larger when the system is far away from the equilibrium environment.
In Fig. 6(b), $\mathcal{F}_{\phi}$ is plotted as a function of the
dimensionless quantity $\lambda t$ for the different nonequilibrium parameter
$a$ in the non-Markovian dynamic region ($\nu=4\lambda$). It is seen that the
initial value of $\mathcal{F}_{\phi}$ has the different value of the
nonequilibrium parameter $a$. $\mathcal{F}_{\phi}$ appears oscillating
recovery phenomenon with time $t$ due to the feedback effects of the non-
Makovian environment. Similar to the case of the inset of Fig. 6(a), the
initial value of $\mathcal{F}_{\phi}$ will be larger as $|a|$ increases, as
shown the inset of Fig. 6(b).

In Fig. 6(c), $\mathcal{C}_{l}$ is plotted as a function of the dimensionless
quantity $\lambda t$ for different the $a$ in the Markovian region
($\nu=0.8\lambda$). In Fig. 7 (d), we plot the time evolution of the
$\mathcal{C}_{l}$ for $\theta=\frac{\pi}{2}$, $\phi=0$ and $\nu=4\lambda$. It
can be seen that the $\mathcal{C}_{l}$ oscillates damply to zero. This figure
shows that the $\mathcal{C}_{l}$ in the nonequilibrium environment is greater
than the equilibrium environment in the short time. But the $\mathcal{C}_{l}$
will be close to zero in a long evolution time. Obviously, the nonequilibrium
parameter $a$ plays the important role in the dynamic revolutions of the
$\mathcal{F}_{\phi}$ and $\mathcal{C}_{l}$. When $|a|$ is greater, the initial
value of coherence is larger, as shown the inset of Fig. 6(d).

As a comparison, when the environment is out of equilibrium, the
$\mathcal{C}_{l}$ and $\mathcal{F}_{\phi}$ will be delay in both Markovian and
non-Markovian regions, which indicates that the bigger $\nu$, the slower down
the evolution of $\mathcal{C}_{l}$ and $\mathcal{F}_{\phi}$. For example,
comparing Fig. 6(a) and Fig. 6(c), it can be seen that the tendency of
$\mathcal{C}_{l}$ and $\mathcal{F}_{\phi}$ is very similar in
$\nu=0.8\lambda$. The tendency of Fig. 6(b) and Fig. 6(d) is like in
$\nu=4\lambda$.

Figure 7: (Color online) Left: QFI $\mathcal{F}_{\phi}$ as a function of
$\lambda t$ when $a=0$, $\kappa=0$, $\theta=\frac{\pi}{2}$ and under different
values of $\nu$: $\nu=0.2\lambda$(red line), $\nu=0.4\lambda$(blue line),
$\nu=0.6\lambda$(black line), $\nu=0.8\lambda$(green line). Right: QFI
$\mathcal{F}_{\phi}$ as a function of $\lambda t$ when $a=0$, $\kappa=0$,
$\nu=0.8\lambda$ and under different values of $\theta$:
$\theta=\frac{\pi}{2}$(red line), $\theta=\frac{\pi}{3}$(blue line),
$\theta=\frac{\pi}{4}$(black line), $\theta=\frac{\pi}{5}$(green line).

In the above Figures 1-6, we have discussed in detail the influences of
different noise parameters on the dynamical behaviors of QFI and QC of the
qubit in a nonequilibrium environments. In order to better understand the
difference between nonequilibrium and equilibrium environments, we also draw
the dynamical curves of the QFI of the qubit in the equilibrium environment,
as shown in the Figures 7(a) and (b). Figures 7(a) shows that these curves
share the same beginning for different $\nu$ values and Figures 7(b) indicates
that these curves also share the same beginning for different initial states
when the nonequilibrium environment parameter($a=0$) and the memory decay
rate($\kappa=0$). In the equilibrium environment, the dynamical curves of
$\mathcal{C}_{l}(t)$ are similar to that of the $\mathcal{F}_{\phi}$ so we
omit it in order to reduce the space.

Comparing Figure 7 and Figures 1-6, we can know that, in the equilibrium
environment, the different noise parameters only alter their time evolution
but not their initial values. However, in the nonequilibrium environment, the
different noise parameters alter not only their time evolution but also their
initial values.

##  V Conclusion

In summary, we have investigated the non-Markovianity, QFI and quantum
coherence of a single qubit coupled to a nonequilibrium environment and have
obtained the expressions of QFI and quantum coherence as well as the
relationship between QFI and quantum coherence. Their relationship ( Eq. (16))
indicates that the quantum coherence can enlarge the QFI and can effectively
enhance the quantum metrology, which gives us a novel understanding of the
relationship among quantum resources. We have also discussed in detail the
influences of the different noise parameters on the non-Markovianity, QFI and
quantum coherence. The results show that, the suitable environment parameters,
including the nonequilibrium parameter, the memory decay rate and the jumping
rate, are beneficial to enhance the non-Markovianity and slower the decaying
of QFI and quantum coherence. And we utilized the non-Markovianity to explain
the dynamical behaviors of the QFI and the quantum coherence. Besides, in the
equilibrium environment, the different noise parameters only alter their time
evolution but not their initial values. However, in the nonequilibrium
environment, the different noise parameters alter not only their time
evolution but also their initial values.

These results give us an active way to suppress decoherence, which is quite
significant in quantum information processing and quantum metrology. These
would be a very interesting issue that needs further attention. We hope that
our work is valuable to understand the common feature of quantum resources and
the relationship between QFI and quantum coherence. In future work, we will
study the essential relations between quantum resources and try to provide a
unified framework of them.

## Acknowledgments

Project supported by the Scientific Research Project of Hunan Provincial
Education Department, China (Grant No 16C0949) and the National Science
Foundation of China (Grant No 11374096).

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# Order Cancellation Law in a Semigroup of Closed Convex Sets

Jerzy Grzybowski111Corresponding author and Hubert Przybycień

###### Abstract

In this paper generalize Robinson’s version of an order cancellation law for
subsets of vector spaces in which we cancel by unbounded sets. We introduce
the notion of weakly narrow sets in normed spaces, study their properties and
prove the order cancellation law where the canceled set is weakly narrow. Also
we prove the order cancellation law for closed convex subsets of topological
vector space where the canceled set has bounded Hausdorff-like distance from
its recession cone. We topologically embed the semigroup of closed convex sets
sharing a recession cone having bounded Hausdorff-like distance from it into a
topological vector space. This result extends Bielawski and Tabor’s
generalization of Rådström theorem.

2010 Mathematics Subject classification. 52A07, 18E20, 46A99.

Key words and phrases. semigroup of convex sets, order cancellation law,
recession cone, Minkowski–Rådström–Hörmander space.

##  1 Introduction

The order cancellation law or Rådström cancellation theorem [22, 24],
investigated also in [14, 17, 19], enables an embedding of a semigroup of
convex sets into a quotient vector space called a Minkowski–Rådström–Hörmander
space [11, 27]. This space plays a crucial role in differentiation of
nonsmooth functions (quasidifferential calculus of Demyanov and Rubinov [1, 6,
7, 10, 13, 20]) and in integration and differentiation in the theory of
multifunctions [4, 5]. The cancellation by unbounded sets is especially
challenging. It is usually impossible. However, in this paper we significantly
extend the range of applicability of this law.

In order to embed a given semigroup of subsets of a vector space into some
other vector space, we need the law of cancellation: $A+B=B+C\Longrightarrow
A=C$. This embedding is very important in set-valued analysis. Let us recall
that the cancellation law holds true in the semigroup $\mathcal{B}(X)$ of
nonempty bounded closed convex sets in real Hausdorff topological vector space
$X$, where the addition $\dot{+}$ is defined by $A\dot{+}B=$ cl$(A+B)=$
cl$\\{a+b:a\in A,b\in B\\}$. Minkowski studied the semigroup of compact convex
subsets of Euclidean spaces, therefore, the algebraic or vector addition $A+B$
is also called the Minkowski addition. In infinite dimentional topological
vector spaces, we need to consider $A\dot{+}B$ instead of $A+B$. Rådström [24]
applied the cancellation law in order to embed the convex cone
$\mathcal{B}(X)$, where $X$ is a normed vector space, into a normed vector
space. Hörmander [18] generalized the Rådström’s result to the locally convex
spaces and Urbański [27] to the arbitrary topological vector spaces. However,
the order law of cancellation which is applied in the aforementioned cased is
based on the following theorem.

######  Theorem 1.1.

(see for example Proposition 2.1 in [27].)  Let $X$ be a Hausdorff topological
vector space and $A,B,C\subset X$. If $B$ is nonempty and bounded and $C$ is
closed and convex then $\textup{cl}(A+B)\subset\textup{cl}(B+C)\Longrightarrow
A\subset C$.

In [17] the assumption of closedness of $C$ is weakend. In this paper we drop
the assumption of boundedness of $B$.

Since the cancellation law does not hold in the semigroup $\mathcal{C}(X)$ of
nonempty closed convex sets, this semigroup cannot be embedded into a vector
space. However, Robinson [25] proved the cancellation law for the family
$\mathcal{C}_{V}(\mathbb{R}^{n})$ of nonempty closed convex subsets of
$\mathbb{R}^{n}$ sharing common recession cone $V$. He also embedded
$\mathcal{C}_{V}(\mathbb{R}^{n})$ into a topological vector space (but not
into a normed space). Bielawski and Tabor [3] restricted the family
$\mathcal{C}_{V}(X)$, where $X$ is a normed vector space to such closed convex
sets $A$ that the Hausdorff distance $d_{H}(A,V)$ is finite. It enabled them
to embed restricted $\mathcal{C}_{V}(X)$ into a normed vector space,
generalizing in this case the Rådström’s result [24].

In this paper, we investigate the possibility of cancellation ($A+B\subset
B+C\Longrightarrow A\subset C$) for some unbounded sets $B$. In particular, we
generalize in Theorem 4.7 and 4.11 the following Robinson’s result.

