# Superdiffusion of vortices in two-component quantum fluids of light

M. Roldão<EMAIL_ADDRESS>J. L. Figueiredo  P. Monteiro  J.
T. Mendonça  H. Terças<EMAIL_ADDRESS>GoLP — Instituto de
Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa,
Portugal

###### Abstract

The quantum diffusion of a vortex in a two-component quantum fluid of light is
investigated. In these systems, the Kerr nonlinearity promotes interactions
between the photons, displaying features that are analogue of a Bose-Einstein
condensates. Quantum fluids of light have the advantage of simulating matter-
wave phenomena at room temperatures. While the analogy is true at the mean
field level, the full quantum dynamics of an impurity in quantum fluids of
light of, and therefore the ability of featuring genuine quantum noise, has
never been considered. We numerically solve the problem by simulating a
vortex-like impurity in the presence of noise with the Bogoliubov spectral
density, and show that the vortex undergoes superdiffusion. We support our
results with a theory that has been previously developed for the brownian
motion of point-like particles.

Quantum Fluids of Light, Quantum Polaron, Brownian Motion, Superdiffusion,
Quantum Vortex.

††preprint: APS/123-QED

Introduction.— The first observations of diffusive processes in nature are due
to Brown in 1827, who reported on the erratic motion of wildflower pollen
suspended in water [1, 2]. Theoretical understanding of the brownian motion
remained poor until 1905, with Einstein’s and Smoluchowski’s kinetic
explanations, proposing that the heavy pollen particules are propelled by the
sucessive kicks by the light water molecules [3, 4]. The success of the
kinetic theory of the brownian motion led to Perrin’s estimation of Avogadro’s
number in 1908 [5]. An intriguing variation of the phenomenon appears in the
quantum regime [6, 7], with profound consequences in condensed matter systems,
such as Anderson localization, which has been observed for both matter and
light waves. Quantum brownian motion may originate from random external
potentials such as disorder or via the coupling to a random reservoir. In the
latter case, the Caldeira-Legget model provides a semi-empirical treatment of
the coupling between a quantum particle and the environment, overcoming the
difficulties of obtaining a Langevin equation from the standard procedures of
quantization [8, 9, 10]. Depending on the specific form of the particle-bath
coupling, the system may display normal or anomalous diffusion [11, 12]. Over
the recent years, a great deal of attention has been given to the
investigation of quantum diffusion in cold atoms, with the intention of
implementing the polaron problem by immersing quantum impurities in a Bose-
Einstein condensate (BEC) [13, 14, 15, 16, 17, 18, 19]. Recently, the quantum
diffusion of a Bose polaron in BECs has been suggested as a route towards
high-sensitivity measurements of temperature in the sub-nanokelvin regime [20,
21].

Another interesting physical system in which quantum brownian motion could
possibly occur, albeit lacking investigation, are quantum fluids of light
(QFL), often referred as the optical analogues of Bose-Einstein condensates
[22, 23]. Quantum fluids of light are platforms where photon-photon
interactions are possible thanks to the Kerr nonlinearity, encompassed in the
nonlinear part of the susceptibility, $\displaystyle\chi^{(3)}$. The fluidic
characteristic of a light beam has been first discussed by Brambilla [24] and
Staliunas [25], but have received a great deal of attention in the more recent
years [26, 27, 28, 29, 30]. At the mean-field level, paraxial QFLs are
governed by an equation of the Ginzburg-Landau type, which contains a
biunivocal correspondence with the Gross-Pitaevskii equation [23]. Such
correspondence has allowed the emulation many physical effects taking place in
BECs, such as solitons and vortex formation [31], quantum turbulence [32, 33,
34, 35, 36] and spectral condensation [37, 38], Rayleigh-Taylor instabilities
[39], and have even been pointed as analogue (also known as hydrodynamic, or
Unruh [40]) black holes linked to the existence of a local sound speed [41,
42, 43, 44, 45]. Moreover, spin-orbit coupling in QFLs are possible in
nonparaxial configurations, making possible the quantum simulation with multi-
component BECs [46].

Figure 1: (color online): Brownian motion of a vortex in a quantum fluid of
light with two orthogonal polarizations. Left panel : The bath excitations are
introduced in the first polarization of the electric field and are randomly
initiated according to the Bogoliubov spectral density,
$\displaystyle\mathcal{J}(k)$ in the inset. The cutoff wavevector has been
chosen to verify the condition $\displaystyle\lambda_{\text{min}}=2\pi/k_{\rm
max}=16\xi$. Right panel: The vortex in the second component plays the role of
a quantum impurity. It is initially located at the center
$\displaystyle(x,y)=(0,0)$ and is propelled by the action of the noise. In
both panels, we have set $\displaystyle t=25\tau$.

Another crucial benchmark of the viability of QFLs as quantum simulators of
BECs has been the recent observation of the Bogoliubov spectrum, an aspect
that is intimately related to the bosonic nature of the quantum excitations
[47, 48]. Therefore, testing the ability of the latter to produce
superdiffusion would constitute a critical test to the full quantum mechanical
nature of QFLs, therefore opening the door towards the simulation of open
quantum systems with fully optical setups.

In this Letter, we theoretically investigate the brownian motion of a quantum
vortex in a two-component quantum fluid of light propagating in a rubidium
vapor, which is an appealing platform for applications quantum technologies
[49, 50]. In our scheme, the two components correspond to the different
polarizations of the electric field [34]. We show that a quantum vortex
undergoes superdiffusion, which is compatible with the superohmic nature of
the Bogoliubov spectrum.

Two-component quantum fluids of light.— Quantum fluids of light (QFL) refer to
the collective behavior of photons in nonlinear media. Considering a
centrosymmetric media, such as alkali metals, the polarization field
$\displaystyle\bm{P}$ can be expanded to the third-order in the electric field
$\displaystyle\bm{E}$ as

$\bm{P}=\epsilon_{0}\left(\chi^{(1)}\bm{E}+\chi^{(3)}|\bm{E}|^{2}\bm{E}\right)+\mathcal{O}(E^{5}),$
(1)

where $\displaystyle\epsilon_{0}$ is the dielectric permittivity and
$\displaystyle\chi^{(1)}=\sqrt{n_{1}^{2}-1}$, with $\displaystyle
n_{1}=c/(\partial_{k}\omega_{L})$ is the (linear) refractive index. The term
$\displaystyle\chi^{(3)}$ is commonly designated as the nonlinear optical Kerr
effect [51] and leads to a dependence of the index of refraction on the
applied electric field, i.e. the intensity of the photon beam. For rubidium,
the Kerr susceptibility $\displaystyle\chi^{(3)}={\rm
Re}\chi^{(3)}(\delta,T)+i\rm{Im}\chi^{(3)}(\delta,T)$ is a function of the
detuning $\displaystyle\delta=\omega_{L}-\omega_{0}$, where
$\displaystyle\omega_{L}$ is the laser frequency and $\displaystyle\omega_{0}$
is the D2-transition frequency and $\displaystyle T$ is the temperature [52].
In what follows, we decompose the electric field into two orthogonal
polarizations $\displaystyle\bm{\varepsilon}_{\alpha}$, with
$\displaystyle\alpha=\\{1,2\\}$, as
$\displaystyle\bm{E}(\bm{r},t)=\sum_{\alpha}E_{\alpha}(\bm{r}_{\perp},t)\bm{\varepsilon}_{\alpha}e^{ik_{L}z-i\omega_{L}t}$,
with $\displaystyle\bm{r}_{\perp}=(x,y)$ denoting the perpendicular direction
and $\displaystyle z$ the propagation direction. Under the paraxial
approximation, each polarization is governed by the following equation [53,
54]

$i\dfrac{\partial E_{\alpha}}{\partial
z}=-\frac{1}{2k_{L}}\nabla_{\perp}^{2}E_{\alpha}-\frac{3}{2}n_{2}k_{L}|E_{\alpha}|^{2}E_{\alpha}-n_{2}k_{L}|E_{\beta}|^{2}E_{\alpha},$
(2)

where the nonlinear refractive index is given by $\displaystyle
n_{2}\equiv\chi^{(3)}/n_{0}$, with $\displaystyle n_{0}$ being the rubidium
vapor density (which, in turn, is controlled through the temperature
$\displaystyle T$). Recent quantitative measurements show that its absolute
value $\displaystyle|n_{2}|$ increase with the vapour density and decrease
with both $\displaystyle|\delta|$ and beam intensity
$\displaystyle|\bm{E}|^{2}$ [55]. As $\displaystyle n_{2}$ is also frequency
dependent, the frequency detuning $\displaystyle\delta$ may be such that
$\displaystyle n_{2}<0$, which corresponds to a self-defocusing medium. This
is the conditions we consider in the remainder of this manuscript.

For a more quantitative analysis, it is convenient to map the optical problem
into a Bose-Einstein condensate. In the zero temperature limit, BECs can be
described by a macroscopic order parameter $\displaystyle\Psi$ that obeys the
Gross-Pitaevskii equation. This phenomenon of condensation involves a dominant
occupation of the lowest quantum state, which motivates writing the
$\displaystyle\hat{\Phi}$ operator as a sum of two terms,
$\displaystyle\hat{\Phi}(\bm{r},t)=\Psi(\bm{r},t)+\delta\hat{\varphi}(\bm{r},t)$.
The first term
$\displaystyle\Psi(\bm{r},t)=\bra{0}\hat{\Phi}(\bm{r},t)\ket{0}$ is the mean
field order parameter and refers to the condensed fraction of the system,
while $\displaystyle\delta\hat{\varphi}(\bm{r},t)$ corresponds to the
fluctuations (or excitations), the fraction that is not condensed in the GS.
Since we are considering a weakly interacting gas ($\displaystyle
a_{s}\ll\lambda_{dB}$, the de Broglie wavelength), excitations are expected to
be small. Neglecting the excitations, we have the Gross-Pitaevskii equation
for the condensate

$i\hbar\dfrac{\partial\Psi}{\partial
t}=\left[-\dfrac{\hbar^{2}\nabla^{2}}{2m}+V_{\text{trap}}(\bm{r})+g|\Psi|^{2}\right]\Psi.$
(3)

This equation is valid for Bose-Einstein condensates if the temperature of the
system is much lower than the critical temperature of condensation,
$\displaystyle T_{c}$. For these systems, the Bogoliubov dispersion relation
is written as

$\omega^{2}=c^{2}k^{2}+\frac{\hbar^{2}k^{4}}{4m^{2}},$ (4)

where $\displaystyle c_{s}=\sqrt{gn_{0}/m}=\xi/\tau$ is the speed of sound
[56]. Equations (2) and (3) have strong similarities between their terms, this
being what justifies the parallelism established between nonlinear optics and
the BEC physics.

Diffusion and non-Markovianity.— We intend to simulate the quantum dynamics of
an impurity in a Bose-Einstein condensate. This system is ruled by a system of
coupled Gross-Pitaevskii equations. For quantum fluids of light, interactions
between bosons have a fixed relation of $\displaystyle\sigma\equiv
g_{21}/g_{11}=2/3$ [53, 54]; space and time are scaled such that
$\displaystyle\tau=t/t^{\prime}=\hbar/\mu$,
$\displaystyle\xi_{i}=r/r^{\prime}=\hbar/\sqrt{g_{ii}m_{i}n_{0}}$ and
$\displaystyle g_{ii}=\mu_{i}/n_{0}$ with $\displaystyle i=\\{1,2\\}$. We are
interested in the case $\displaystyle m_{1}$=$\displaystyle m_{2}$ and
$\displaystyle g_{11}=g_{22}$, such that Eqs. (2) read

$\begin{cases}i\dfrac{\partial\Psi_{1}}{\partial
t^{\prime}}=\left[-\dfrac{1}{2}\nabla_{\perp}^{2}+|\Psi_{1}|^{2}+\sigma|\Psi_{2}|^{2}\right]\Psi_{1}\\\
\\\ i\dfrac{\partial\Psi_{2}}{\partial
t^{\prime}}=\left[-\dfrac{1}{2}\nabla_{\perp}^{2}+\sigma|\Psi_{1}|^{2}+|\Psi_{2}|^{2}\right]\Psi_{2}.\end{cases}$
(5)

To solve the system in Eq. (5), we follow a finite-difference, Cranck-
Nicholson scheme. A Runge-Kutta method of $\displaystyle 4^{\text{th}}$ order
in the time domain is used. The simulation parameters were chosen to guarantee
convergence [57] ($\displaystyle dx^{2}/dt\geq 2$), $\displaystyle dx=0.1\xi$,
$\displaystyle dt=0.005\tau$. The wavefunction $\displaystyle\Psi_{1}$ is the
background that holds the bosonic excitations, while $\displaystyle\Psi_{2}$
is the impurity, a vortex-like defect in density, which we initialize with the
help of a imaginary-time evolution method. Dirichlet boundary conditions were
used with a squared box of side $\displaystyle 120\xi.$ The thermal bath
corresponding to the Bogoliubov excitations has the prescribed spectral
density. The cutoff frequency $\displaystyle\Lambda_{c}$, or the
$\displaystyle\lambda_{\text{min}}$, selected the dominant frequency in the
noise spectrum. A snapshot of the simulation can be found in Fig. 1.

Figure 2: (color online) Diffusion coefficient $\displaystyle\mathcal{D}$ as a
function of the cutoff frequency $\displaystyle\Lambda_{c}$. For a point-like
impurity, we find a maximum for $\displaystyle\tau\Lambda_{c}=2$.

As described in [20], the dynamics of an impurity in a BEC implies the
interactions between this particle and the thermal heat bath where it is
immersed. The total Hamiltonian is

$H=H_{I}+H_{B}+H_{BB}+H_{IB}.$ (6)

The spectral density of the heat bath is calculated through the tensor
$\displaystyle\underline{\underline{g_{\bm{k}}}}=\underline{g_{\bm{k}}}\cdot\underline{g_{\bm{k}}}^{T}$
as

$\underline{\underline{J}}(\omega)=\sum\limits_{\bm{k}\neq
0}\hbar\underline{\underline{g_{\bm{k}}}}\delta(\omega-\omega_{\bm{k}}).$ (7)

For the generic d-dimensional case the spectral density is a diagonal tensor
with entries $\displaystyle\mathcal{J}_{d}$ (due to spatial symmetry of the
noise). For a point-like particle in two dimensions (QFL analogy),

$\mathcal{J}_{2}(\omega)=\frac{\sigma^{2}\omega_{0}^{4}m_{B}^{2}}{g_{B}n_{0}^{2}\sqrt{\frac{\omega^{2}}{\omega_{0}^{2}}+1}}\left(\sqrt{\frac{\omega^{2}}{\omega_{0}^{2}}+1}-1\right)^{2}.$
(8)

Here $\displaystyle m_{B}$ is the bosonic mass, $\displaystyle n_{0}$ is the
untrapped density, and $\displaystyle\omega_{0}=\tau^{-1}=\mu/\hbar$. Other
important aspect of the impurity dynamics is due to the Markovianity of its
motion. Formally, Markovianity is tackled through the Langevin equation for
the motion of the impurity where the damping kernel $\displaystyle\Gamma(t)$
is obtained from the Spectral density $\displaystyle\underline{\underline{J}}$
as

$\underline{\underline{\Gamma}}(t)=\frac{1}{m_{I}}\int_{0}^{\infty}d\omega\frac{1}{\omega}\;\underline{\underline{J}}(\omega)\cos(\omega
t).$ (9)

When the damping kernel is not zero, the dynamics of the system has a memory
term. Only when the damping kernel is zero the dynamics is Markovian: in terms
of the Spectral density, when $\displaystyle J(\omega)=\eta\omega$.

The mean square displacement of the impurity (MSD), for the long time limit in
the two dimensional case, is

$\langle[x(t)-x(0)]^{2}\rangle=\left[\langle\dot{x}(0)^{2}\rangle+\frac{\hbar\tau^{2}\Lambda_{c}^{3}}{6m_{I}}\right]\left(\frac{t}{\alpha_{2}}\right)^{2},$
(10)

where $\displaystyle\alpha_{2}=1+\frac{1}{4}(\Lambda_{c}\tau)^{2}$.
Considering the characteristic healing length
$\displaystyle\xi=\hbar/\sqrt{\mu m}$ and $\displaystyle\tau=\hbar/\mu$, being
the chemical potential $\displaystyle\mu=gn_{0}$, we get

$\frac{\text{MSD}(t)}{\xi^{2}}=\frac{m_{B}}{m_{I}}\frac{\tau\Lambda_{c}^{3}}{6}\left(\frac{t}{\alpha_{2}}\right)^{2},$
(11)

choosing $\displaystyle\langle\dot{x}(0)^{2}\rangle=0$. The ratio
$\displaystyle m_{B}/m_{I}$ influences the amplitude of the movement, such as
the cutoff frequency $\displaystyle\Lambda_{c}$. A superdiffusion regime have
therefore been predicted and the system should contain non-Markovian effects
for a Bogoliubov heat bath acting on an impurity [20]. The diffusion
coefficient is defined as
$\displaystyle\mathcal{D}_{2}=\hbar\tau^{2}\Lambda_{c}^{3}/(6m_{I}\alpha_{2}^{2})$.
Comparing with the numerical results we assessed if the behaviour of the
vortex was similar to the one of the localized particle or if their different
wavefunction distributions are relevant. In Fig. 2, the non-monotony of the
diffusion coefficient, with respect to $\displaystyle\Lambda_{c}$, is shown.
We should remind ourselves that these results were obtained for a localized
impurity.

We may integrate the expression for the MSD numerically to predict the
behavior of an impurity in the quantum bath. In Fig. 2, we see the behaviour
of the nonlinear diffusion coefficient $\displaystyle\mathcal{D}$, such as the
values of $\displaystyle\lambda_{\text{min}}$ used in the simulation. To
observe the effect of $\displaystyle\Lambda_{c}$, the simulation was run with
4 different values (table 1). As it is shown in figure 3, the initial growth
of the MSD is similar to the prediction, but started diverging after
$\displaystyle t=10\tau$. The analytical prediction for the MSD is in
disagreement with our results by a scaling factor of 200. This can be related
to the mass of the vortex being distinct to the bosonic mass, therefore larger
than the previously considered point-like impurity.

$\displaystyle\lambda_{\text{min}}[\xi]$ |  $\displaystyle\Lambda_{c}^{-1}[\tau]$ |  $\displaystyle N_{\text{runs}}$ |  $\displaystyle t_{\text{max}}[\tau]$
---|---|---|---
16 |  2.50 |  20 |  25
8 |  1.18 |  25 |  40
6 |  0.85 |  20 |  25
4 |  0.50 |  20 |  25
Table 1: Parameters of the simulation in its characteristic units
$\displaystyle\xi=\hbar/\sqrt{\mu m}$ and $\displaystyle\tau=\hbar/\mu$.

An oscillation with a period of $\displaystyle\sim 10\tau$ is present in the
results in accord with the theoretical model [20]. It predicts that this
oscillation becomes negligible for large time scales as a quadratic dependency
in time emerges (see Eq. 10). The simulation results do not confirm this
quadratic dependency, as time increased. Comparing the different
$\displaystyle\lambda_{\text{min}}$ runs, it is clear that the MSD increases
in general for larger values of this parameter, being
$\displaystyle\lambda_{\text{min}}=16$ the largest value that was tested.
Aiming to simulate an untrapped impurity with Dirichlet boundary conditions,
the $\displaystyle\lambda_{\text{min}}$ was kept below a size comparable with
$\displaystyle 60\xi=(x_{\text{max}}-x_{\text{min}})/2$, to prevent boundary
effects.

Figure 3: (color online) Superdiffusion of a quantum vortice driven by the
quantum noise. Left panel: the comparison between the predicted behaviour of
mean-square displacement (MSD) (black dashed curve) and the results for
$\displaystyle\lambda_{\text{min}}=8\xi$ \- grey lines are the individual
runs, the blue line is the averaged MSD. Right panel: Vortex mean-square
displacements for different values of the cutoff.

The relation between $\displaystyle\mathcal{D}$ and $\displaystyle\Lambda_{c}$
is not in agreement with the theoretical prediction (see figures 2 and 3).
While the prediction for this range of $\displaystyle\Lambda_{c}$, the
diffusion coefficient should be increasing with $\displaystyle\Lambda_{c}$.
The simulation results indicate otherwise, as we would be on the decreasing
regime of the curve in 2 ($\displaystyle\tau\Lambda_{c}>2$).

These disagreements suggest that the behaviour of a vortex in the heat bath is
quite distinct from the prediction for a point-like particle. Therefore the
spectral density was recalculated following the same method of Ref. [20], with
a vortex-like density profile

$\rho_{I}(\bm{q})=\int_{-\infty}^{\infty}d\bm{r}^{\prime}e^{-i\bm{q}\cdot\bm{r}^{\prime}}\delta(\bm{r}^{\prime}-\bm{r})\frac{(\bm{r}-\bm{r^{\prime}})^{2}}{\sqrt{(\bm{r}-\bm{r^{\prime}})^{2}+\xi^{2}}}.$
(12)

This change in the radial density of the vortex with an aperture of the order
of the healing length $\displaystyle\xi$, gives an expression for the spectral
density such as

$\mathcal{J}(\omega)=\frac{n_{0}g_{IB}^{2}}{2h\xi}\frac{\omega/\Lambda_{c}\left(\sqrt{1+\frac{\omega^{2}}{\Lambda_{c}^{2}}}-1\right)^{1/2}}{\left(1+\frac{\omega^{2}}{\Lambda_{c}^{2}}\right)^{3/4}},$
(13)

different from Eq. 8. This small change changes the influence of the
excitations on the impurity dynamics. In Fig. 4, the different behaviour of
the coupling coefficient $\displaystyle|g_{\bm{k}}|$ and spectral density
diagonal elements $\displaystyle\mathcal{J}$ are shown. While for a particle,
as far as the frequency $\displaystyle\omega$ increase, the coupling
increases, and the higher frequency dominates, for a localized vortex, the
interactions saturate to a fixed value $\displaystyle\mathcal{J}_{\infty}$.

As we increase the excitations frequency $\displaystyle\omega$, the relevance
of each frequency to the vortex stabilizes and all the frequencies tend to
have the same importance. This means none will resonate to displace the
vortex, as we can see from Eq. 7. A particle will always have a higher
$\displaystyle\omega$ to the wandering motion so the diffusive coefficient
should have a behaviour with $\displaystyle\Lambda_{c}$ such as shown in
figure 2. In these systems, we observe superdiffusive motion, as the
Bogoliubov spectrum of the quantum noise selects specific frequencies to be
dominant in the spectrum. The Bogoliubov spectrum is not homogeneous.
Ultimately, looking at equation 9, superdiffusion emerges from the fact that
this resonant behaviour between the quantum noise and the impurity, is not
exactly random, in the Brownian sense, but is responsible for this faster
diffusive effect, which has the characteristic fingerprint of a time
dependence such as MSD$\displaystyle\sim t^{\alpha}$, with
$\displaystyle\alpha>1$ the superdiffusion index. Of course, this
superdiffusion is highly dependent on the noise spectrum. If we introduce
white noise to the system, we expect that the damping kernel does not select
any specific frequency and the memory term in Langevin equation disappears and
we recover the Brownian motion. This is also what happens in the high
temperature regime $\displaystyle\hbar\omega/k_{B}T\xrightarrow{}0$, as we
lose the memory effects. These conclusions are deeply connected to the
markovianity of the system. The noise works as a memory external bath through
the noise kernel $\displaystyle\underline{\underline{\Gamma}}(t)$, so the
markovianity of the system is highly excitation-dependent, i.e. dependent on
the spectrum of the bath.