######  Theorem 1.2.

(see Lemma 1 in [25].)  Let $X=\mathbb{R}^{n}$ and $A,B,C\subset X$. If $B,C$
are nonempty closed and convex and ${\rm recc}\,B={\rm recc}\,C$ then
$A+B\subset B+C\Longrightarrow A\subset C.$

In Theorem 5.4 we generalize the following Bielawski-Tabor’s order law of
cancellation.

######  Theorem 1.3.

(see Proposition 1 in [3].) Let $X$ be a Banach space, $V$ a closed convex
cone in $X$, $d_{H}$ a Hausdorff distance and
$C_{V}=\\{A\in\mathcal{C}(X):d_{H}(A,V)<\infty\\}.$ If $A,B,C\in C_{V}$ then
$\textup{cl}(A+B)\subset\textup{cl}(B+C)\Longrightarrow A\subset C.$

Finally, we generalize topologically in Theorem 5.6 Bielawski-Tabor’s
embedding theorem.

######  Theorem 1.4.

(see Theorem 2 in [3].) Let $X$ be a Banach space, $V$ a closed convex cone in
$X$, $d_{H}$ a Hausdorff distance and
$C_{V}=\\{A\in\mathcal{C}(X):d_{H}(A,V)<\infty\\}.$ Then the abstract convex
cone $C_{V}$ can be embedded isometrically and isomorphically as a closed
convex cone into a Banach space.

In Section 2 we prove an order cancellation law in an ordered semigroup with
an operator of convergence. This result is applied in Section 5 for convex
sets in topological vector spaces having certain kind of ”finite distance”
between a set and its recession cone.

In Section 3 we give some properties of asymptotic and recession cones in
infinite dimensional spaces. We also present examples of unbounded convex sets
with trivial recession cones and of a recession cone of sum of sets greater
than a sum of recession cones of summands.

In Section 4 we introduce the notion of narrow unbounded sets. We prove in
Theorem 4.7 that in the inclusion cl $(A+B)\subset$ cl$(B+C)$ we can cancel a
narrow set $B$ with an asymptotic cone contained in a recession cone of closed
convex set $C$. We also present properties of addition of narrow sets that
help us in Theorem 4.16 to show that the family of all weakly narrow closed
convex subsets of normed vector space sharing a given pointed recession cone
is closed with respect to Minkowski addition.

##  2 Cancellation law in an ordered semigroup with an operator of convergence

In this section, we prove the order cancellation law for, respectively,
bounded and convex elements in ordered semigroups. The results of this section
will be applied in Section 5 where we give a generalization of Bielawski-Tabor
result [3] to topological vector spaces.

Let $\mathbb{Q}_{2}^{+}$ be the set of all non-negative dyadic rationals and
let us consider a system $S=(S,+,\cdot\,\,,\leqslant,\,\lim)$, where ’$+$’ and
’$\cdot$’ are binary operations from $S\times S$ and $\mathbb{Q}_{2}^{+}\times
S$ into $S$ respectively, $\leqslant$ is partial order on $S$ and $\lim$ is an
limit operator turning $S$ into $\mathcal{L}^{\ast}$-space according to [12],
1.7.18, p. 90. Specifically, (1) if $a_{n}=a$ then $\lim a_{n}=a$, (2) if
$\lim a_{n}=a$ then $\lim a_{k_{n}}=a$ for every subsequence $(a_{k_{n}})$,
(3) if $a_{n}$ does not converge to $a$ than there exists subsequence
$(a_{k_{n}})$ such that no subsequence of $(a_{k_{n}})$ converges to $a$.
Moreover, let the following conditions be satisfied:

(S1)

$a+(b+c)=(a+b)+c$, for all $a,b,c\in S$,

(S2)

$a+b=b+a$, for all $a,b\in S$,

(S3)

there exists an element $0\in S$ such that $a+0=a$, for all $a\in S$,

(S4)

If $a\leqslant b$ then $a+c\leqslant b+c$ for all $a,b,c\in S$,

(S5)

$1\cdot a=a$, for all $a\in S$,

(S6)

if $a\leqslant b$ then $\alpha\cdot a\leqslant\alpha\cdot b$, for all $a,b,\in
S,\alpha\in\mathbb{Q}_{2}^{+}$,

(S7)

$\alpha(\beta\cdot a)=(\alpha\beta)\cdot a$, for all
$\alpha,\beta\in\mathbb{Q}_{2}^{+},a\in S$,

(S8)

$\alpha\cdot(a+b)=\alpha\cdot a+\alpha\cdot b$, for all
$\alpha\in\mathbb{Q}_{2}^{+},a,b\in S$,

(S9)

$(\alpha+\beta)\cdot a\leqslant\alpha\cdot a+\beta\cdot a$, for all
$\alpha,\beta\in\mathbb{Q}_{2}^{+},a\in S$,

(S10)

for all sequnces $(a_{n}),(b_{n})\subset S$ if
$\lim_{n\to\infty}a_{n}=a\wedge\lim_{n\to\infty}b_{n}=b$ and
$\forall_{n\in\mathbb{N}}a_{n}\leqslant b_{n}$ then $a\leqslant b$,

(S11)

for all sequnces $(a_{n})\subset S$ and $b\in S$ if $\lim_{n\to\infty}a_{n}=a$
then
$\lim_{n\to\infty}(a_{n}+b)=a+b.$

######  Definition 2.1.

We say that an element $a\in S$ is bounded if
$\lim_{n\to\infty}(2^{-n}\cdot a)=0.$

######  Definition 2.2.

We say that an element $a\in S$ is convex if $(\alpha+\beta)\cdot
a=\alpha\cdot a+\beta\cdot a$ for all $\alpha,\beta\in\mathbb{Q}_{2}^{+}$.

######  Lemma 2.3.

Let $S$ be a semigroup satisfying conditions (S1–11). If $a,b,c\in S,$
$\lim_{n\to\infty}(2^{-n}b)$ exists , $c$ is convex and $a+b\leqslant b+c,$
then $a+\lim_{n\to\infty}(2^{-n}b)\leqslant c+\lim_{n\to\infty}(2^{-n}b).$

###### Proof.

For $k\in\mathbb{N}$ we have

$\begin{split}2^{k}\cdot a+b&\leqslant\underbrace{a+a+...+a}_{2^{k}-{\rm
times}}+b\leqslant\underbrace{a+a+...+a}_{2^{k}-1-{\rm times}}+b+c\leqslant\\\
&\leqslant\underbrace{a+a+...+a}_{2^{k}-2{\rm
times}}+b+c+c\leqslant...\leqslant b+\underbrace{c+c+...+c}_{2^{k}-{\rm
times}}=\\\ &=b+2^{k}\cdot c.\end{split}$

Last equality follows from the convexity of an element $c$. Multiplying both
sides of the inequality by $2^{-k}$ and using (S6) and (S7) we obtain
$a+2^{-k}\cdot b\leqslant 2^{-k}\cdot b+c.$ Now by (S11) we obtain
$a+\lim_{n\to\infty}(2^{-n}b)=\lim_{k\to\infty}(a+2^{-k}\cdot b)$ and
$c+\lim_{n\to\infty}(2^{-n}b)=\lim_{k\to\infty}(c+2^{-k}\cdot b)$ hence by
(S10) we get $a+\lim_{n\to\infty}(2^{-n}b)\leqslant
c+\lim_{n\to\infty}(2^{-n}b).$ ∎

######  Theorem 2.4.

(Order law of cancellation) Let $S$ be a semigroup satisfying conditions
(S1–11). If $a,b,c\in S,$ $b$ is bounded, $c$ is convex and $a+b\leqslant
b+c,$ then $a\leqslant c.$

###### Proof.

The teorem is a direct corollary from Lemma 2.3. ∎

Let $S_{bc}\subset S$ be a subsemigroup of bounded convex elements. If the
topology in $S_{bc}$ is generated by a uniformity in which the addition is
strongly uniformly continuous then there exists a topological embedding of the
semigroup $S_{bc}$ into a quotient group $\widetilde{S_{bc}}=S_{bc}\times
S_{bc}/_{\sim}$, where $(a,b)\sim(c,d)\Longleftrightarrow a+d=b+c$ (see
Proposition 1.2(i) [27] and Proposition 3.5 [15]).

Theorem 2.4 will be applied in the last section of this paper. The following
trivial example shows a semigroup $S$ satisfying all conditions (S1-11) and
containing only convex elements.

######  Example 2.5.

Let $S=\mathbb{Q}^{+}_{2}\cup\\{\infty\\}$, $\mathbb{Q}_{2}\cup\\{\infty\\}$,
$\mathbb{R}_{+}\cup\\{\infty\\}$ or $\mathbb{R}\cup\\{\infty\\}$. Let addition
be the normal addition and $S$ be ordered by natural order induced from the
reals and $0\cdot A=\\{0\\}$ for all elements $A$ of $S$. We define a limit
$\lim a_{n}=a$ in usual way. Obviously, the conditions (S1-11) are satisfied.
Only the element $\infty$ is not bounded and all elements of $S$ are convex.

The next example shows a semigroup $S$ satisfying all conditions (S1-11) and
containing unbounded and non-convex elements.

######  Example 2.6.

Let $S$ be the family of all nonempty finite subsets of
$\mathbb{Q}^{+}_{2}\cup\\{\infty\\}$, $\mathbb{Q}_{2}\cup\\{\infty\\}$,
$\mathbb{R}_{+}\cup\\{\infty\\}$ or $\mathbb{R}\cup\\{\infty\\}$. Let addition
be natural algebraic addition and $S$ be ordered by inclusion and $0\cdot
A=\\{0\\}$ for all elements $A$ of $S$. We define a limit $\lim A_{n}=A$ if
and only if every neighborhood of $A$ contains all but finite number of sets
$A_{n}$ and

$A\subset\bigcap_{n_{0}=1}^{\infty}\textup{cl}(\bigcup_{n\geqslant
n_{0}}A_{k_{n}})$

for every increasing sequence $(k_{n})$ of integers. It is easy to observe
that the conditions (S1-11) are satisfied. Bounded elements of $S$ are sets
not containing the element $\infty$. Convex elements of $S$ are all sets
containing at most one number other than $\infty$.