The accuracy of the mean-squared displacement, being a statistical quantity,
is increased with the number of runs. Its error will increase in time for
diffusive processes and will require more runs to predict more accurately its
behaviour for larger time scales.

Figure 4: (color online) Particle (red) and vortex (blue) spectral densities
diagonal element. $\displaystyle\mathcal{J}_{\infty}=n_{0}g_{IB}^{2}/2h\xi$.

Conclusions and future work.— Quantum fluids of light were introduced as an
analogy between nonlinear optics and Bose-Einstein condensation, a formal
analogy that may be of extreme importance and relevance in both areas, as well
as for real quantum computing applications, such as the development of quantum
memories. Regarding the study of diffusive effects on Bose-Einstein
condensates, a numerical simulation involving the interaction between a vortex
solution, as an impurity and a thermal Bogoliubov bath was studied. All the
parameters were discussed, as well as the important and relevant details from
the numerical point of view, such as the conditions to achieve numerical
stability, but also physically, for example, the imaginary time evolution
procedure. The results were analyzed and motivated a deeper study of the
assumptions that were made, leading to a detailed discussion of the impurity
density profile and its role in the interactions. The regimes of
superdiffusion or standard Brownian diffusion were evaluated and their
relation with the loss of markovianity was presented as a direct consequence
of the interactions of the impurity with the quantum noise bath, in
particular, a direct consequence of the noise spectrum that is used. The
spectrum of the noise was discussed to have immense importance to the
behaviour of the system and its super or subdiffusive regime. We concluded
that in the situation where a white noise spectrum is included, or we reach
the high-temperature regime, superdiffusion is lost, as well as the memory
effects intrinsic to non-markovianity. Aiming to reach more diverse results in
this matter, the next steps were discussed as well as other interesting
configurations that might be studied with the same algorithm.

Acknowledgements.— The authors acknowledge the financial support from
Fundacção para a Ciência e a Tecnologia (FCT-Portugal) through Contract No.
CEECIND/00401/2018, Project No. PTDC/FIS-OUT/3882/2020, and Grant No
UI/BD/151557/2021.

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representatives of the isomorphism classes of simple right $R$-modules. It is
easily seen that the assignment $U\mapsto r_{R}(U)$ defines a bijection from
${\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}}_{R}$ to the set
${\mathbf{P}\mathbf{r}\mathbf{i}\mathbf{m}}_{R}$ of right primitive ideals of
$R$; consequently we are enabled to consider the partial ordering
$\preccurlyeq$ in ${\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}}_{R}$ (which we
called in [9] the _natural partial order_) by prescribing that $U\preccurlyeq
V$ exacty when $r_{R}(U)\subset r_{R}(V)$. Given
$U,V\in{\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}}_{R}$, we have that
$U\preccurlyeq V$ if and only if $h(U)\leqslant h(V)$ and, for every
$A\in\mathbf{F}\mathbf{P}_{R}$ such that
$A/A\cdot\operatorname{Soc}_{h(V)-1}(R)\simeq V$, it is true that $U\lesssim
A/A\cdot\operatorname{Soc}_{h(U)-1}(R)$ (see [9, Theorem 2.2]); moreover it
follows from [10, Theorem 2.1] (333 We take here an opportunity to remark that
a mistake has been left to stand in the final and published version of the
quoted paper. Precisely, in the quoted theorem and its proof ten instances
appear of a factor module of the form $(***+L_{\beta+1})/L_{\beta}$, which
must be read as $(***+L_{\beta})/L_{\beta}$ instead, as one can easily infer
from the context.) that $V\prec U$ if and only if $h(U)<h(V)$ and, if
$A\in\mathbf{F}\mathbf{P}_{R}$ is such that
$A/A\cdot\operatorname{Soc}_{h(V)-1}(R)\simeq V$, then
$A/A\cdot\operatorname{Soc}_{h(U)-1}(R)$ contains an infinite direct sum of
copies of $U$.

Let $A\in\mathbf{E}\mathbf{S}\mathbf{F}\mathbf{P}_{R}$ and let us consider the
simple module $V=A/AL_{h(A)-1}$. We say that $A$ is _order-selective_ (see
[11]) if, for every $U\in{\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}}_{R}$, the
property $U\lesssim A/AL_{h(U)-1}$ is equivalent to $U\preccurlyeq V$. By the
above $A$ is order-selective if and only if, given any
$U\in{\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}}_{R}$ such that $h(U)<h(V)$,
the condition $U\lesssim A/AL_{h(U)-1}$ implies that $A/AL_{h(U)-1}$ contains
an infinite direct sum of copies of $U$. Finally, we say that a complete and
irredundant subset
$\mathbf{A}\subset\mathbf{E}\mathbf{S}\mathbf{F}\mathbf{P}_{R}$ is _order
representative_ if all its elements are order-selective; if it is the case, by
[11, Corollary 3.2] the property $A\lesssim B$ is equivalent to
$A/A\cdot\operatorname{Soc}_{h(A)-1}(R)\preccurlyeq
B/B\cdot\operatorname{Soc}_{h(B)-1}(R)$ for every $A,B\in\mathbf{A}$.

######  Proposition 5.6.

Let $J$ be an artinian and finitely sheltered subset of $I$ and, with the
above notations, let us consider the set
$\mathbf{U}_{J}\,\,\colon\\!\\!=\\{\mathbf{u}_{j}H(I,J)\mid j\in J\\}$. Then:

  1. (1)

$\mathbf{U}_{J}$ is an order representative subset of
$\mathbf{E}\mathbf{S}\mathbf{F}\mathbf{P}_{H(I,J)}$;

  2. (2)

$\mathbf{K}_{0}(H(I,J))$ is free with $[\mathbf{U}_{J}]$ as a basis and

$\mathbf{K}_{0}(H(I,J))^{+}=M([\mathbf{U}_{J}]).$

###### Proof.

(1) We first observe that if $j\in J$ and $\alpha$ is the unique ordinal such
that $j\in J_{\alpha+1}\setminus J_{\alpha}$, then
$h(\mathbf{u}_{j}H(I,J))=\alpha+1$ and

$(\mathbf{u}_{j}+H(I,J_{\alpha}))[H(I,J)/H(I,J_{\alpha})]\simeq\mathbf{u}_{j}H(I,J)/\mathbf{u}_{j}H(I,J_{\alpha})]\simeq\mathbf{u}_{j}H(I,j)$

is a simple right $H(I,J)/H(I,J_{\alpha})$-module, showing that
$\mathbf{u}_{j}H(I,J)$ is eventually simple.

Next, let $V$ be any simple right $H(I,J)$-module and set $\beta+1=h(V)$. Then

$V=V\cdot H(I,J_{\beta+1})\quad\text{ and }\quad V\cdot H(I,J_{\beta})=0,$

so that $V$ is a simple right module over $H(I,J\setminus J_{\beta})\simeq
H(I,J)/H(I,J_{\beta})$ and, as such, we have

(5.2) $V=V\cdot[H(I,J_{\beta+1}\setminus
J_{\beta})]=V\cdot\operatorname{Soc}(H(I,J\setminus J_{\beta})).$

Now the homogeneous components of $\operatorname{Soc}(H(I,J\setminus
J_{\beta}))$ are precisely the $H(I,k)$ for $k\in J_{\beta+1}\setminus
J_{\beta}$, thus it follows from (5.2) that $V$ is isomorphic to
$\mathbf{u}_{k}H(I,k)=\mathbf{u}_{k}H(I,J_{\beta+1}\setminus J_{\beta})$ for a
unique $k\in J_{\beta+1}\setminus J_{\beta}$. Finally, by using Theorem 4.1,
Remark 4.2 and Theorem 3.6, (3), we get

$\displaystyle V\simeq\mathbf{u}_{k}H(I,k)=\mathbf{u}_{k}H(I,J\setminus
J_{\beta})\simeq\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}H(I,J_{\beta})$
$\displaystyle=\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}\operatorname{Soc}_{\beta}(H(I,J)).$

We conclude that $\mathbf{U}_{J}$ is a complete and irredundant subset of
$\mathbf{E}\mathbf{S}\mathbf{F}\mathbf{P}_{H(I,J)}$.

In order to complete the proof of (1), it remains to show that every member of
$\mathbf{U}_{J}$ is order selective. Thus, given $k\in J$, let $U$ be any
simple right $H(I,J)$-module such that $h(U)<h(\mathbf{u}_{k}H(I,J))$, set
$\alpha+1=h(U)$, assume that
$U\lesssim\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}\operatorname{Soc}_{\alpha}(H(I,J))$
and let us show that
$\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}\operatorname{Soc}_{\alpha}(H(I,J))$
contains an infinite direct sum of copies of $U$. As we proved above, there is
a unique $j\in J$ such that
$U\simeq\mathbf{u}_{j}H(I,J)/\mathbf{u}_{j}\operatorname{Soc}_{\alpha}(H(I,J))$,
thus

$\mathbf{u}_{j}H(I,J)/\mathbf{u}_{j}\operatorname{Soc}_{\alpha}(H(I,J))\lesssim\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}\operatorname{Soc}_{\alpha}(H(I,J)),$

that is

(5.3)
$\mathbf{u}_{j}H(I,J)/\mathbf{u}_{j}H(I,J_{\alpha})\lesssim\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}H(I,J_{\alpha}).$

It follows that $\mathbf{u}_{j}H(I,J\setminus
J_{\alpha})\lesssim\mathbf{u}_{k}H(I,J\setminus J_{\alpha})$ and, by using
Lemma 5.4, we infer that

$\displaystyle H(I,j)=H(I,J\setminus J_{\alpha})\mathbf{u}_{j}H(I,J\setminus
J_{\alpha})$ $\displaystyle\subset H(I,J\setminus
J_{\alpha})\mathbf{u}_{k}H(I,J\setminus J_{\alpha})\subset H(I,\\{\leqslant
k\\}\cap J).$

As a consequence $j<k$ and, by choosing any $Z\in\mathcal{R}_{k}$, we have
that $\mathbf{u}_{k}H(I,k)\simeq\mathbf{e}_{Z}H(I,k)$ and hence
$\mathbf{u}_{k}H(I,J)\simeq\mathbf{e}_{Z}H(I,J)$ by Lemma 5.3. According to
Lemma 2.8, (3), there is a subset $\mathcal{S}$ of $\mathcal{R}_{j}$ such that
$|\mathcal{S}|=\boldsymbol{\aleph}$ and $Y\subset Z$ for every
$Y\in\mathcal{S}$. Consequently

$\bigoplus_{Y\in\mathcal{S}}\mathbf{e}_{Y}H(I,J)\subset\mathbf{e}_{Z}H(I,J)\simeq\mathbf{u}_{k}H(I,J),$

so that

$\bigoplus_{Y\in\mathcal{S}}(\mathbf{e}_{Y}H(I,J)/\mathbf{e}_{Y}H(I,J_{\alpha}))\lesssim\mathbf{u}_{k}H(I,J)/\mathbf{u}_{k}H(I,J_{\alpha})\simeq\mathbf{u}_{k}H(I,J)$

and we are done, because
$U\simeq\mathbf{e}_{Y}H(I,J)/\mathbf{e}_{Y}H(I,J_{\alpha})$ for every
$Y\in\mathcal{S}$.

(2) is now a consequence of [11, Theorems 3.6 and 4.3]. ∎

We are now in a position to state and prove the main theorem of this section.

######  Theorem 5.7.

Let $I$ be an $\boldsymbol{\aleph}$-poset having a finite cofinal subset. With
the above notations, the group $\mathbf{K}_{0}(\mathfrak{B}(I))$ is free with
$[\mathbf{U}]$ as a basis and

(5.4) $\mathbf{K}_{0}(\mathfrak{B}(I))^{+}=M([\mathbf{U}]).$

Consequently the assignment $i\mapsto[\mathbf{u}_{i}\mathfrak{B}(I)]$ induces
an isomorphism

$\rho(I)\colon G(I)\longrightarrow\mathbf{K}_{0}(\mathfrak{B}(I))$

of partially ordered abelian groups.

###### Proof.

We first prove that $[\mathbf{U}]$ is a $\mathbb{Z}$-independent subset of
$\mathbf{K}_{0}(\mathfrak{B}(I))$. Let $J$ and $K$ be two disjoint and finite
subsets of $I$, assume that $(a_{j})_{j\in J}$ and $(b_{k})_{k\in K}$ are
families of non negative integers such that

(5.5) $\sum_{j\in J}a_{j}[\mathbf{u}_{j}\mathfrak{B}(I)]=\sum_{k\in
K}b_{k}[\mathbf{u}_{k}\mathfrak{B}(I)],$

and let us prove that, consequently, $a_{j}=0=b_{k}$ for all $j\in J$ and
$k\in K$. The statement is trivial if $J\cup K=\emptyset$ (it cannot be
false…). By proceeding inductively on $n=|J\cup K|$, let $n>0$ and assume that
the statement is true whenever $|J\cup K|<n$. Inasmuch as $\mathfrak{B}(I)$ is
unit-regular, (5.5) is equivalent to the existence of an isomorphism (444If
$\mathfrak{I}$ is a right ideal of some ring $R$ and $n$ is a natural number,
we use here the notation $\mathfrak{I}^{n}$ to mean the direct sum of $n$
copies of $\mathfrak{I}$, not their product!)

(5.6) $\bigoplus_{j\in
J}(\mathbf{u}_{j}\mathfrak{B}(I))^{a_{j}}\simeq\bigoplus_{k\in
K}(\mathbf{u}_{k}\mathfrak{B}(I))^{b_{k}}.$

Consequently, if $J=\emptyset$ (resp. $K=\emptyset$), then $b_{k}=0$ for all
$k\in K$ (resp. $a_{j}=0$ for all $j\in J$) and induction works. Thus, assume
that $J\neq\emptyset\neq K$. If we set $L=J\cup K$, it follows from (2) and
(3) of Theorem 3.6 that

$\mathbf{u}_{i}\mathfrak{B}(I)=\mathbf{u}_{i}H(I,\\{\leqslant L\\})$

for every $i\in L$, therefore (5.6) can be rewritten as

(5.7) $\bigoplus_{j\in J}(\mathbf{u}_{i}H(I,\\{\leqslant
L\\}))^{a_{j}}\simeq\bigoplus_{k\in K}(\mathbf{u}_{k}H(I,\\{\leqslant
L\\}))^{b_{k}}.$

Let $L^{\prime}$ be the set of all maximal elements of $L$ and note that
$\\{<L^{\prime}\\}$ is a lower subset of $\\{\leqslant L\\}$, therefore
$H(I,\\{<L^{\prime}\\})$ is an ideal of the ring $H(I,\\{\leqslant L\\})$ and
we have a $\mathbb{K}$-algebra isomorphism

$H(I,\\{\leqslant L\\})/H(I,\\{<L^{\prime}\\})\simeq H(I,L^{\prime})$

(see Remark 4.2). After tensoring (over $H(I,\\{\leqslant L\\})$) by

$H(I,\\{\leqslant L\\})/H(I,\\{<L^{\prime}\\}),$

the isomorphism (5.7) gives rise to an isomorphism

(5.8) $\bigoplus_{j\in J\cap
L^{\prime}}(\mathbf{u}_{j}H(I,L^{\prime}))^{a_{j}}\simeq\bigoplus_{k\in K\cap
L^{\prime}}(\mathbf{u}_{k}H(I,L^{\prime}))^{b_{k}}.$

Since $H(I,L^{\prime})=\bigoplus_{i\in L^{\prime}}H(I,i)$ is a (not
necessarily artinian) semisimple ring having each $H(I,i)$ as a homogeneous
component, then the $\mathbf{u}_{i}H(I,i)$ are pairwise non-isomorphic minimal
right ideals of $H(I,L^{\prime})$ for $i$ ranging in $L^{\prime}$, thus (5.8)
entails that $a_{j}=0=b_{k}$ for all $j\in J\cap L^{\prime}$ and $k\in K\cap
L^{\prime}$. As a result we have in $\mathbf{K}_{0}(\mathfrak{B}(I))$ the
equality

(5.9) $\sum_{j\in J\setminus
L^{\prime}}a_{j}[\mathbf{u}_{j}\mathfrak{B}(I)]=\sum_{k\in K\setminus
L^{\prime}}b_{k}[\mathbf{u}_{j}\mathfrak{B}(I)]$

and induction applies, because $|(J\setminus L^{\prime})\cup(K\setminus
L^{\prime})|<n$.

We have sofar shown that $[\mathbf{U}]$ is a $\mathbb{Z}$-linearly independent
subset of $\mathbf{K}_{0}(\mathfrak{B}(I))$. Let $\mathcal{F}(I)$ be the set
of all finite subsets of $I$ which contain all maximal elements of $I$ and
note that $H(I,J)$ is a unital subring of $\mathfrak{B}(I)$ for every
$J\in\mathcal{F}(I)$. Inasmuch as $\mathcal{F}(I)$ is directed under the
ordering by inclusion and $\mathfrak{B}(I))$ is the union of its unital
subrings $H(I,J)$, for $J\in\mathcal{F}(I)$, one might conclude the proof by
using the fact that the functor $\mathbf{K}_{0}(-)$ preserves direct limits
(see [19, Proposition 15.11]), together with Proposition 5.6. Nonetheless, we
wish to display here a direct and relatively fast argument. Thus, let us
denote by $\mu_{J}\colon H(I,J)\to\mathfrak{B}(I)$ the inclusion map for every
$J\in\mathcal{F}(I)$ and let us first show that

(5.10)
$\mathbf{K}_{0}(\mathfrak{B}(I))=\bigcup_{J\in\mathcal{F}(I)}\mathbf{K}_{0}(\mu_{J})(H(I,J)).$

Remember that the group $\mathbf{K}_{0}(\mathfrak{B}(I))$ is generated by the
elements of the form $[\mathbf{e}\mathfrak{B}(I)]$, where $\mathbf{e}$ is an
idempotent of $\mathfrak{B}(I)$, because $\mathfrak{B}(I)$ is a regular ring.
Now, if $\mathbf{e}$ is any idempotent of $\mathfrak{B}(I)$, since

$\mathfrak{B}(I)=\bigcup_{J\in\mathcal{F}(I)}H(I,J)$

we can choose some $J\in\mathcal{F}(I)$ such that $\mathbf{e}\in H(I,J)$. We
have

$[\mathbf{e}\mathfrak{B}(I)]=[\mathbf{e}H(I,J)\otimes_{H(I,J)}\mathfrak{B}(I)]=\mathbf{K}_{0}(\mu_{J})([\mathbf{e}H(I,J)]),$

consequently (5.10) immediately follows.

Given $J\in\mathcal{F}(I)$, by Proposition 5.6 the group
$\mathbf{K}_{0}(H(I,J))$ is free with the set
$\mathbf{U}_{J}=\\{[\mathbf{u}_{j}H(I,J)]\mid j\in J\\}$ as a basis. Thus
(5.10), enable us to conclude that

$[\mathbf{U}]=\bigcup_{J\in\mathcal{F}(I)}\\{[\mathbf{u}_{i}\mathfrak{B}(I)]\mid
i\in
J\\}=\bigcup_{J\in\mathcal{F}(I)}\\{\mathbf{K}_{0}(\mu_{J})([\mathbf{u}_{i}H(I,J)])\mid
i\in J\\}$

generates $\mathbf{K}_{0}(\mathfrak{B}(I))$ and hence is a basis.

Next, let $\mathbf{0}\neq\mathbf{e}=\mathbf{e}^{2}\in\mathfrak{B}(I)$ and
choose some $J\in\mathcal{F}(I)$ such that $\mathbf{e}\in H(I,J)$. By
Proposition 5.6 there is a family $(a_{j})_{j\in J}$ of integers such that

$[\mathbf{e}H(I,J)]=\sum_{j\in J}a_{j}[\mathbf{u}_{j}H(I,J)]$

in $\mathbf{K}_{0}(H(I,J))$ and $a_{j}>0$ whenever $j$ is a maximal element of
$J$. Then in $\mathbf{K}_{0}(\mathfrak{B}(I))$ we have

$[\mathbf{e}\mathfrak{B}(I)]=\mu_{J}([\mathbf{e}H(I,J)])=\sum_{j\in
J}a_{j}\mu_{J}([\mathbf{u}_{j}H(I,J)])=\sum_{j\in
J}a_{j}[\mathbf{u}_{j}\mathfrak{B}(I)].$

This is enough to conclude that the first member of (5.4) is contained in the
second. Conversely, let $(a_{i})_{i\in I}$ be any family of integers having
finite support $F$, such that $a_{i}>0$ whenever $i$ is a maximal element of
$F$, and choose any $J\in\mathcal{F}(I)$ such that $F\subset J$. By
Proposition 5.6 there is some $B\in\mathbf{F}\mathbf{P}_{H(I,J)}$ such that

$[B]=\sum_{j\in J}a_{j}[\mathbf{u}_{j}H(I,J)].$

Consequently we have

$\displaystyle\sum_{i\in I}a_{i}([\mathbf{u}_{j}\mathfrak{B}(I)])=\sum_{j\in
J}a_{j}\mathbf{K}_{0}(\mu_{J})([\mathbf{u}_{j}H(I,J)])=\mathbf{K}_{0}(\mu_{J})([B])$
$\displaystyle=[B\otimes_{H(I,J)}\mathfrak{B}(I)].$

This shows the reverse inclusion in (5.4) and hence the equality. The last
statement follows from Proposition 5.5. ∎

##  6\. Primeness versus primitivity. The primitive spectrum.

In this section we investigate which properties of the
$\boldsymbol{\aleph}$-poset $I$ are related with the primeness and the
primitivity of the algebra $\mathfrak{B}(I)$. As we shall see, while if
$|I|=\boldsymbol{\aleph}_{0}$ primeness of $\mathfrak{B}(I)$ always implies
primitivity, for every cardinal $\boldsymbol{\aleph}>\boldsymbol{\aleph}_{0}$
there exists some $\boldsymbol{\aleph}$-poset $I$ such that $\mathfrak{B}(I)$
is prime but not primitive. As we know (see Remark 4.2), every factor ring of
$\mathfrak{B}(I)$ is isomorphic to $H(I,J)$ for a unique upper subset $J$ of
$I$; the main result of this section is that the ring $H(I,J)$ is prime if and
only if $J$ is downward directed, while the existence of a coinitial chain in
$J$ is necessary and sufficient for $H(I,J)$ to be primitive.

At any rate, based on the results so far obtained, we can exhibit easily here
a first special set of primitive factor rings of $\mathfrak{B}(I)$, namely
those of the form $H(I,i\leqslant)$. In fact, given $i\in I$, we have from
Theorem 4.4 that

$\operatorname{Soc}(H(I,i\leqslant))=H(I,i)\neq\mathbf{0}$

is homogeneous. If $\mathfrak{I}$ is any nonzero ideal of $H(I,i\leqslant)$
and $L$ is the unique lower subset of $\\{i\leqslant\\}$ such that
$\mathfrak{I}=H(I,L)$, it follows from Theorem 3.6, (3) that
$H(I,i)H(I,l)\neq\mathbf{0}\neq H(I,l)H(I,i)$ for every $l\in L$, meaning that
$H(I,i)\mathfrak{I}\neq\mathbf{0}\neq\mathfrak{I}H(I,i)$. This shows that
$H(I,i\leqslant)$ is a primitive ring and so $H(I,i\nleqslant)$ is a primitive
ideal of $\mathfrak{B}(I)$. As we shall see, those of the form
$H(I,i\nleqslant)$, for $i\in I$, are the only primitive ideals of
$\mathfrak{B}(I)$ exactly when $I$ is artinian.

In order to reach our goal we rely on the following three results, the first
of which is a characterization of right primitive rings which was already used
in [2] and seems to have been observed for the first time by Formanek in [18]
(see also [27, Lemma 11.28]).