Another example is provided by a family of open domains.

######  Example 2.7.

Let $X$ be a real Hausdorff topological vector space and $S^{\ast}$ be the
family of all open domains of $X$, that is of all interiors of closed subsets
of $X$, and $S=S^{\ast}\cup\\{\\{0\\}\\}$. Let addition be defined by
$A\ddot{+}B=$ int cl$(A+B)$, the family $S$ be ordered by inclusion and
$0\cdot A=\\{0\\}$. We define a limit $\lim A_{n}=A$ if and only if for any
neighborhood $U$ of origin there exists $n_{0}$ such that $A_{n}\subset A+U$,
$A\subset A_{n}+U$ for all $n\geqslant n_{0}$. The conditions (S1-11) are
satisfied. Bounded elements of $S$ are bounded sets. Convex elements of $S$
are convex sets.

######  Definition 2.8.

A quartet $(S,\mathbb{R}_{+},+,\cdot)$ satisfying equalities $(S1-3,5,7-8)$,
$(\alpha+\beta)a=\alpha a+\beta a$ and $0\cdot a=0$ for all
$\alpha,\beta\geqslant 0,a,b,c\in S$ is called an abstract convex cone.

##  3 Assymptotic and recession cones

Our main results are placed in sections $4$ and $5$. Since considered sets in
those sections are unbounded and convex, here we discuss some properties of
the recession cone of convex set.

######  Definition 3.1.

For a Hausdorff topological vector space $X$ and $A\subset X$ the set
$A_{\infty}=\\{\lim\limits_{n\rightarrow\infty}t_{n}x_{n}\,|\,(x_{n})\subset
A,(t_{n})\subset\mathbb{R},t_{n}\downarrow 0\,\\}$ is called an assymptotic
cone of $A$. A set recc$A=\\{x\in X\,|\,\forall a\in
A,t\in\mathbb{R}_{+}:a+tx\in A\,\\}$ is called a recession cone of $A$.

In case of ambiguity about exact topology we put $A^{\tau}_{\infty}$,
$A^{*}_{\infty}$ or $A^{\|\cdot\|}_{\infty}$. Obviously, a recession cone of a
set is contained in an assymptotic cone of this set. We are going to express
assymptotic and recession cones in terms of Minkowski subtraction.

######  Definition 3.2.

For a vector space $X$ and subsets $A,B\subset X$ the Minkowski difference of
sets $A$ and $B$ is defined by $A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}B=\\{x\in X:x+B\subset A\\}.$

The following obvious proposition gives us a useful formula of Minkowski
difference of sets.

######  Proposition 3.3.

If $A,B\subset X$ then $A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}B=\bigcap_{b\in B}(A-b).$

Let $X$ be a vector space and let $\mathcal{A}\subset 2^{X}$ be a family of
subsets of $X.$ We say that $\mathcal{A}$ is closed under translations  if for
every $A\in\mathcal{A}$ and for every $x\in X$ we have $x+A\in\mathcal{A}.$ We
also say that $\mathcal{A}$ is closed under intersections  if for any
subfamily $\\{A_{t}\\}_{t\in T}\subset\mathcal{A}$ the intersection
$\bigcap_{t\in T}A_{t}$ belongs to $\mathcal{A}.$

The following proposition is an intermediate corollary from Proposition 3.3.

######  Corollary 3.4.

Let $X$ be a vector space and let $\mathcal{A}\subset 2^{X}$ be a family of
subsets of $X.$ If $\mathcal{A}$ is closed under intersections and
translations then $\mathcal{A}$ is closed under the Minkowski difference.

It is well known that recc$A=A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}A$ for any nonempty convex set $A$.

######  Lemma 3.5.

Let $X$ be a topological vector space and let $A\subset X$ be a closed and
convex set then $A_{\infty}=A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}A$.

###### Proof.

Let $u\in A_{\infty},x\in A$. By Definition 3.1 we have
$u+x=\lim_{n\to\infty}(t_{n}x_{n}+x)=\lim_{n\to\infty}(t_{n}x_{n}+(1-t_{n})x)\in
A$ for some $(x_{n})\subset A,(t_{n})\subset\mathbb{R}$ such that
$t_{n}\downarrow 0$. Hence $u+A\subset A.$ Therefore, $A_{\infty}\subset
A\mathbin{\ooalign{\hss\raise 4.30554pt\hbox{.}\hss\cr$-$}}A.$

On the other hand, we have $A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}A={\rm recc}A\subset A_{\infty}$. ∎

######  Corollary 3.6.

Let $X$ be a topological vector space. If a subset $A\subset X$ is closed and
convex then also the cone $A_{\infty}$ is closed and convex.

The corollary follows from the fact that convexity and closedness are
preserved under translation and intersection.

It is well known (Theorem 8.4 in [26]) that if $X$ is finite dimensional
normed space and $A\subset X$ is closed convex subset then the condition
$A_{\infty}=\mbox{recc}A=\\{0\\}$ implies that the set $A$ is bounded. Now, we
show that this is no longer true in infinite dimensional normed space. To do
this we need the following two lemmas.

######  Lemma 3.7.

Let $X$ be a separable infinite dimensional normed space. Then there exists a
sequence $(f_{n})\subset X^{*}$ such that:

(i)

$\|f_{n}\|=1$ for $n\in\mathbb{N},$

(ii)

for any $x\in X,$ if $f_{n}(x)=0$ for every $n\in\mathbb{N},$ then $x=0.$

###### Proof.

Let $A=\\{x_{n}\,|\,n\in\mathbb{N}\\}$ be a dense and countable set in $X.$ By
Hahn-Banach theorem there exists a sequence $(f_{n})\subset X^{*}$ of
continuous linear functionals such that $\|f_{n}\|=1$ and
$f_{n}(x_{n})=\|x_{n}\|.$ Take any $x\in X$ and suppose that $f_{n}(x)=0$ for
every $n\in\mathbb{N}.$ Let $(x_{n_{k}})\subset A$ be a sequence such that
$\|x-x_{n_{k}}\|<\dfrac{1}{k}$. We have
$\|x_{n_{k}}\|=f_{n_{k}}(x_{n_{k}}-x)\leqslant\|x-x_{n_{k}}\|<\dfrac{1}{k}.$
Hence $x=\lim_{k\to\infty}x_{n_{k}}=0.$ ∎

######  Lemma 3.8.

Let $X$ be infinite dimensional a separable normed space then there exists
$A\subset X$ such that $A$ is closed convex unbounded and
$A\mathbin{\ooalign{\hss\raise 4.30554pt\hbox{.}\hss\cr$-$}}A=\\{0\\}.$

###### Proof.

Let $(f_{n})\subset X^{*}$ be a sequence such as in the proof of Lemma 3.7.
Let $A:=\\{x\in X\,|\,|f_{n}(x)|\leqslant n,n\in\mathbb{N}\\}.$ The set $A$ is
closed and convex. Now, we show that $A$ is not bounded. Suppose contrary that
there exists a $\sigma\in\mathbb{R}$ such that for any $x\in A$ we have
$\|x\|\leqslant\sigma.$ Let us define a functional
$\xi:X\rightarrow\mathbb{R}$,
$\xi(x):=\sup\limits_{n\in\mathbb{N}}|f_{n}(x)|n^{-1}.$ It is easy to see that
$\xi$ is some new norm and $A=\\{x\in X\,|\,\xi(x)\leqslant 1\\}$ is a unit
ball in this norm. Thus for any $x\in X$ we have $\|x\|\leqslant\sigma\xi(x).$
Now take $m\in\mathbb{N}$ such that $m>\sigma$ and let
$u\in\bigcap\limits_{1\leqslant j\leqslant m}\mbox{\rm ker}f_{j},u\neq 0$ then
we have

$\|u\|\leqslant\sigma\xi(u)=\sigma\sup\limits_{n>m}|f_{n}(x)|n^{-1}\leqslant\sigma
m^{-1}\|u\|<\|u\|$

but this is impossible. Hence $A$ is not bounded.
Yet $A\mathbin{\ooalign{\hss\raise 4.30554pt\hbox{.}\hss\cr$-$}}A=\\{0\\}$,
since $A$ is bounded in some other norm. ∎

In the following example a set $A$ has trivial recession cone, while no linear
functional is bounded on $A$.

######  Example 3.9.

Let $c_{00}$ be a space of all sequences of real numbers with finite amount of
nonzero terms. Let $\tau$ be any topology in $c_{00}$ in which all projections
on axes of coordinates are continuous functionals. Let
$A:=$conv$\bigcup\limits_{n=1}^{\infty}([n,2n]^{n}\times\\{0_{\mathbb{R}^{\infty}}\\})$.
Let $A_{\tau}:=$cl${}_{\tau}A$. Let $\sigma$ be a product topology in
$c_{00}$. Then $\sigma$ is weaker than $\tau$. Notice that
$A_{\sigma}\cap(\mathbb{R}^{n}\times\\{0_{\mathbb{R}^{\infty}}\\})=$conv$\bigcup\limits_{k=1}^{n}([k,2k]^{k}\times\\{0_{\mathbb{R}^{\infty}}\\})$.
Hence $A_{\tau}=A$. The set $A$ contains no half-line, and recc$A=\\{0\\}$.
However, for any functional $f\in c_{00}^{*}$ the image $f(A)$ is unbounded.
Hence $A$ is unbounded in any topology which separates points in $c_{00}$.

######  Theorem 3.10.

Let $X$ be an infinite dimensional normed space. Then there exists a subset
$A\subset X$ such that $A$ is closed convex unbounded and
$A\mathbin{\ooalign{\hss\raise 4.30554pt\hbox{.}\hss\cr$-$}}A=\\{0\\}.$

###### Proof.

Since every infinite dimensional normed space $X$ contains some separable
infinite dimensional subspace $Y$, our theorem follows from Lemma 3.8. ∎

The following theorem shows that in any infinitely dimensional Banach space
the equality recc$A\,\dot{+}\,$recc$B\,=\,$recc$(A\,\dot{+}\,B)$ is false for
some closed convex subsets $A$ and $B$.