######  Proposition 6.1.

A ring $R$ is right primitive if and only if there is a proper right ideal
$\mathfrak{I}$ such that $\mathfrak{I}+\mathfrak{A}=R$ for every ideal
$\mathfrak{A}\neq 0$.

The second is a generalization of [2, Proposition 3.4]; this latter was used
by Abrams, Bell and Rangaswamy in order to establish sufficient conditions on
a graph $E$ for its Leavitt algebra $L(E)$ (over some field) to be primitive.
In a similar manner, our generalization will enable us to state that the
existence of a coinitial chain in $J$ is a sufficient condition for $H(I,J)$
to be primitive.

######  Proposition 6.2.

Let $R$ be a unital ring, let $A$ be any totally ordered set and assume that
there is a family $(e_{i})_{i\in A}$ of nonzero idempotents of $R$ such that,
given $i,j\in A$ with $i\leqslant j$, then $e_{i}e_{j}=e_{i}$ (resp.
$e_{j}e_{i}=e_{i}$). If every nonzero ideal of $R$ contains some $e_{i}$, then
$R$ is right (resp. left) primitive.

###### Proof.

The proof is totally the same as that of [2, Proposition 3.4], however we
repeat it for completeness. Concerning rigth primitivity, it follows from the
assumption that if $i,j\in A$ and $i\leqslant j$, then $Re_{i}\subset Re_{j}$
and so $(1-e_{i})R\supset(1-e_{j})R$, therefore

$\\{(1-e_{i})R\mid i\in A\\}$

is a chain of proper right ideals of $R$ and so
$\mathfrak{I}=\bigcup\\{(1-e_{i})R\mid i\in A\\}$ is a proper right ideal of
$R$. Let $\mathfrak{A}$ be any nonzero ideal of $R$ and let $i\in A$ be such
that $e_{i}\in\mathfrak{A}$. Then

$1=e_{i}+(1-e_{i})\in\mathfrak{A}+\mathfrak{I}$

and therefore $\mathfrak{A}+\mathfrak{I}=R$. We conclude from Proposition 6.1
that $R$ is right primitive. The proof relative to left primitivity is just
specular. ∎

Finally, the following lemma is the key to establish that the existence of a
coinitial chain in $J$ is a necessary condition for $H(I,J)$ to be primitive.

######  Lemma 6.3.

Given a partially ordered set $I$, the following conditions are equivalent:

  1. (1)

$I$ has a coinitial chain;

  2. (2)

There is a family $(F_{i})_{i\in I}$ of finite subsets of $I$ such that
$F_{i}\subset\\{\leqslant i\\}$ for every $i\in I$ and, given $i,j\in I$, at
least one element of $F_{i}$ is comparable with some element of $F_{j}$.

###### Proof.

(1)$\Rightarrow$(2) If $A$ is a coinitial chain of $I$, for every $i\in I$
choose $k_{i}\in A$ such that $k_{i}\leqslant i$ and define
$F_{i}=\\{k_{i}\\}$. Clearly the family $(F_{i})_{i\in I}$ meets the
requirements stated in (2).

(2)$\Rightarrow$(1) Assume (2) and observe that, consequently, $I$ is downward
directed. Suppose that no chain of $I$ is coinitial; of course, we may assume
that $I$ is not empty, otherwise there is nothing to prove. As a first
consequence $I$ must be infinite and has no minimal element, otherwise $I$
would admit a smallest element $m$ and $\\{m\\}$ would be a coinitial chain.
We claim that, for every ordinal $\lambda$, there exists a family
$(i_{\alpha})_{\alpha<\lambda}$ of elements of $I$ such that if
$\beta<\alpha<\lambda$, then $i_{\beta}>i_{\alpha}$ and $i_{\alpha}$ is a
lower bound for $F_{i_{\beta}}$. Thus $\\{i_{\alpha}\mid\alpha<\lambda\\}$
would be a subset of $I$ having the same cardinality of $\lambda$ and this
leads to a contradiction as soon as the cardinality of $\lambda$ exceeds that
of $I$. By proceeding recursively on $\alpha<\lambda$, let us choose any
$i_{0}\in I$. Suppose that $0<\alpha<\lambda$ and assume that we have defined
$i_{\beta}$ for every $\beta<\alpha$, in such a way that if
$\gamma<\beta<\alpha$, then $i_{\beta}<i_{\gamma}$ and $i_{\beta}$ is a lower
bound for $F_{i_{\gamma}}$. If $\alpha=\beta+1$ for some $\beta$, we choose a
lower bound $j$ for $F_{i_{\beta}}\cup\\{i_{\beta}\\}$ and, since $j$ cannot
be a minimal element, we choose $i_{\alpha}<j$, so that $i_{\beta}>i_{\alpha}$
and $i_{\alpha}$ is a lower bound for $F_{i_{\beta}}$. As a consequence, if
$\gamma<\beta$ then $i_{\alpha}$ is a lower bound for $F_{i_{\gamma}}$ too.
Suppose that $\alpha$ is a limit ordinal and let us consider the chain
$C=\\{i_{\beta}\mid\beta<\alpha\\}$. Since $C$ is not coinitial, there is some
$h\in I$ such that $h\not\in\\{C\leqslant\\}$. Let $k$ be a lower bound for
$F_{h}$ and let us prove that $k<i_{\beta}$ whenever $\beta<\alpha$. Given
$\beta<\alpha$, by the assumption (2) there are two elements $u\in
F_{i_{\beta}}$ and $v\in F_{h}$ which are comparable. It is not the case that
$u\leqslant v$ otherwise, by the inductive assumption, $i_{\beta+1}\leqslant
u\leqslant v\leqslant h$ and hence $h\in\\{C\leqslant\\}$, because
$i_{\beta+1}\in C$. Thus $v<u$ and so $k\leqslant v<u\leqslant i_{\beta}$, as
wanted. Finally, let us define define $i_{\alpha}\,\,\colon\\!\\!=k$; by the
above and the inductive assumption, $i_{\alpha}$ is a lower bound for
$F_{i_{\delta}}$ for all $\delta<\alpha$. Thus our claim is true and the proof
is complete. ∎

######  Theorem 6.4.

Let $I$ be a nonempty $\boldsymbol{\aleph}$-poset having a finite cofinal
subset and let $J$ be an upper subset of $I$. Then we have:

  1. (1)

$H(I,J)$ is prime if and only if $J$ is downward directed;

  2. (2)

$H(I,J)$ is right primitive if and only if $J$ has a coinitial chain, if and
only if $H(I,J)$ is left primitive.

###### Proof.

Fiest note that $H(I,J)$ has a multiplicative unit $\mathbf{u}$, by the
assumption and Proposition 3.8.

(1) According to Theorem 4.1 the ring $H(I,J)$ is prime if and only
if$H(I,\\{\leqslant i\\})H(I,\\{\leqslant j\\})\neq 0$ for all $i,j\in I$. In
view of Theorem 3.6 the latter condition is equivalent to requiring that,
given $i,j\in I$, there are two comparable elements $h,k\in I$ such that
$h\leqslant i$ and $k\leqslant j$; this exactly means that $\\{i,j\\}$ has a
lower bound.

(2) Let $C$ be a coinitial chain of $J$; by the Hausdorff Maximal Principle we
may assume that $C$ is a maximal chain of $J$. For every $i\in C$ let us
denote by $\emptyset_{i}$ the constant map $k\mapsto\emptyset$ from
$\\{i\leqslant\\}$ to $\boldsymbol{\aleph}$ and let us consider the element

$Y_{(\emptyset_{i},C\cap\\{i\leqslant\\})}=\left\\{(p,B)\in
X(I)\left|\,\,p|_{\\{i\leqslant\\}}\right.=\emptyset_{i}\text{ and
}B\cap\\{i\leqslant\\}=C\cap\\{i\leqslant\\}\right\\}\in\mathcal{R}_{i}$

and set
$\mathbf{e}_{i}=\mathbf{e}_{Y_{(\emptyset_{i},C\cap\\{i\leqslant\\})}}\in
H(I,i)\subset H(I,J)$. If $i,j\in C$ and $i<j$, then
$Y_{(\emptyset_{i},C\cap\\{i\leqslant\\})}\subset
Y_{(\emptyset_{j},C\cap\\{j\leqslant\\})}$ and hence

(6.1)
$\mathbf{e}_{i}\mathbf{e}_{j}=\mathbf{e}_{i}=\mathbf{e}_{j}\mathbf{e}_{i};$

moreover $\mathbf{e}_{i}\neq\mathbf{0}$ for every $i\in C$, because
$\emptyset\not\in\mathcal{R}_{i}$. We have now the family
$(\mathbf{e}_{i})_{i\in C}$ of idempotents of the unital ring $H(I,J)$,
therefore right primitivity of $H(I,J)$ will follow from 6.2 once we prove
that every nonzero ideal of $H(I,J)$ contains $\mathbf{e}_{i}$ for some $i\in
C$. Given a nonzero ideal $\mathfrak{A}$ of $H(I,J)$, it follows from Theorem
4.1 that $\mathfrak{A}=H(I,L)$ for some nonempty lower subset $L$ of $J$.
Given any $l\in L$, by the assumption there is some $i\in C$ such that
$i\leqslant l$. Then necessarily $i\in L$, therefore $\mathbf{e}_{i}\in
H(I,L)\subset\mathfrak{M}$ and we are done. Clearly, since (6.1) holds
whenever $i,j\in C$ and $i<j$, then $H(I,J)$ is left primitive as well.

Conversely, assume that $H(I,J)$ is right primitive. By Proposition 6.1 there
is a proper right ideal $\mathfrak{I}$ of $H(I,J)$ such that
$\mathfrak{I}+\mathfrak{A}=H(I,J)$ for every nonzero ideal $\mathfrak{A}$ of
$H(I,J)$. In particular, in view of Theorem 4.1 we have that
$\mathfrak{I}+H(I,\\{\leqslant i\\})=H(I,J)$ for every $i\in J$. As a result,
for every $i\in J$ we can choose a nonzero element $\mathbf{a}_{i}\in
H(I,\\{\leqslant i\\})$ such that $\mathbf{u}-\mathbf{a}_{i}\in\mathfrak{I}$
and there is a subset $F_{i}=\\{h_{i1},\ldots,h_{in_{i}}\\}$ of $\\{\leqslant
i\\}$ and nonzero elements $\mathbf{b}_{i1}\in
H(I,h_{i1}),\ldots,\mathbf{b}_{in_{i}}\in H(I,h_{in_{i}})$ such that

$\mathbf{a}_{i}=\mathbf{b}_{i1}+\cdots+\mathbf{b}_{in_{i}}.$

The family $(F_{i})_{i\in J}$ satisfies the condition stated in (2) of Lemma
6.3. To see this, observe that if $i,j\in J$, then necessarily
$\mathbf{a}_{i}\mathbf{a}_{j}\neq\mathbf{0}$, otherwise we would get

$\mathbf{a}_{j}=(\mathbf{u}-\mathbf{a}_{i})\mathbf{a}_{j}\in\mathfrak{I}$

and hence $\mathbf{u}\in\mathfrak{I}$. Consequently
$\mathbf{b}_{ih}\mathbf{b}_{jk}\neq\mathbf{0}$ for some $h\in F_{i}$ and $k\in
F_{j}$ and so $H(I,h)H(I,k)\neq 0$; by Theorem 3.6 this means that $h$ and $k$
are comparable. Now it follows from Lemma 6.3 that $J$ admits a coinitial
chain.

Finally, it is clear that a completely specular argument applies by assuming
that $H(I,J)$ is left primitive. ∎

As a particular case we get the following

######  Corollary 6.5.

Let $I$ be a nonempty $\boldsymbol{\aleph}$-poset having a finite cofinal
subset. Then we have:

  1. (1)

$\mathfrak{B}(I)$ is prime if and only if $I$ is downward directed;

  2. (2)

$\mathfrak{B}(I)$ is right primitive if and only if $I$ has a coinitial chain,
if and only if $\mathfrak{B}(I)$ is left primitive.