######  Theorem 3.11.

Let $X$ be an infinite dimensional Banach space. Then there exists subsets
$A,B\in\mathcal{C}(X)$ with trivial reccesion cones such that
recc$(A\dot{+}B)$ is a half-line.

###### Proof.

Let $X$ be an infinitely dimensional Banach space. By the Mazur theorem there
exists a closed subspace $Y$ of $X$ having Schauder basis $(e_{i}),i=0,1,...$.
Moreover (see [8]), there exists a constant $C\geqslant 1$ such that
$|x_{i}|\leqslant C\|x\|$ for all $x=\sum\limits_{j=0}^{\infty}x_{j}e_{j}\in
Y,i=0,1,...$. Let $A={\rm
cl\,conv}\\{ne_{0}+2^{n}ne_{n}\,|\,n=0,1,2,\ldots\\}$, $B={\rm
cl\,conv}\\{ne_{0}-2^{n}ne_{n}\,|\,n=0,1,2,\ldots\\}$. Notice that the ray
$\\{te_{0}\,|\,t\geqslant 0\\}$ is contained in $A+B$. Hence this ray is
contained in the recession cone recc$(A+B)$.

Let us assume that the set $A\dot{+}B$ contains some ray $\\{tv\,|\,t\geqslant
0\\},v\neq 0$, where $v\in X$, $v=\sum\limits_{n=0}^{\infty}v_{n}e_{n}$. Let
$y\in A+B$. Then
$y=\sum\limits_{n=0}^{\infty}\alpha_{n}(ne_{0}+2^{n}ne_{n})+\sum\limits_{n=0}^{\infty}\beta_{n}(ne_{0}-2^{n}ne_{n})\textbf{}$,
where $\alpha_{n},\beta_{n}\geqslant
0,\sum\limits_{n=0}^{\infty}\alpha_{n}=\sum\limits_{n=0}^{\infty}\beta_{n}=1$
and the set $\\{n\,|\,\max(\alpha_{n},\beta_{n})>0\\}$ is finite. Notice that

$C\|tv-y\|\geqslant|tv_{n}-2^{n}n(\alpha_{n}-\beta_{n})|\geqslant
t|v_{n}|-2^{n}n$

for all $n\geqslant 1$. If $v_{n}\neq 0$ then $\|tv-y\|>1$ for all
$t>\dfrac{2^{n}n+C}{|v_{n}|}$. Hence dist$(tv,A\dot{+}B)\geqslant 1$ which
implies that $v_{n}=0$ for all $n\geqslant 1$, and we obtain the equality
recc$\,(A\dot{+}B)=[0,\infty)e_{0}$.

We know that recc$\,A\,\cup\,$recc$\,B\subset\,\,$recc$\,(A\dot{+}B)$. Does
$e_{0}$ belong to recc $A$? If yes then $e_{0}\in A$. Assume that there exists
$y\in$ conv$\\{ne_{0}+2^{n}ne_{n}\,|\,n=0,1,2,\ldots\\}$ such that
$\|e_{0}-y\|<\dfrac{1}{3C}$. Let
$y:=\sum_{n=1}^{\infty}\alpha_{n}\cdot(ne_{0}+2^{n}ne_{n})$. Then

$\max\big{(}\big{|}1-\sum_{n=1}^{\infty}\alpha_{n}n\big{|},\sup\limits_{n\geqslant
1}(\alpha_{n}2^{n}n)\big{)}\leqslant C\|e_{0}-y\|<\dfrac{1}{3}.$

Hence we obtain the inequalities $\alpha_{n}n<\dfrac{1}{3\cdot
2^{n}},n\geqslant 1$. Then
$\sum\limits_{n=1}^{\infty}\alpha_{n}n<\dfrac{1}{3}$. But

$1-\dfrac{1}{3}<\big{|}1-\sum\limits_{n=1}^{\infty}\alpha_{n}n\big{|}\leqslant
C\|e_{0}-y\|<\dfrac{1}{3}.$

Hence we obtain a contradiction. Then $e_{0}\not\in A\supset\,$recc$\,A$.
Therefore, the recession cone of $A$ and, similarly, of $B$ is trivial. ∎

######  Corollary 3.12.

Let $X$ be an infinite dimensional Banach space. Then there exists subsets
$A,B\in\mathcal{C}(X)$ such that
recc$\,(A\dot{+}B)\neq\textup{recc}\,A\,\dot{+}\,\textup{recc}\,B$.

######  Remark 3.13.

It is not difficult to see that in finite dimensional spaces a family of
closed convex sets with a fixed common recession cone is closed with respect
to Minkowski addition $A\dot{+}B=\textup{cl}(A+B)$ [25]. Theorem 3.11 shows
that in infinite dimensional spaces, even Hilbert spaces, a family of closed
convex sets with fixed common recession cone need not to be closed with
respect to Minkowski addition. In order to obtain such a family (with common
recession cone and closed with respect to Minkowski addition), we need to
introduce a condition expressed not in terms of recession cones.

######  Remark 3.14.

Theorem 3.11 shows that in any Banach space $X$ we cannot cancel by sets with
trivial recession cone. For example let $A,B\subset X$ be sets from theorem
3.11. Then $A\dot{+}B\dot{+}\textup{\,recc\,}(A\dot{+}B)=A\dot{+}B$. However,
$A\dot{+}\textup{\,recc\,}(A\dot{+}B)\neq A$. Hence the condition for
canceling by convex sets in Banach spaces cannot be expressed just in terms of
recession cones.

######  Theorem 3.15.

Let $X$ be an infinite dimensional Banach space. Then there exists subsets
$A,B\in\mathcal{C}(X)$ with trivial reccesion cones such that the sum
$A\dot{+}B$ is not closed.

If the space $X$ is not reflexive then we can always find two bounded closed
convex sets, with not closed Minkowski sum. In fact, possibility of finding
such two bounded closed convex sets is equivalent to non-reflexivity of the
space $X$ [23].

###### Proof.

Let $X$ be an infinitely dimensional Banach space. Again by the Mazur theorem
we have a closed subspace $Y$ of $X$ with Schauder basis $(e_{i}),i=0,1,...$
and a constant $C\geqslant 1$ such that $|x_{i}|\leqslant C\|x\|$ for all
$x=\sum\limits_{j=0}^{\infty}x_{j}e_{j}\in Y,i=0,1,...$. Let $A={\rm
cl\,conv}\\{\dfrac{e_{0}}{n}+a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}$, $B={\rm
cl\,conv}\\{\dfrac{e_{0}}{n}-a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}$, $a_{n}>2^{n}$.
Notice that the origin belongs to the closure of $A+B$. We are going to prove
that the origin does not belong to $A+B$. If
$x=\sum\limits_{j=0}^{\infty}x_{j}e_{j}\in A$ then $x_{0}\geqslant 0$. Also
$x_{0}\geqslant 0$ for all $x\in B$. It is enough to show that $x_{0}>0$ for
all $x\in A$. If $x\in{\rm
conv}\\{\dfrac{e_{0}}{n}+a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}$ then there exists a
number $k>0$ such that $x_{k}\geqslant 1$. Hence for all $x\in A$ we have
$\|x\|\geqslant\dfrac{1}{C}$. Therefore, the origin does not belong to $A$.
Suppose, $y=\sum\limits_{j=0}^{\infty}y_{j}e_{j}\in A$ and $y_{0}=0$. Then
$y_{k}>0$ for some $k$. There exists a sequence $(x^{n})\subset{\rm
conv}\\{\dfrac{e_{0}}{n}+a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}$ such that $x^{n}$
tends to $y$. Then $x^{n}_{k}$ tends to $y_{k}$ for all $k$. We can represent
$x^{n}$ as convex combination
$x^{n}=\sum\limits_{j=0}^{\infty}\alpha^{n}_{j}(\dfrac{e_{0}}{j}+a_{j}e_{j})$.
Hence $x^{n}_{k}=\alpha^{n}_{k}a_{k}$ tends to $y_{k}$, and $\alpha^{n}_{k}$
tends to $\dfrac{y_{k}}{a_{k}}$. Therefore,
$x^{n}_{0}\geqslant\dfrac{\alpha^{n}_{k}}{k}$ where
$\dfrac{\alpha^{n}_{k}}{k}$ tends to $\dfrac{y_{k}}{ka_{k}}>0$.

Notice that ${\rm cl}(A+B)={\rm cl}({\rm
conv}\\{\dfrac{e_{0}}{n}+a_{n}e_{n},\,|\,n=0,1,2,\ldots\\}+{\rm
conv}\\{\dfrac{e_{0}}{n}-a_{n}e_{n}\,|\,n=0,1,2,\ldots\\})$. Since for all
$x\in{\rm conv}\\{\dfrac{e_{0}}{n}+a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}+{\rm
conv}\\{\dfrac{e_{0}}{n}-a_{n}e_{n}\,|\,n=0,1,2,\ldots\\}$ we have
$-a_{k}\leqslant x_{k}\leqslant a_{k}$, the same holds true for all $x\in{\rm
cl}(A+B)$ and $k>0$. Similarly, $0\leqslant x_{0}\leqslant 2$ for all
$x\in{\rm cl}(A+B)$. Therefore, if $v$ belongs to the recc ${\rm cl}(A+B)$
then $v=0$. Hence both $A$ and $B$ have a trivial recession cone. ∎

Notice that the sets $A$ and $B$ from the last example cannot be finitely
well-positioned because in a reflexive Banach space no unbounded closed convex
set with trivial recession cone is finitely well-positioned.

##  4 Order cancellation law in normed spaces.

In this section, we generalize the order law of cancellation for closed convex
sets with common recession cone proved by Robinson in [25] for finite
dimensional spaces.