If $|I|=\boldsymbol{\aleph}_{0}$ and $I$ is downward directed, then it is easy
to see that $I$ admits a coinitial chain. It seems worth to emphasize that the
condition $|I|=\boldsymbol{\aleph}_{0}$ does not make $I$ an
$\boldsymbol{\aleph}_{0}$-poset, because $I$ might have uncountably many
maximal chains. As an example, let $I=A\cup B$, where $A$ and $B$ are disjoint
countable sets, together with two bijections $h\mapsto a_{h}$ and $h\mapsto
b_{h}$ from $\mathbb{Z}$ to $A$ and $B$ respectively. Then the binary relation

$\\{(a_{h},a_{k})\mid h\leqslant k\\}\cup\\{(b_{h},b_{k})\mid h\leqslant
k\\}\cup\\{(a_{h},b_{k})\mid h<k\\}\cup\\{(b_{h},a_{k})\mid h<k\\}$

is a partial order in $I$ with the following Hasse diagram:

$\begin{split}\begin{picture}(100.0,110.0)(0.0,0.0)\put(25.0,80.0){\line(0,-1){22.0}}
\put(25.0,50.0){\line(0,-1){22.0}} \put(60.0,80.0){\line(0,-1){22.0}}
\put(60.0,50.0){\line(0,-1){22.0}}
\put(25.0,95.0){\makebox(0.0,0.0)[b]{$\vdots$}}
\put(60.0,95.0){\makebox(0.0,0.0)[b]{$\vdots$}}
\put(25.0,80.0){\makebox(0.0,0.0)[b]{$a_{1}$}}
\put(60.0,80.0){\makebox(0.0,0.0)[b]{$b_{1}$}}
\put(25.0,50.0){\makebox(0.0,0.0)[b]{$a_{0}$}}
\put(60.0,50.0){\makebox(0.0,0.0)[b]{$b_{0}$}}
\put(25.0,20.0){\makebox(0.0,0.0)[b]{$a_{-1}$}}
\put(60.0,20.0){\makebox(0.0,0.0)[b]{$b_{-1}$}}
\put(25.0,5.0){\makebox(0.0,0.0)[b]{$\vdots$}}
\put(60.0,5.0){\makebox(0.0,0.0)[b]{$\vdots$}}
\put(30.0,50.0){\line(1,-1){22.0}} \put(52.0,50.0){\line(-1,-1){22.0}}
\put(30.0,80.0){\line(1,-1){22.0}} \put(52.0,80.0){\line(-1,-1){22.0}}
\end{picture}\end{split}$

If $X$ is any subset of $\mathbb{Z}$, then

$\\{a_{h}\mid h\in X\\}\cup\\{b_{h}\mid h\in\mathbb{Z}\setminus X\\}$

is a maximal chain of $I$ and every maximal chain of $I$ arises in this way.
Thus $I$ is not an $\boldsymbol{\aleph}_{0}$-poset, but it is an
$\boldsymbol{\aleph}_{1}$-poset.

In any case, if $|I|=\boldsymbol{\aleph}_{0}$, then $I$ is an
$\boldsymbol{\aleph}$-poset for every cardinal
$\boldsymbol{\aleph}>\boldsymbol{\aleph}_{0}$ and $\mathfrak{B}(I)$ is prime
if and only if it is primitive.

Now, let $X$ be a set with $\boldsymbol{\aleph}=|X|>\boldsymbol{\aleph}_{0}$
and let $I$ be the lattice of all cofinite subsets of $X$ partially ordered by
inclusion. Then $I$ is downward directed and admits $X$ as the greatest
element. Since every chain in $I$ is at most countable, the set of all maximal
chains of $I$ has cardinality $\boldsymbol{\aleph}$ and so $I$ is an
$\boldsymbol{\aleph}$-poset. It is not difficult to see that no chain of $I$
is coinitial, therefore $\mathfrak{B}(I)$ is prime but it is not primitive.

We know from Theorems 4.1 that each ideal of $\mathfrak{B}(I)$ is of the form
$H(I,L)$ for a unique lower subset $L$ of $I$ and $\mathfrak{B}(I)/H(I,L)$ is
in a natural way isomorphic to $H(I,I\setminus L)$ (see Remark 4.2). As a
consequence of Theorem 6.4 we have the following immediate description of the
primitive spectrum
${\mathbf{P}\mathbf{r}\mathbf{i}\mathbf{m}}_{\mathfrak{B}(I)}$ of
$\mathfrak{B}(I)$, which is at the same time right and left.

######  Proposition 6.6.

The primitive ideals of $\mathfrak{B}(I)$ are those of the form
$H(I,\\{A\nleqslant\\})$ for some chain $A$ of $I$.

In other words the assignment $A\mapsto H(I,\\{A\nleqslant\\})$ defines a
surjective map from the set of all chains of $I$ to
${\mathbf{P}\mathbf{r}\mathbf{i}\mathbf{m}}_{\mathfrak{B}(I)}$; clearly, it is
also injective if and only if $I$ is an antichain. Among the chains in $I$ we
have the singletons and we have seen directly, at the beginning of the present
section, that each $i\in I$ gives rise to the primitive ideal
$H(I,i\nleqslant)$. Thus we may consider the map

$\Psi\colon
I\longrightarrow{\mathbf{P}\mathbf{r}\mathbf{i}\mathbf{m}}_{\mathfrak{B}(I)}$

defined by

$\Psi(i)=H(I,\\{i\nleqslant\\})$

for all $i\in I$. Of course $\Psi(i)\subset\Psi(j)$ if and only if $i\leqslant
j$, so that $\Psi$ is an order imbedding. With the next result we describe the
remarkable instance in which $\Psi$ is an order isomorphism.

######  Proposition 6.7.

With the above notations, the following conditions are equivalent:

  1. (1)

$\Psi$ is an order isomorphism;

  2. (2)

$I$ is artinian;

  3. (3)

$\mathfrak{B}(I)$ is semiartinian.

###### Proof.

(1)$\Rightarrow$(2) If $\Psi$ is bijective and $C$ is a non-empty chain of
$I$, then the ideal $H(I,C\nleqslant)$ is primitive and so there is some $i\in
I$ such that $H(I,C\nleqslant)=\Psi(i)=H(I,i\nleqslant)$. Consequently
$\\{C\nleqslant\\}=\\{i\nleqslant\\}$ and therefore $i$ is the smallest
element of $C$. This shows that $I$ is artinian.

(2)$\Rightarrow$(3): see Theorem 4.4.

(3)$\Rightarrow$(1) Assume that $\mathfrak{B}(I)$ is semiartinian, let
$\mathfrak{P}$ be a primitive ideal of $\mathfrak{B}(I)$ and let $L$ be the
unique lower subset of $I$ such that $\mathfrak{P}=H(I,L)$. We have that
$H(I,I\setminus L)\simeq\mathfrak{B}(I)/\mathfrak{P}$ is primitive and
semiartinian, therefore

$\operatorname{Soc}(H(I,I\setminus L))=H(I,(I\setminus L)_{1})$

(see Theorem 4.4) is homogeneous and is a minimal ideal of $H(I,I\setminus
L)$. As a consequence $(I\setminus L)_{1}=\\{m\\}$ for some $m\in I\setminus
L$ and $m$ must be the smallest element of $I\setminus L$. We conclude that
$I\setminus L=\\{m\leqslant\\}$ and so $L=\\{m\nleqslant\\}$, proving that
$\mathfrak{P}=\Psi(m)$. ∎

##  7\. Making $G(-)$ and $\mathfrak{B}(-)$ functors: the category
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$.

In this last section we show that each of the assignments $I\mapsto G(I)$ and
${I\mapsto\mathfrak{B}(I)}$ admits a covariant and a contravariant functorial
extension to appropriate categories of partially ordered sets. Precisely, if
we consider the category ${\mathbf{P}\mathbf{o}\mathbf{s}}$ whose objects are
all posets, while the morphisms are all maps which are order imbeddings (555
We recall that if $I$, $J$ are partially ordered sets, a map $f\colon I\to J$
is an _(order) imbedding_ if, for every $i,i^{\prime}\in I$, the properties
$i\leqslant i^{\prime}$ and $f(i)\leqslant f(i^{\prime})$ are equivalent; if
it is the case, $f$ is obviously injective.) of the domain as an upper subset
of the codomain, then the assignment $I\mapsto G(I)$ extends to a pair of
functors $G(-)$, $G^{*}(-)$, the first covariant and the second contravariant,
from ${\mathbf{P}\mathbf{o}\mathbf{s}}$ to the category
${\mathbf{P}\mathbf{o}\mathbf{A}\mathbf{b}}$ whose objects are all partially
ordered abelian groups and morphisms are all order preserving group
homomorphisms. Next we show that, given an infinite cardinal
$\boldsymbol{\aleph}$ and a field $\mathbb{K}$, the assignment
$I\mapsto\mathfrak{B}(I)$ extends to a pair of functors $\mathfrak{B}(-)$,
$\mathfrak{B}^{*}(-)$, the first covariant and the second contravariant, from
the subcategory ${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$ of
${\mathbf{P}\mathbf{o}\mathbf{s}}$ of all $\boldsymbol{\aleph}$-posets _having
a finite cofinal subset_ to the category
${\mathbf{A}\mathbf{l}\mathbf{g}}_{\mathbb{K}}$ of _unital_
$\mathbb{K}$-algebras and _not necessarily unital_ $\mathbb{K}$-algebra
homomorphisms. Finally, we will see that these four functors match with the
functor $\mathbf{K}_{0}(-)$, in the sense that there are natural equivalences
$\rho(-)\colon G(-)\approx\mathbf{K}_{0}(-)\cdot\mathfrak{B}(-)$ and
$\rho^{*}(-)\colon G^{*}(-)\approx\mathbf{K}_{0}(-)\cdot\mathfrak{B}^{*}(-)$.

Let $I$ and $J$ be partially ordered sets and let $f\colon I\to J$ be an
isotone map. As $G(I)$ and $G(J)$ are the free abelian groups generated by $I$
and $J$ respectively, the most “natural” group homomorphism $G(f):G(I)\to
G(J)$ to be assigned to $f$ is clearly the unique homomorphism which extends
$f$ to $G(I)$, that is

$G(f)\left(\sum_{i\in I}x_{i}i\right)=\sum_{i\in I}x_{i}f(i).$

Of course we have that $G(1_{I})=1_{G(I)}$ and, if $g$ is a second isotone map
and $fg$ is defined, then $G(fg)=G(f)G(g)$. We would like $G(f)$ to be
isotone, that is

(7.1) $G(f)(M(I))\subset M(J),$

However this may fail if $f$ is not injective. For example, suppose that
$I=\\{h,k\\}$, where $h<k$, and $J=\\{j\\}$; then the unique map $f:I\to J$ is
obviously isotone. If we consider the element $x=-2h+k\in M(I)$, we have that
$G(f)(x)=-j\not\in M(J)$. Fortunately (7.1) holds if $f$ is injective. In
fact, assume that $x\in M(I)$ and let $m$ be a maximal element of
$\operatorname{Supp}(G(f)(x))=f(\operatorname{Supp}(x))$. Then the unique
element ${l\in M(I)}$ such that $m=f(l)$ is maximal in
$\operatorname{Supp}(x)$. In fact, if $i\in\operatorname{Supp}(x)$ and
${l\leqslant i}$, then $m=f(l)\leqslant f(i)$ and
$f(i)\in\operatorname{Supp}(G(f)(x))$, therefore $m=f(i)$ and hence $l=i$.
Consequently $G(f)(x)_{m}=x_{l}>0$ and this proves that $G(f)(x)\in M(J)$.

We are now allowed to speak about the covariant functor $G(-)$ from the
category of all partially ordered sets and _injective isotone maps_ to the
category ${\mathbf{P}\mathbf{o}\mathbf{A}\mathbf{b}}$.