######  Definition 4.1.

Let $X$ be a normed space and let $A$ be a subset of $X.$ Let $\tau$ be a
linear topology weaker than norm topology. By $\mathfrak{B}_{A}^{\tau}$, we
define the set $\left\\{\xi\in X\,|\,\text{there exists }(x_{n})\subset
A\text{ with }\|x_{n}\|\rightarrow\infty\text{ and
}\dfrac{x_{n}}{\|x_{n}\|}\rightharpoonup\xi\right\\}$ where the symbol
”$\rightharpoonup$” denotes the $\tau$-convergence in $X.$

Obviously, the set $\mathfrak{B}_{A}^{\tau}$ is a subset of an asymptotic cone
$A^{\tau}_{\infty}$. Also, the cone $A^{\tau}_{\infty}$ is a cone generated by
$\mathfrak{B}_{A}^{\tau}$. The definition of the set $\mathfrak{B}_{A}$ can be
found in [21] for the weak topology $\tau$. We denote by
$\mathfrak{B}_{A}^{\|\cdot\|}$, $\mathfrak{B}_{A}$, $\mathfrak{B}_{A}^{*}$ the
set $\mathfrak{B}_{A}^{\tau}$ for, respectively, norm, weak and *-weak
topology in $X$. We have
$\mathfrak{B}^{s}_{A}\subset\mathfrak{B}_{A}\subset\mathfrak{B}^{\ast}_{A}$.
The following proposition is obvious.

######  Proposition 4.2.

Let $X$ be a normed space and let $A\subset X$ and $\tau$ be a linear topology
weaker than norm topology. Then
$\mathfrak{B}^{\tau}_{\textup{cl}A}=\mathfrak{B}^{\tau}_{A}$ for the closure
$\textup{cl}A$ in the norm topology.

In [21] it was shown that recc$\,A=A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}A=\text{cone}\,\mathfrak{B}_{A}$ if the set $A$
is closed and convex. Now we show the following proposition.

######  Proposition 4.3.

Let $X$ be a normed space and let $A\subset X$ be a closed and convex subset
of $X.$ If $\,{\rm recc}\,A\neq\\{0\\}$ then ${\rm
recc}\,A=\textup{cone}\,\mathfrak{B}^{\|\cdot\|}_{A}.$

###### Proof.

Let $\xi\in{\rm recc}\,A,$ $\xi\neq 0$. By Lemma 3.5 there exist sequences
$(x_{n})\subset A$ and $(t_{n})\subset\mathbb{R},$ $t_{n}\downarrow 0$ such
that $t_{n}x_{n}\rightarrow\xi,$ hence
$\dfrac{x_{n}}{\|x_{n}\|}=\dfrac{t_{n}x_{n}}{t_{n}\|x_{n}\|}\rightarrow\dfrac{\xi}{\|\xi\|}.$
Therefore, $\dfrac{\xi}{\|\xi\|}\in\mathfrak{B}^{\|\cdot\|}_{A},$ so ${\rm
recc}\,A\subset\text{cone}\,\mathfrak{B}_{A}^{\|\cdot\|}.$ ∎

######  Remark 4.4.

Let us notice that $\mathfrak{B}^{\|\cdot\|}_{A}\neq\emptyset$ implies that
recc$\,A=A\mathbin{\ooalign{\hss\raise
4.30554pt\hbox{.}\hss\cr$-$}}A\neq\\{0\\}$. Equivalently, recc$\,A=\\{0\\}$
implies that $\mathfrak{B}^{\|\cdot\|}_{A}=\emptyset$

######  Definition 4.5.

Let $X$ be a normed space, $A\subset X$ and $\tau$ be a linear topology weaker
than norm topology. We say that $A$ is $\tau$-narrow if for any sequence
$(x_{n})\subset A\text{ with }\|x_{n}\|\rightarrow\infty$ we can choose a
subsequence $(x_{n_{k}})$ with $\dfrac{x_{n_{k}}}{\|x_{n_{k}}\|}$ converging
in the topology $\tau$ to a point other than the origin.

The following proposition is obvious implication of Proposition 4.2.

######  Proposition 4.6.

Let $X$ be a normed space and let $A\subset X$ and $\tau$ be a linear topology
weaker than norm topology. Let the set $A$ be $\tau$-narrow. Then the closure
$\textup{cl}A$ in the norm topology is also $\tau$-narrow.

###### Proof.

Let $(x_{k})\subset\textup{cl}A$ be a sequence such that $\|x_{k}\|$ tends to
$\infty$. Then $x_{k}=a_{k}+u_{k}$ for some sequences $(a_{k})\subset A$ and
$(u_{k})\subset X$ such that $\|u_{k}\|$ tends to $0$. Hence
$\|a_{k}\|=\|x_{k}-u_{k}\|$ tends to $\infty$. There exists a subsequence
$(a_{k_{l}})$ such that $\dfrac{a_{k_{l}}}{\|a_{k_{l}}\|}$ tends to some point
$a\neq 0$ in the topology $\tau$. We have the difference
$\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}-\dfrac{a_{k_{l}}}{\|a_{k_{l}}\|}=\dfrac{u_{k_{l}}}{\|x_{k_{l}}\|}+\dfrac{a_{k_{l}}}{\|a_{k_{l}}\|}\dfrac{\|a_{k_{l}}\|-\|x_{k_{l}}\|}{\|x_{k_{l}}\|}$
tending to the origin. Therefore, $\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}$ tends to
$a\neq 0$. ∎

If a unit ball in $X$ is $\tau$-sequentially compact then a set $A$ is
$\tau$-narrow if and only if $0\notin\mathfrak{B}^{\tau}_{A}$

######  Theorem 4.7.

Let $X$ be a normed space and $\tau$ be a linear topology in $X$ weaker than
the norm topology. Let $A,B,C\subset X$, $C$ be closed in $\tau$ and convex,
$\textup{recc}C$ be pointed, $B$ be $\tau$-narrow and
$B^{\tau}_{\infty}\subset\textup{recc}C$. Then the inclusion $A+B\subset B+C$
implies that $A\subset C.$

###### Proof.

Assume that $0\in B$ and $A=\\{0\\}$. Then by induction we can prove that for
every $n\in\mathbb{N}$ we have $0\in B\subset B+\underbrace{C+...+C}_{n}=B+nC$
hence $0\in\bigcap\limits_{n\in\mathbb{N}}\Big{(}\dfrac{B}{n}+C\Big{)}.$ Then
there exist $b_{n}\in B$ and $c_{n}\in C$ such that
$0=\dfrac{b_{n}}{n}+c_{n}.$

Case 1. The sequence $\dfrac{b_{n}}{n}$ is norm-bounded. If $(b_{n})$ is norm-
bounded than $\dfrac{b_{n}}{n}$ tends to $0$ and $0$ is the limit of the
sequence $(c_{n})$. If $(b_{n})$ is not norm-bounded then, applying the
assumption that the set $B$ is $\tau$-narrow, for some subsequence
$(b_{n_{k}})$ the sequence $\dfrac{b_{n_{k}}}{\|b_{n_{k}}\|}$ tends in the
topology $\tau$ to some $b_{0}\in B^{\tau}_{\infty}\setminus\\{0\\}$. The
number sequence $\dfrac{\|b_{n_{k}}\|}{{n_{k}}}$ is bounded and contains a
subsequence convergent to $t\geqslant 0$. Hence some subsequence
$\dfrac{b_{n_{k_{l}}}}{n_{k_{l}}}=\dfrac{b_{n_{k_{l}}}}{\|b_{n_{k_{l}}}\|}\dfrac{\|b_{n_{k_{l}}}\|}{n_{k_{l}}}$
tends in the topology $\tau$ to $tb_{0},t\geqslant 0$. Then
$\lim_{l\rightarrow\infty}c_{n_{k_{l}}}=-tb_{0}\in C.$ Notice that
$tb_{0}\in\,B^{\tau}_{\infty}\subset\textup{ recc}\,C$, and
$0=-tb_{0}+tb_{0}\in C.$

Case 2. The sequence $\dfrac{b_{n}}{n}$ is not bounded. We may assume that
$\|\dfrac{b_{n}}{n}\|\longrightarrow\infty$ and write
$0=\dfrac{b_{n}}{\|b_{n}\|}+\dfrac{nc_{n}}{\|b_{n}\|}.$ By assumption there
exists a subsequence $\dfrac{b_{n_{k}}}{\|b_{n_{k}}\|}$ tending in the
topology $\tau$ to some $b_{0}\neq 0$ from $B^{\tau}_{\infty}$ and,
consequently,
$\lim_{k\rightarrow\infty}\dfrac{n_{k}c_{n_{k}}}{\|b_{n_{k}}\|}=-b_{0}.$ We
obtain $b_{0}\in B^{\tau}_{\infty}\subset{\rm recc}\,C$ and, by Lemma 3.5,
also $-b_{0}\in C^{\tau}_{\infty}={\rm recc}\,C$ which is impossible, since
recc C is pointed.

We just have proved that the theorem holds in the case of $0\in B$ and
$A=\\{0\\}$. In general case let $A+B\subset B+C$ and $b\in B$. Consider any
$x\in A$. Then $x+(B-b)\subset(B-b)+C$, and $\\{0\\}+(B-b)\subset(B-b)+(C-x)$.
The sets $\\{0\\}$, $(B-b)$ and $(C-x)$ satisfy assumptions of the theorem in
its proved case. Hence $0\in(C-x)$, and $x\in C$. Therefore, $A\subset C$. ∎

######  Lemma 4.8.

Let $X$ be a locally convex space. Let for $B\subset X$ and any $A,C\subset
X$, such that $C$ be closed and convex, the inclusion $A+B\subset B+C$ imply
that $A\subset C.$ Then the inclusion $A+B\subset\textup{cl}(B+C)$ imply that
$A\subset C.$

###### Proof.

Notice that the implication $A+B\subset B+$cl$(C+U)\Longrightarrow
A\subset$cl$(C+U)$ holds true for any convex neighborhood $U$ of the origin.
Assume that $A+B\subset\textup{cl}(B+C)$. We obtain that $A+B\subset
B+C+U\subset B+$cl$(C+U)$ and, by the assumptions of the theorem,
$A\subset$cl$(C+U)\subset C+U+U.$ Then $A+B\subset\textup{cl}(B+C)$ implies
that $A$ is contained in the intersection of all $C+2U$. Therefore, $A\subset
C$. ∎

######  Corollary 4.9.

Let $X$ be a normed space. Assume that there exists a locally convex topology
$\tau$, weaker than the norm topology. Let $A,B,C\subset X$, $C$ be closed in
$\tau$ and convex, $B$ be $\tau$-narrow, $\textup{recc}C$ be pointed and
$B^{\tau}_{\infty}\subset\textup{recc}C$. Then the inclusion
$A+B\subset\textup{cl}_{\tau}(B+C)$ implies that $A\subset C.$

A normed space $X$ is said to be a wcf space (see [2]) if it contains a weakly
compact subset $K$, such that $X={\rm cl\,lin}K$.