Passing to consider the contravariant side, let again $I$ and $J$ be partially
ordered sets and $f\colon I\to J$ an isotone map. Again, in order to assign to
$f$ an order preserving group homomorphism from $G(J)$ to $G(I)$ we have at
our disposal a “natural” path to follow. In fact, by identifying $G(J)$ as the
subgroup $\mathbb{Z}^{(J)}$ of $\mathbb{Z}^{J}$ of all maps from $J$ to
$\mathbb{Z}$ having finite support, we see that $f$ induces the homomorphism
of groups

$f^{*}\colon\mathbb{Z}^{(J)}\longrightarrow\mathbb{Z}^{I}$

defined by $f^{*}(x)=x\circ f$ for all $x\in G(J)$, that is
$(f^{*}(x))_{i}=x_{f(i)}$ for all $i\in I$. Given $x\in G(J)$, we have that

$\operatorname{Supp}(f^{*}(x))=f^{-1}(\operatorname{Supp}(x)),$

therefore $f^{*}(x)\in G(I)$ for all $x\in G(J)$ if and only if $f^{-1}(j)$ is
finite for each $j\in J$. But, even if this latter condition holds, if $f$ is
not injective the homomorphism $f^{*}$ may fail to be an isotone map, as the
following example shows.

######  Example 7.1.

Let $I=\\{i,j,k\\}$ and $J=\\{u,v\\}$ with $i<j,i<k,j\nleqslant k,k\nleqslant
j$ and $u<v$; next define $f\colon I\to J$ with $f(i)=f(j)=u$ and $f(k)=v$. If
we consider $x=v-u\in M(J)$, then we see that $f^{*}(x)=-i-j+k\not\in M(I)$,
because $j$ is maximal in $I$, therefore $f^{*}$ is not isotone.

Thus, also for the contravariant case we must restrict our attention to the
injective isotone maps and, as we are going to see with the next result, only
part of them are allowed. We observe that if $f\colon I\to J$ is injective,
then $f^{*}$ is an epimorphism, as it is the projection of $\mathbb{Z}^{(J)}$
onto $\mathbb{Z}^{(f(I))}\simeq\mathbb{Z}^{(I)}$ with kernel
$\mathbb{Z}^{(J\setminus f(I))}$. Moreover, for every $i\in I$ and $j\in J$ we
have that

$(f^{*}(j))_{i}=\begin{cases}1,\text{ if $j=f(i)$;}\\\ 0,\text{
otherwise.}\end{cases}$

Consequently:

(7.2) $\text{for every $i\in I$ and $j\in J$,}\qquad
f^{*}(j)=\begin{cases}i,\text{ if $j=f(i)$;}\\\ 0,\text{
otherwise.}\end{cases}$

######  Proposition 7.2.

Let $I,J$ be partially ordered sets and let $f\colon I\to J$ be an injective
and isotone map. Then the homomorphism $f^{*}:G(J)\to G(I)$ is isotone, namely

(7.3) $f^{*}(M(J))\subset M(I),$

if and only if $f$ is an order imbedding and $f(I)$ is an upper subset of $J$.

###### Proof.

Suppose that (7.3) holds and let $i,j\in I$ be such that $f(i)\leqslant f(j)$.
Then $f(j)-f(i)\in M(J)$ and consequently, by using (7.2), we see that
$j-i=f^{*}(f(j)-f(i))\in M(I)$. As a result $i\leqslant j$ and this proves
that $f$ is an imbedding. Next, let $i\in I$, $j\in J$ and assume that
$f(i)\leqslant j$ but $j\not\in f(I)$. Then $-i=f^{*}(j-f(i))\in M(I)$ again
by (7.2), which is impossible. Thus necessarily $j\in f(I)$ and so $f(I)$ is
an upper subset of $J$.

Conversely, assume that $f$ is an imbedding and $f(I)$ is an upper subset of
$J$. Given $x\in M(J)$, $x\neq 0$, and a maximal element $m$ of
$\operatorname{Supp}(f^{*}(x))$, we claim that $(f^{*}(x))_{m}>0$. Suppose, on
the contrary, that $(f^{*}(x))_{m}\leqslant 0$. Then
$x_{f(m)}=(f^{*}(x))_{m}\leqslant 0$ and so $f(m)$ is a non-maximal element of
$\operatorname{Supp}(x)$. Let $j\in\operatorname{Supp}(x)$ be such that
$f(m)<j$; then $j\in f(I)$ by the assumption. If we let $i\in I$ be such that
$f(i)=j$, again from the assumption we have that $m<i$. But
$(f^{*}(x))_{i}=x_{f(i)}=x_{j}\neq 0$, hence
$i\in\operatorname{Supp}(f^{*}(x))$ and this contradicts the maximality of $m$
in $\operatorname{Supp}(f^{*}(x))$. Thus $(f^{*}(x))_{m}>0$ and therefore
$f^{*}(x)\in M(I)$. ∎

We are led to consider the category whose objects are all partially ordered
sets and morphisms are those maps which are order imbedding of the domain as
an upper subset of the codomain, which we denote by
${\mathbf{P}\mathbf{o}\mathbf{s}}$. It is obvious that $1_{I}^{*}=1_{G(I)}$
for every object $I$ of ${\mathbf{P}\mathbf{o}\mathbf{s}}$, while if $f\colon
I\to J$ $g\colon J\to K$ are morphisms in ${\mathbf{P}\mathbf{o}\mathbf{s}}$,
then $(gf)^{*}=f^{*}g^{*}$. Thus we are legitimate to define the contravariant
functor $G^{*}(-)$ from the category ${\mathbf{P}\mathbf{o}\mathbf{s}}$ to the
category ${\mathbf{P}\mathbf{o}\mathbf{A}\mathbf{b}}$.

The condition of being an order imbedding of the domain as an upper subset of
the codomain, for isotone maps between partially ordered sets, is very close
to a known condition on morphisms between directed graphs; we feel it is worth
to enlighten that connection.

We recall that a _(directed) graph_ is an ordered 4-tuple

$E=(E^{0},E^{1},s_{E},r_{E}),$

where $E^{0},E^{1}$ are two disjoint sets and

$s_{E}\colon E^{1}\to E^{0},\qquad r_{E}\colon E^{1}\to E^{0}$

are maps. The elements of $E^{0}$ are called the _vertices_ and those of
$E^{1}$ the _arrows_ (or _edges_) of $E$. A _graph morphism_ from a graph $E$
to a graph $F$ is an ordered pair $f=(f^{0},f^{1})$ consisting of two maps
$f^{0}:E^{0}\to F^{0}$ and $f^{1}:E^{1}\to F^{1}$ such that
$s_{F}f^{1}=f^{0}s_{E}$ and $r_{F}f^{1}=f^{0}r_{E}$. We denote by
$\mathbf{D}\mathbf{i}\mathbf{G}\mathbf{r}$ the category of all directed graphs
and arbitrary graph morphisms.

Every partially ordered set $I$ gives rise to its _associated graph_
$(I^{0},I^{1},s_{I},r_{I})$, where $I^{0}=I$, $I^{1}=\\{(i,j)\in I\times I\mid
i<j\\}$ and $s_{I}((i,j))=i$, $r_{I}((i,j))=j$ for all $(i,j)\in I^{1}$. If
$I,J$ are partially ordered sets and $f:I\to J$ is an isotone map, then we may
consider the two maps $f^{0}:I^{0}\to J^{0}$, $f^{1}:I^{1}\to J^{1}$, where
$f^{0}(i)=f(i)$ for all $i\in I$ and, if $i<j$ in $I$, then
$f^{1}((i,j))=(f^{0}(i),f^{0}(j))$; we can check immediately that the pair
$(f^{0},f^{1})$ is a graph morphism from $(I^{0},I^{1},s_{I},r_{I})$ to
$(J^{0},J^{1},s_{J},r_{J})$. Conversely, if
$(f^{0},f^{1})\colon(I^{0},I^{1})\to(J^{0},J^{1})$ is any graph morphism, then
the map $f\,\,\colon\\!\\!=f^{0}\colon I\to J$ is isotone. We say that
$(f^{0},f^{1})$ is the _graph morphism associated_ to $f$.

In seeking for an appropriate subcategory of
$\mathbf{D}\mathbf{i}\mathbf{G}\mathbf{r}$ to which one can extend, as a
functor, the map which assigns to every directed graph $E$ its path
$\mathbb{K}$-algebra $\mathbb{K}E$ and its Leavitt path algebra
$L_{\mathbb{K}}(E)$, in [22] Goodearl defines a graph morphism
$f=(f^{0},f^{1})\colon E\to F$ to be a $CK$-_morphism_ (short for _Cuntz-
Krieger morphism_) if it satisfies the following conditions:

  1. (1)

$f^{0}$ and $f^{1}$ are both injective;

  2. (2)

For each $v\in E^{0}$ which is neither a sink nor an infinite emitter, $f^{1}$
induces a bijection from $s_{E}^{-1}(v)$ to $s_{F}^{-1}(f^{0}(v))$

(for row-finite graphs, meaning that each vertex emits at most finitely many
arrows, $CK$-morphisms are precisely the _complete morphisms_ of [5], p. 161);
then he shows that the assignments $E\mapsto\mathbb{K}E$ and $E\mapsto
L_{\mathbb{K}}(E)$ admit natural functorial extensions to the subcategory
$\mathbf{C}\mathbf{K}\mathbf{G}\mathbf{r}$ of
$\mathbf{D}\mathbf{i}\mathbf{G}\mathbf{r}$ of all graphs together with
$CK$-morphism. Let us say that $f$ is a _strict $CK$-morphism_ if it satisfies
the above condition (1) together with the following stronger form of (2):

  1. (3)

For _every_ $v\in E^{0}$, $f^{1}$ induces a bijection from $s_{E}^{-1}(v)$ to
$s_{F}^{-1}(f^{0}(v))$.

######  Proposition 7.3.

Let $I,J$ be partially ordered sets and let $f\colon I\to J$ be an isotone
map. Then $f$ is an order imbedding and $f(I)$ is an upper subset of $J$ if
and only if the associated graph morphism
$(f^{0},f^{1})\colon(I^{0},I^{1})\to(J^{0},J^{1})$ is a strict $CK$-morphism.

###### Proof.

Suppose that $f$ is an order imbedding and $f(I)$ is an upper subset of $J$.
Then both $f^{0}$, $f^{1}$ are clearly injective. Given $i\in I$, we have that
$f^{1}$ induces an injective map from $s_{I}^{-1}(i)$ to
$s_{J}^{-1}(f^{0}(i))$, which is also surjective. In fact, given $(k,l)\in
s_{J}^{-1}(f^{0}(i))$, necessarily $k=f^{0}(i)<l$. As $f^{0}(I)=f(I)$ is an
upper subset of $J$, there is some $j\in I$ such that $l=f(j)$ and, since $f$
is an order imbedding, we have that $i<j$, thus $(i,j)\in I^{1}$ and hence
$(k,l)=f^{1}(i,j)$.

Conversely, suppose that $(f^{0},f^{1})$ is a strict $CK$-morphism. Then
$f=f^{0}$ is injective. Let $i,j\in I$ and assume that $f(i)<f(j)$, that is
$(f(i),f(j))\in I^{1}$. As $(f(i),f(j))\in s_{J}^{-1}(f^{0}(i))$, by the
assumption there is a unique $(i^{\prime},j^{\prime})\in s_{I}^{-1}(i)$ such
that $(f(i),f(j))=f^{1}(i^{\prime},j^{\prime})$. Necessarily $i^{\prime}=i$,
$j^{\prime}=j$, so that $i<j$. Next, let $i\in I$ and $k\in J$ be such that
$f(i)<k$. Then $(f(i),k)\in s_{J}^{-1}(f^{0}(i))$ and, again from the
assumption, we infer that $(f(i),k)=f^{1}((i,j))$ for a unique $j\in I$. Thus
$k=f(j)\in f(I)$ and we are done. ∎

We now proceed to find the promised two functorial extensions of the
assignment $I\mapsto\mathfrak{B}(I)$.