######  Corollary 4.10.

Let $A,B,C$ be subsets of a dual $X^{\ast}$ to a wcf space $X$, $C$ be
*-weakly closed and convex, $B$ be nonempty and $B_{\infty}^{*}$ be asymptotic
cone with respect to *-weak topology. Assume that
$0\notin\mathfrak{B}^{\ast}_{B}$, and $B_{\infty}^{*}\subset\text{\rm
recc}\,C$. If the cone $\text{\rm recc}\,C$ is pointed then the inclusion
$A+B\subset\textup{cl}^{*}(B+C)$ implies that $A\subset C.$

###### Proof.

Since $X$ is wcf, by Corollary 2 in [2] and Eberlein–Smulian theorem [28] the
unit ball in $X^{\ast}$ is *-weakly sequentially compact, and $B$ is *-weakly-
narrow. Applying Corollary 4.9 we obtain our corollary. ∎

Since every reflexive Banach space $X$ is a dual of a wcf space and its *-weak
topology coincides with the weak topology, the next theorem follows from
Corollary 4.10.

######  Theorem 4.11.

Let $A,B,C$ be subsets of a reflexive Banach space $X$, $C$ be closed and
convex, $B$ be nonempty and $B_{\infty}$ be asymptotic cone with respect to
weak topology. Assume that $0\notin\mathfrak{B}_{B}$,
$B_{\infty}\subset\text{\rm recc}\,C=V$ and $V$ is a pointed cone. Then the
inclusion $A+B\subset\textup{cl}_{\textup{weak}}(B+C)$ implies that $A\subset
C.$

The theorem generalizes Robinson’s result (Lemma 1 in [25]) in
$\mathbb{R}^{n}$. The condition $0\notin\mathfrak{B}_{B}$ in the theorem is
crucial. Without it the theorem is false in every infinite dimensional Banach
space. In Theorem 3.11 we have recc$\,(A\,\dot{+}\,B)=[0,\infty)e_{0}$. Then
$\big{(}[0,\infty)e_{0}+B\big{)}+A\subset A\,\dot{+}\,B$. Yet,
$\big{(}[0,\infty)e_{0}+B\big{)}\not\subset B$. The reason is that even though
recc$\,A=$ recc$\,B=\\{0\\}$ and $\mathfrak{B}_{A}^{s}=\emptyset$, we have
$0\in\mathfrak{B}_{A}$.

######  Proposition 4.12.

Let $X$ be a reflexive Banach space. Let $B,C\subset X$, $B,C$ be closed and
convex, $(\textup{recc}B)\cap(-\textup{recc}C)=\\{0\\}$ and
$0\notin\mathfrak{B}_{B}$($B$ be weakly-narrow). Then the sum $B+C$ is closed.

###### Proof.

Let $x\in\textup{cl}(B+C)$. Then there exist sequences $(b_{n})\subset B$,
$(c_{n})\subset C$ and $(u_{n})\subset X,\|u_{n}\|\downarrow 0$ such that
$b_{n}+c_{n}+u_{n}=x$. First, if $\|b_{n}\|$ is bounded then some subsequence
$(b_{n_{k}})$ weakly tends to $b\in B$. Hence $c_{n_{k}}$ tends to $x-b\in C$,
and $x\in B+C$. Second, if $\|b_{n}\|$ tends to infinity then some subsequence
$\dfrac{b_{n_{k}}}{\|b_{n_{k}}\|}$ weakly tends to some $b\in$ recc $B,b\neq
0$. Hence $\dfrac{c_{n_{k}}}{\|c_{n_{k}}\|}$ tends to some $-b\in$ recc $C$,
which contradicts the assumption that the cone $V$ is pointed. ∎

Proposition 4.12 is also a consequence of Dieudonné’s theorem [9].

######  Proposition 4.13.

Let $X$ be a normed vector space. Let $B,C\subset X$, $B,C$ be closed and
convex, $\textup{recc}B=\textup{recc}C$ be pointed, and $B$ be weakly-narrow.
Then $\textup{recc cl}(B+C)=\textup{recc}C$.

###### Proof.

Let $x\in\textup{recc cl}(B+C),x\neq 0$. Then
$b_{0}+c_{0}+[0,\infty)x\subset\textup{cl}(B+C)$ for some $b_{0}\in B,c_{0}\in
C$. There exist sequences $(b_{n})\subset B$, $(c_{n})\subset C$ and
$(u_{n})\subset X,\|u_{n}\|\downarrow 0$ such that
$b_{n}+c_{n}+u_{n}=b_{0}+c_{0}+nx$. Notice that

$x=\dfrac{b_{n}}{n}+\dfrac{c_{n}}{n}+\dfrac{u_{n}}{n}-\dfrac{b_{0}}{n}-\dfrac{c_{0}}{n}.$

First, if $\dfrac{\|b_{n}\|}{n}$ tends to 0 then $\dfrac{c_{n}}{n}$ tends to
$x$, and $x\in C_{\infty}=$ recc $C$. Second, if $\dfrac{\|b_{n}\|}{n}$ tends
to $\beta\in(0,\infty)$ then some subsequence $\dfrac{b_{n_{k}}}{n_{k}}$ tends
to $\beta b\in$ recc $B$, where $\dfrac{b_{n_{k}}}{\|b_{n_{k}}\|}$ weakly
tends to some point $b\neq 0$. Hence $\dfrac{c_{n_{k}}}{n_{k}}$ tends to
$x-\beta b\in$ recc $C$, and $x\in$ recc $B+$ recc $C=$ recc $C$.

Third, if $\dfrac{\|b_{n}\|}{n}$ tends to infinity then again some subsequence
$\dfrac{b_{n_{k}}}{\|b_{n_{k}}\|}$ weakly tends to a nonzero element $b\in$
recc $B$. Notice that also
$\dfrac{b_{0}+c_{0}-u_{n_{k}}+n_{k}x-b_{n_{k}}}{\|b_{n_{k}}\|}$ tends to $-b$
while the fraction
$\dfrac{\|b_{0}+c_{0}-u_{n_{k}}+n_{k}x-b_{n_{k}}\|}{\|b_{n_{k}}\|}$ tends to
$1$. Then
$\dfrac{c_{n_{k}}}{\|c_{n_{k}}\|}=\dfrac{b_{0}+c_{0}-u_{n_{k}}+n_{k}x-b_{n_{k}}}{\|b_{0}+c_{0}-u_{n_{k}}+n_{k}x-b_{n_{k}}\|}$
tends to $-b$ and $-b\in$ recc $C$, which contradicts the assumption that the
cone recc $C$ is pointed. ∎

Obviously, the assumption of weak-narrowness of one of the sets in the last
proposition is essential.

######  Theorem 4.14.

Let $X$ be a normed space and $\tau$ be a linear topology weaker than the norm
topology. Let subsets $B,C\subset X$ be $\tau$-narrow and
$B^{\tau}_{\infty}\cap(-C^{\tau}_{\infty})=\\{0\\}$. Then the set
$\textup{cl}_{\|\cdot\|}(B+C)$ is $\tau$-narrow.

###### Proof.

Let a $(x_{k})\subset B+C$, $\|x_{k}\|$ tends to $\infty$. For some sequences
$(b_{k})\subset B$ and $(c_{k})\subset C$ we have $x_{k}=b_{k}+c_{k}$.

First, assume that the sequence $(\|b_{k}\|)$ is bounded. Hence
$\|c_{k}\|\longrightarrow\infty$. For some subsequence $(c_{k_{l}})$ terms
$\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}$ tend to some $c\neq 0$ in the topology
$\tau$. Also
$\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}-\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}=\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}-\dfrac{c_{k_{l}}}{\|x_{k_{l}}\|}+\dfrac{c_{k_{l}}}{\|x_{k_{l}}\|}-\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}$
$=\dfrac{b_{k_{l}}}{\|x_{k_{l}}\|}+\dfrac{\|c_{k_{l}}\|-\|x_{k_{l}}\|}{\|x_{k_{l}}\|}\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}$.
Since the sequence $(\|x_{k_{l}}-c_{k_{l}}\|=\|b_{k_{l}}\|)$ is bounded, a
term $\dfrac{\|c_{k_{l}}\|-\|x_{k_{l}}\|}{\|x_{k_{l}}\|}$ tends to 0. Since
$\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}$ is convergent in the topology $\tau$, the
sequence $\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}-\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}$
tends to 0. Then the sequence $\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}$ tends to
$c\neq 0$.

Second, let $\|b_{k}\|\longrightarrow\infty$ and
$\|c_{k}\|\longrightarrow\infty$. By assumption there exists a sequence
$(k_{l})$ such that $\dfrac{b_{k_{l}}}{\|b_{k_{l}}\|}\longrightarrow b\neq 0$,
$\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}\longrightarrow c\neq 0$ and
$\dfrac{\|b_{k_{l}}\|}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}\longrightarrow t\in[0,1]$.
If $\dfrac{\|b_{k_{l}}\|+\|c_{k_{l}}\|}{\|x_{k_{l}}\|}\longrightarrow\infty$
then
$\dfrac{x_{k_{l}}}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}=\dfrac{b_{k_{l}}}{\|b_{k_{l}}\|}\dfrac{\|b_{k_{l}}\|}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}+\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}\dfrac{\|c_{k_{l}}\|}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}$
is convergent and tends to $0=tb+(1-t)c$. Then $b=0$ or $c=0$ or $0\neq
tb=(t-1)c\in B^{\tau}_{\infty}\cap(-C^{\tau}_{\infty})$ which contradicts the
assumptions.

Therefore, we have
$\dfrac{\|b_{k_{l}}\|+\|c_{k_{l}}\|}{\|x_{k_{l}}\|}\longrightarrow
r\in[1,\infty)$. Then the term
$\dfrac{x_{k_{l}}}{\|x_{k_{l}}\|}=\left(\dfrac{b_{k_{l}}}{\|b_{k_{l}}\|}\dfrac{\|b_{k_{l}}\|}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}+\dfrac{c_{k_{l}}}{\|c_{k_{l}}\|}\dfrac{\|c_{k_{l}}\|}{\|b_{k_{l}}\|+\|c_{k_{l}}\|}\right)\dfrac{\|b_{k_{l}}\|+\|c_{k_{l}}\|}{\|x_{k_{l}}\|}$
tends to $r(tb+(1-t)c)\neq 0$.

We have just proved that $B+C$ is $\tau$-narrow. By Proposition 4.6 the set
cl${}_{\|\cdot\|}(B+C)$ is $\tau$-narrow. ∎

######  Corollary 4.15.

Let $X$ be a reflexive Banach space. Let $B,C\subset X$ be closed and convex
sets sharing a pointed recession cone $\textup{recc}B=\textup{recc}C$. Assume
that $0\notin\mathfrak{B}_{B}\cup\mathfrak{B}_{C}$. Then
$0\notin\mathfrak{B}_{\textup{cl}(B+C)}$, where cl is the weak closure.

###### Proof.

Notice that for reflexive spaces the unit ball in $X$ is weakly sequentially
compact. Hence the condition $0\notin\mathfrak{B}_{B}\cup\mathfrak{B}_{C}$
implies that $B$ and $C$ are weakly narrow. Since their asymptotic cones are
contained in one pointed recession cone, the assumptions of Theorem 4.14 are
fulfilled for the weak topology $\tau$. Therefore, the set
$\textup{cl}_{\|\cdot\|}(B+C)=\textup{cl}(B+C)$ is weakly narrow and
$0\notin\mathfrak{B}_{\textup{cl}(B+C)}$. ∎

######  Theorem 4.16.

Let $X$ be a normed vector space and $V\subset X$ be a closed convex pointed
cone. Then the family of all weakly narrow closed convex subsets of $X$
sharing the recession cone $V$ with modified Minkowski addition $\dot{+}$
defined by $A\dot{+}B:=\textup{cl}\,(A+B)$ and multiplication by non-negative
numbers $($we assume $0\cdot A:=V)$ is an abstract convex cone having a
neutral element $V$ and satisfying cancellation law.

###### Proof.

First, notice that for any two weakly narrow closed convex sets $A$ and $B$
with a common pointed recession cone $V$ the set $\textup{cl}\,(A+B)$ is
weakly narrow by Theorem 4.14 and the recession cone of $\textup{cl}\,(A+B)$
is equal to $V$ by Proposition 4.13. Then the family of all weakly narrow
closed convex sets is closed with respect to the addition $\dot{+}$. Checking
conditions for an abstract convex cone is straightforward. The cancellation
law in the family of all weakly narrow closed convex sets follows from Theorem
4.11. ∎

In view of Proposition 4.12 we can formulate the following Corollary.

######  Corollary 4.17.

Let $X$ be a reflexive Banach space and $V$ be a closed convex pointed cone.
Then the family of all subsets $A\subset X$ sharing the recession cone $V$ and
satisfying the condition $0\notin\mathfrak{B}_{A}$ with Minkowski addition and
multiplication by non-negative numbers $($we assume $0\cdot A:=V)$ is an
abstract convex cone having a neutral element $V$ and satisfying cancellation
law.

Let $X$ be a normed vector space and $V\subset X$ be a closed convex pointed
cone. Then by Theorem 4.16 we can embed the family
$\mathcal{C}_{V}^{\textup{n}}(X)$ of all weakly narrow closed convex subsets
of $X$ sharing the recession cone $V$ in a Minkowski–Rådström–Hörmander vector
space $(\mathcal{C}_{V}^{\textup{n}}(X))^{2}/_{\sim}$. Moreover, by Theorem
2.5 in [16] if, additionally, $X$ is a reflexive Banach space and the set
$\\{x\in V\,|\,\|x\|=1\\}$ can be strictly separated from the origin then
every quotient class $[A,B]\in(\mathcal{C}_{V}^{\textup{n}}(X))^{2}/_{\sim}$
has a minimal representative.

##  5 Order cancellation law in topological vector spaces

In this section we prove order cancellation law for topological vector space
where we cancel by closed and convex sets . These results generalize theorems
obtained by and Tabor, Bielawski in [3].

Let $X$ be a topological vector space and let $V\subset X$ be a closed and
convex cone. By $\mathcal{C}(X)$ we denote the family of al nonempty, closed
and convex subsets of $X$ and by $\mathcal{B}(X)$ the family of all nonempty,
bounded, closed and convex subsets of $X.$ Now let
$\mathcal{C}_{V}^{0}(X)=\\{A\in\mathcal{C}(X)\,|\,V\subset\mbox{recc}\,A\\}$
and

$\mathcal{C}_{V}(X)=\\{A\in\mathcal{C}(X)\,|\,A\subset
V\,\dot{+}\,B,\,\,V\subset A\,\dot{+}\,B\mbox{ for some
}B\in\mathcal{B}(X)\\}.$ (5.1)

We may look at elements of $\mathcal{C}_{V}(X)$ as at sets of bounded
Hausdorff-like distance from the cone $V$. The family $\mathcal{C}_{V}(X)$ is
a subfamily of all ’bounded’ elements of $\mathcal{C}_{V}^{0}(X)$. For a
sequence $(A_{n})\subset\mathcal{C}_{V}^{0}(X)$ we define the limit operator
as follows: $\lim(A_{n}):=A\in\mathcal{C}_{V}^{0}(X)$ if and only if for every
neighborhood $U$ of zero in $X$ there exists $k\in\mathbb{N}$ such that
$A\subset A_{n}+U\mbox{ and }A_{n}\subset A+U$ for $n\geqslant k.$

Moreover, the family $\mathcal{C}_{V}^{0}(X)$ with the addition
$A\,\dot{+}\,B=\textup{cl}(A+B)$ and the multiplication by non-negative
numbers defined by $\lambda A:=\\{\lambda a\,|\,a\in A\\},$ $0\cdot A:=V$ is
an abstract convex cone.

######  Proposition 5.1.

The algebraic system $(\mathcal{C}_{V}^{0}(X),\cdot\,,\dot{+},\subset,\lim)$
satisfies the axioms (S1–11).

###### Proof.

Obviously, the axioms (S1–2) and (S4–9) are satisfied. So we need to prove
only the axioms (S3) and (S10–11). Since $V$ is a neutral element in
$\mathcal{C}_{V}^{0}(X)$, the axiom (S–3) holds true.

To prove (S10) let us assume that $\lim(A_{n})=A$, $\lim(B_{n})=B$ and
$A_{n}\subset B_{n}$ for all $n\in\mathbb{N}.$ Take any neighborhood $U$ of
the origin in $X$ then $A\subset A_{k}+U\subset B_{k}+U\subset B+U+U$ for
sufficiently large $k$. Hence

$A\subset\bigcap\\{(B+U+U)\,|\,U\textup{ is a neighborhood of
}0\\}=\textup{cl}B=B,$

so the axiom (S10) is satisfied.

To prove (S11), let us assume that $\lim(A_{n})=A$, $\lim(B_{n})=B.$ Take any
neighbourhood $U$ of zero in $X$ and let $W$ be a neighborhood of zero in $X$
such that $W+W+W\subset U.$ Let $k$ be a natural number such that $A\subset
A_{n}+W$, $A_{n}\subset A+W$, $B\subset B_{n}+W$ and $B_{n}\subset B+W$ for
$n\geqslant k.$ Then

$A\,\dot{+}\,B\subset A+B+W\subset A_{n}+B_{n}+W+W+W\subset
A_{n}+B_{n}+U\subset A_{n}\,\dot{+}\,B_{n}+U$

and, analogously, $A_{n}\dot{+}B_{n}\subset A\dot{+}B+U$ for $n\geqslant k$.
Hence $\lim(A_{n}\dot{+}B_{n})=A\dot{+}B.$ Therefore, the axiom (S11) is
satisfied. ∎

Now, we prove two lemmas concernig the family $\mathcal{C}_{V}(X)$

######  Lemma 5.2.

For any $A\in\mathcal{C}_{V}(X)$ we have ${\mbox{recc}}A=V.$

###### Proof.

Take any $A\in\mathcal{C}_{V}(X)$. First, we prove that
$V\subset\mbox{recc}A$. Let us observe that for every $n\in\mathbb{N}$ we have
$V=\dfrac{1}{n}V\subset\dfrac{1}{n}A\,\dot{+}\,\dfrac{1}{n}B$ for some
$B\in\mathcal{B}(X).$ Now, take arbitrary $a\in A,v\in V$. For any
neighborhood $U$ of the origin let $W$ be another neighborhood of the origin
such that $W+W+W\subset U.$ Let also $m\in\mathbb{N}$ be large enough that
$\dfrac{a}{m}\in W$ and $\dfrac{1}{m}B\subset W.$ Then for some $a_{1}\in A$
and $b_{1}\in B$ we have

$a+v\in
a+\dfrac{1}{m}a_{1}+\dfrac{1}{m}b_{1}+W=\left[\dfrac{1}{m}a_{1}+\left(1-\dfrac{1}{m}\right)a\right]+\dfrac{1}{m}a+\dfrac{1}{m}b_{1}+W\,\,\,\,\,\,\\\
\subset A+W+W+W\subset A+U.\hskip 85.35826pt$

Hence $A+V\subset\textup{cl}A=A.$ Then $V\subset\mbox{recc}\,A.$

To prove the inverse inclusion observe that $A\subset V\,\dot{+}\,B$ for some
$B\in\mathcal{B}(X)$. Take any $a\in A$. We have

$\mbox{recc}\,A=\mbox{recc}\,(A-a)=\bigcap_{n}\dfrac{A-a}{n}\subset\bigcap_{n}\dfrac{V\dot{+}B-a}{n}=\bigcap_{n}(V\dot{+}\dfrac{B-a}{n})\subset\textup{cl}V=V,$

and the proof is complete. ∎

######  Lemma 5.3.

For any $A\in\mathcal{C}_{V}(X)$ we have $\lim(2^{-n}A)=V.$

###### Proof.

Let $B\in\mathcal{B}(X)$ be such that $A\subset V\,\dot{+}\,B,\,\,V\subset
A\,\dot{+}\,B.$ Take any neighborhood $U$ of zero in $X$ and let $W$ be a
balanced neighborhood of zero such that $W+W\subset U$ and let
$k\in\mathbb{N}$ be such that $2^{-n}B\subset W$ for $n\geqslant k.$ Then we
have $2^{-n}A\subset 2^{-n}(V+B+W)\subset V+W+W\subset V+U$ and, analogously,
$V=2^{-n}V\subset 2^{-n}A+U.$ Thus $\lim(2^{-n}A)=V.$ ∎

The following theorems follow from the above considerations and Section 2.

######  Theorem 5.4.

Let $A,B,C\in\mathcal{C}_{V}^{0}(X)$ and suppose that $\lim(2^{-n}B)$ exists
and is contained in $V$ then the inclusion $A\,\dot{+}\,B\subset
B\,\dot{+}\,C$ implies $A\subset C.$

###### Proof.

In order to apply Lemma 2.3 notice that convexity of a closed convex set $C$
implies the convexity in the sense of Definition 2.2. ∎

The next theorem is a straightforward corollary from Theorem 5.4 and Lemma
5.3.

######  Theorem 5.5.

Let $A,B,C\in\mathcal{C}_{V}(X)$ then the inclusion $A\,\dot{+}\,B\subset
B\,\dot{+}\,C$ implies $A\subset C.$

Tabor and Bielawski [3] proved a version of order cancellation law for closed
convex sets having finite Hausdorff distance from a fixed convex cone $V$ in a
normed space $X$. Theorem 5.5 is another version of order cancellation law
which generalizes to topological spaces a result obtained by Tabor and
Bielawski.

In the Cartesian product $\mathcal{C}_{V}(X)\times\mathcal{C}_{V}(X)$, due to
cancellation law we can introduce an equivalence relation $\sim$ in an usual
way:

$\displaystyle(A,B)\sim(C,D)\Longleftrightarrow A\,\dot{+}\,D=B\,\dot{+}\,C$

Let
$\widetilde{\mathcal{C}}_{V}(X)=\mathcal{C}_{V}(X)\times\mathcal{C}_{V}(X)/_{\sim}$
be the quotient set of the equivalence classes with respect to $\sim\ $. By
$[A,B]\in\widetilde{\mathcal{C}}_{V}(X)$ we denote the equivalence class of an
element $(A,B)\in\mathcal{C}_{V}(X)\times\mathcal{C}_{V}(X).$

The semigroup $\mathcal{C}_{V}(X)$ with the addition $\dot{+}$ and scalar
multiplication defined by $\lambda A:=\\{\lambda a:a\in A\\}$ for $\lambda>0$
and $0\cdot A=V$ is an abstract convex cone. Thanks to Theorem 5.5, it can be
proved that the quotient set $\widetilde{\mathcal{C}}_{V}(X)$ with addition

$\displaystyle[A,B]+[C,D]:=[A\,\dot{+}\,C,B\,\dot{+}\,D]$ (1)

and scalar multiplication defined by

$\displaystyle\lambda[A,B]:=\begin{cases}[\lambda A,\lambda B]\mbox{ if
}\lambda\geqslant 0\\\ [(-\lambda)B,(-\lambda)A]\mbox{ if }\lambda\leqslant
0\end{cases}$ (2)

is a real vector space.

Moreover, the function $j:\mathcal{C}_{V}(X)\to\widetilde{\mathcal{C}}_{V}(X)$
defined by $j(A)=[A,V]$ is a canonical embedding which satisfies
$j(A\,\dot{+}\,B)=j(A)+j(B)\mbox{ and }j(\lambda A)=\lambda j(A)$ for all
$A,B\in\mathcal{C}_{V}(X)$ and $\lambda\geqslant 0.$

Now, let $\mathcal{W}$ be a basis of balanced neighbourhoods of $0$ in
topological vector space $X.$ For $U\in\mathcal{W}$ we define by $W_{U}$ the
set

$\\{(A,B)\in(\mathcal{C}_{V}(X))^{2}:A\subset B\,\dot{+}\,K,B\subset
A\,\dot{+}\,K\mbox{ for some }K\in\mathcal{C}(X),K\subset U\\}.$

The family $\\{W_{U}\\}_{U\in\mathcal{W}}\,$ forms a basis of uniformity on
$\mathcal{C}_{V}(X)$. The uniformity induces a topology in
$\mathcal{C}_{V}(X)$. Let us denote this topology by $\tau_{H}$. The scalar
multiplication $\cdot$ and the addition $\dot{+}$ in $\mathcal{C}_{V}(X)$ are
continuous in the topology $\tau_{H}$.

We define a basis of neighbourhoods of $0$ in the space
$\widetilde{\mathcal{C}}_{V}(X)$ as follows. Let $U\in\mathcal{W}$ and let

$\mathcal{T}_{U}=\\{[A,B]\in\widetilde{\mathcal{C}}_{V}(X):A\subset
B\,\dot{+}\,K,B\subset A\,\dot{+}\,K\mbox{ for some
}K\in\mathcal{C}(X),K\subset U\\}.$

The family $\\{\mathcal{T}_{U}\\}_{U\in\mathcal{W}}\,\,$ forms a basis of
neighbourhoods of $0$ in the space $\widetilde{\mathcal{C}}_{V}(X)$ and
defines a linear topology on the space the space
$\widetilde{\mathcal{C}}_{V}(X)$. Denote this topology by $\sigma_{H}.$

Now, we can express the following theorem which generalizes embedding theorem
obtained by Tabor and Bielawski for normed spaces [3].

######  Theorem 5.6.

The canonical mapping $j:\mathcal{C}_{V}(X)\to
j(\mathcal{C}_{V}(X))\subset\widetilde{\mathcal{C}}_{V}(X)$ is an isomorphic
embedding of a topological abstract convex cone
$(\mathcal{C}_{V}(X),\dot{+},\cdot,\tau_{H})$ into a real topological vector
space $(\widetilde{\mathcal{C}}_{V}(X),+,\cdot,\sigma_{H})$. Moreover the
canonical mapping $j:\mathcal{C}_{V}(X)\to
j(\mathcal{C}_{V}(X))\subset\widetilde{\mathcal{C}}_{V}(X)$ is a
homeomorphism.

The results from this section apply also to topological vector spaces like
$l^{p},L^{p},0<p<1$. In the case of locally convex vector spaces the
introduced topology in $\mathcal{C}_{V}(X)$ coincides with ”Hausdorff”
topology, where sets of families
$\mathcal{T}_{A,U}:=\\{B\in\mathcal{C}_{V}(X)\,|\,A\subset
B\,\dot{+}\,U,B\subset A\,\dot{+}\,U\\},U\in\mathcal{W}$ form basis of
neighborhoods of $A\in\mathcal{C}_{V}(X)$. In spaces which are not locally
convex a problem arises because belonging $B\in\mathcal{T}_{A,U}$ does not
imply that $B_{1}\in\mathcal{T}_{A_{1},U}$ for all $(A_{1},B_{1})\in[A,B]$.
Notice that $A\subset B\,\dot{+}\,U\Rightarrow A\,\dot{+}\,C\subset
B\,\dot{+}\,C\,\dot{+}\,U,C\in\mathcal{C}_{V}(X)$ but having the inclusion
$A\,\dot{+}\,C\subset B\,\dot{+}\,C\,\dot{+}\,U$ we may not be able to cancel
$C$, since the set $B\,\dot{+}\,C\,\dot{+}\,U$ may not be convex. Then the
Minkowski–Rådström–Hörmander space
$\mathcal{C}_{V}(X)\times\mathcal{C}_{V}(X)/_{\sim}$ does not posses a
topology compatible with the topology of neighborhoods $\mathcal{T}_{A,U}$.

##  6 Conclusions

The order law of cancellation $A+B\subset B+C\Longrightarrow A\subset C$ for
subsets of a vector space is a valuable tool in mathematical analysis.
However, proving the order cancellation law for an unbounded set $B$ poses a
difficult challenge. Assuming the inclusion of recession cones
recc$\,B\subset$ recc$\,C$ and the convexity of $C$ is obvious. In section 5
we proved that certain kind of ”finite distance” defined in (5.1) between a
convex set $B$ and recc$\,B\subset$ is sufficient. Moreover, the semigroup of
closed convex sets sharing their recession cones can be topologically embedded
in the quotient topological vector space.

In the case of ”infinite distance” between $B$ and recc$\,B$ we were able to
prove the order cancellation law under conditions that (1) the space $X$ is
normed and (2) the set $B$ is narrow according to Definition 4.5, which means
that the asymptotic cone of $B$ is generated by a subset of the unit sphere
which can separated from the origin. We expect that the semigroup of closed
convex narrow sets sharing their recession cones is metrizable and can be
topologically embedded in the quotient linear metric space.

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Jerzy Grzybowski
Adam Mickiewicz University, Faculty of Mathematics and Computer Science,
Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
E-mail address<EMAIL_ADDRESS>

Hubert Przybycień
Jan Kochanowski University in Kielce, Department of Mathematics, Uniwersytecka
7, 25-406 Kielce, Poland
E-mail address<EMAIL_ADDRESS>