Let $f\colon I\to J$ be a morphism of
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$ and let $i\in I$.
Since $f$ restricts to an order isomorphism from $\\{i\leqslant\\}$ to
$\\{f(i)\leqslant\\}$, the assignment $(u,E)\mapsto(u\cdot
f|_{\\{i\leqslant\\}},f^{-1}(E))$ defines a bijection from $X(f(i)\leqslant)$
to $X(i\leqslant)$ and therefore we get the $\mathbb{K}$-algebra isomorphism

$\overline{f}_{i}\colon\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})\longrightarrow\mathbb{F}\mathbb{R}_{X(f(i)\leqslant)}(\mathbb{K})$

defined by

$\left(\overline{f}_{i}(\mathbf{a})\right)((u,E),(v,F))=\mathbf{a}((u\cdot
f|_{\\{i\leqslant\\}},f^{-1}(E)),(v\cdot f|_{\\{i\leqslant\\}}vf,f^{-1}(F)))$

for all $\mathbf{a}\in\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})$,
$u,v\in\boldsymbol{\aleph}^{(f(i)\leqslant)}$ and
$E,F\in\mathcal{M}(f(i)\leqslant)$. Recall that, by the definition of
$H(I,i)$, the assignment $\mathbf{a}\mapsto\psi_{I,i}(\mathbf{a})$ (see
Proposition 2.5 and the beginning of section 3) defines a $\mathbb{K}$-algebra
isomorphism

$\alpha_{I,i}\colon\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})\longrightarrow
H(I,i)$

when $i$ is not maximal, and an isomorphism

$\alpha_{I,i}\colon\mathbf{1}_{\mathbb{C}\mathbb{F}\mathbb{M}_{X(i\leqslant)}(\mathbb{K})}\mathbb{K}\longrightarrow
H(I,i)=\mathbf{e}_{X(I)_{i}}\mathbb{K}$

when $i$ is maximal; thus we obtain the $\mathbb{K}$-algebra isomorphism

$\widehat{f}_{i}\,\,\colon\\!\\!=\alpha_{J,f(i)}\overline{f}_{i}\alpha_{I,i}^{-1}\colon
H(I,i)\longrightarrow H(J,f(i))$

and hence the $\mathbb{K}$-linear isomorphism

$\widehat{f}\,\,\colon\\!\\!=\bigoplus_{i\in
I}\widehat{f}_{i}\colon\bigoplus_{i\in
I}H(I,i)=\mathfrak{B}(I)\longrightarrow\bigoplus_{i\in I}H(J,f(i))=H(J,f(I)).$

######  Lemma 7.4.

With the above setting and notations, $\widehat{f}$ is a $\mathbb{K}$-algebra
isomorphism.

###### Proof.

Given $i\in I$, let us first compute explicitly $\widehat{f}_{i}(\mathbf{a})$
for each $\mathbf{a}\in H(I,i)$. First observe that, inasmuch as
$H(I,i)\subset S(I,i)$, if $\mathbf{a}\in H(I,i)$ then for every
$s,t\in\boldsymbol{\aleph}^{(\nleqslant i)}$,
$u,v\in\boldsymbol{\aleph}^{(i\leqslant)}$, $C,D\in\mathcal{M}(\leqslant i)$
and $E,F\in\mathcal{M}(i\leqslant)$ we have that

$\mathbf{a}((s\cup u,C\cup E),(s\cup v,C\cup F))=\mathbf{a}((t\cup u,D\cup
E),(t\cup v,D\cup F)).$

This legitimates us to define the map

$\beta\colon
H(I,i)\longrightarrow\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})$

as follows: given $\mathbf{a}\in H(I,i)$, for every
$u,v\in\boldsymbol{\aleph}^{(i\leqslant)}$ and $E,F\in\mathcal{M}(i\leqslant)$

$\beta(\mathbf{a})((u,E),(v,F))\,\,\colon\\!\\!=\mathbf{a}((t\cup u,C\cup
E),(t\cup v,C\cup F)),$

where $t\in\boldsymbol{\aleph}^{(\nleqslant i)}$ and
$C\in\mathcal{M}(\leqslant i)$ are chosen arbitrarily. We claim that

$\alpha_{I,i}^{-1}=\beta.$

Indeed, given $\mathbf{b}\in\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})$,
let $u,v\in\boldsymbol{\aleph}^{(i\leqslant)}$ and
$E,F\in\mathcal{M}(i\leqslant)$. By taking arbitrarily
$t\in\boldsymbol{\aleph}^{(\nleqslant i)}$ and $C\in\mathcal{M}(\leqslant i)$
we have that

$\displaystyle\left[\beta\left(\alpha_{I,i}(\mathbf{b})\right)\right]((u,E),(v,F))$
$\displaystyle=\alpha_{I,i}(\mathbf{b})((t\cup u,C\cup E),(t\cup v,C\cup F))$
$\displaystyle=\mathbf{b}((u,E),(v,F)).$

Since $\alpha_{I,i}$ is an isomorphism, this shows our claim.

Now we observe that, given $r,s\in\boldsymbol{\aleph}^{(f(i)\nleqslant)}$,
$u,v\in\boldsymbol{\aleph}^{(f(i)\leqslant)}$ and $A,B\in\mathcal{M}(J)$, we
have

$\widehat{f}_{i}(\mathbf{a})((r\cup u,A),(s\cup v,B))\neq 0$

only if

$f(i)\in A\cap B,\quad A\cap\\{\leqslant f(i)\\}=B\cap\\{\leqslant
f(i)\\}\quad\text{and}\quad r=s.$

Assume that this is the case, set

$E=f^{-1}(A)\cap\\{i\leqslant\\},\qquad F=f^{-1}(B)\cap\\{i\leqslant\\}$

and choose arbitrarily $t\in\boldsymbol{\aleph}^{(i\nleqslant)}$ and
$C\in\mathcal{M}(\leqslant i)$. Since $f$ is a morphism of
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$, then
$f(\\{i\leqslant\\})=\\{f(i)\leqslant\\}$ and $f(\\{\nleqslant
i\\})\subset\\{\nleqslant f(i)\\}$ and we can compute as follows:

(7.4) $\begin{split}\widehat{f}_{i}(\mathbf{a})&((r\cup u,A),(r\cup
v,B))=\left[\overline{f_{i}}\left(\alpha_{I,i}^{-1}(\mathbf{a})\right)\right]((u,A\cap\\{f(i)\leqslant\\}),(v,B\cap\\{f(i)\leqslant\\}))\\\
&=\left[\alpha_{I,i}^{-1}(\mathbf{a})\right]((u\cdot
f|_{\\{i\leqslant\\}},E),(v\cdot f|_{\\{i\leqslant\\}},F))\\\
&=\mathbf{a}((t\cup(u\cdot f|_{\\{i\leqslant\\}}),C\cup E),(t\cup(v\cdot
f|_{\\{i\leqslant\\}}),C\cup F))\\\ &=\mathbf{a}(((r\cdot
f|_{\\{i\nleqslant\\}})\cup(u\cdot f|_{\\{i\leqslant\\}}),C\cup E),((r\cdot
f|_{\\{i\nleqslant\\}})\cup(v\cdot f|_{\\{i\leqslant\\}}),C\cup F))\\\
&=\mathbf{a}(((r\cup u)f,C\cup E),((r\cup v)f,C\cup F)),\end{split}$

where the last equality comes from the fact that $\mathbf{a}\in Q_{i}$.

Now we can prove that

(7.5)
$\widehat{f}(\mathbf{a}\mathbf{b})=\widehat{f}(\mathbf{a})\cdot\widehat{f}(\mathbf{b})$

for every $\mathbf{a},\mathbf{b}\in\mathfrak{B}(I)$. Thanks to the linearity
of $\widehat{f}$ it is sufficient to assume that $\mathbf{a}\in H(I,i)$ and
$\mathbf{b}\in H(I,j)$ for some $i,j\in I$ with $i\neq j$. Inasmuch as
$\widehat{f}(H(I,i))\subset H(J,f(i))$ for all $i\in I$, if $i$ and $j$ are
not comparable, then both members of (7.5) are zero. Assume that $i<j$. Then
$\mathbf{a}\mathbf{b}\in H(I,i)$ and both members of (7.5) are in $H(J,f(i))$;
consequently, in order to check that (7.5) holds it is sufficient to prove
that if $r\in\boldsymbol{\aleph}^{(f(i)\nleqslant)}$,
$u,v\in\boldsymbol{\aleph}^{(f(i)\leqslant)}$ and $A,B\in\mathcal{M}(J)$ are
such that $f(i)\in A\cap B$ and $A\cap\\{\leqslant f(i)\\}=B\cap\\{\leqslant
f(i)\\}$, the equality

(7.6) $\left[\widehat{f}(\mathbf{a}\mathbf{b})\right]((r\cup u,A),(r\cup
v,B))=\left[\widehat{f}(\mathbf{a})\cdot\widehat{f}(\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))$

holds. By setting

$\displaystyle C=A\cap\\{\leqslant f(i)\\}=B\cap\\{\leqslant f(i)\\},\quad
D=f^{-1}(C)\in\mathcal{M}(\leqslant i),$ $\displaystyle
E=f^{-1}(A)\cap\\{i\leqslant\\},\quad\text{and}\quad
F=f^{-1}(B)\cap\\{i\leqslant\\}$

we obtain from (7.4):

$\displaystyle\left[\widehat{f}(\mathbf{a}\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))$
$\displaystyle\phantom{XXXXXX}=\left[\widehat{f}_{i}(\mathbf{a}\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))$
$\displaystyle\phantom{XXXXXX}=(\mathbf{a}\mathbf{b})(((r\cup u)f,D\cup
E),((r\cup v)f,D\cup F))$ $\displaystyle\phantom{XXXXXX}=\sum_{x\in
X(I)}\mathbf{a}(((r\cup u)f,D\cup E),x)\cdot\mathbf{b}(x,((r\cup v)f,D\cup
F))).$

If we write

$v^{\prime}=v|_{\\{f(i)\leqslant\\}\setminus\\{f(j)\leqslant\\}},$

in order that a single summand of the last sum be nonzero it is necessary that

$x=((r\cup v^{\prime}\cup w)f,D\cup G),$

where $w\in\boldsymbol{\aleph}^{(f(j)\leqslant)}$ and $G$ is a maximal chain
of $\\{i\leqslant\\}$ which belongs to the set

$\mathcal{G}\
=\bigl{\\{}\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\!\\!\\!G\in\mathcal{M}(i\leqslant)\mid
j\in G,\,f(G)\cap[f(i),f(j)]=B\cap[f(i),f(j)]\bigr{\\}}.$

Thus we may write

(7.7) $\displaystyle\left[\widehat{f}(\mathbf{a}\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))$
$\displaystyle\phantom{xxxxx}=\sum_{\begin{subarray}{c}w\in\boldsymbol{\aleph}^{(f(j)\leqslant)}\\\
E\in\mathcal{E}\end{subarray}}\left\\{\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\!\mathbf{a}(((r\cup
u)f,D\cup E),((r\cup v^{\prime}\cup w)f,D\cup G))\right.$
$\displaystyle\phantom{xxxxxxxxxxxxxxx}\cdot\mathbf{b}(((r\cup v^{\prime}\cup
w)f,D\cup G),((r\cup v)f,D\cup F)).$

On the other hand we have

$\displaystyle\left[\widehat{f}(\mathbf{a})\cdot\widehat{f}(\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))$ $\displaystyle\phantom{xxxxxxx}=\sum_{y\in
X(J)}\left\\{\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\![\widehat{f}(\mathbf{a})]((r\cup
u,A),y)\cdot[\widehat{f}(\mathbf{b})](y,(r\cup v,B))\right\\}.$

Here again, a single summand is not zero only if

$y=(r\cup v^{\prime}\cup w,C\cup H),$

where $w\in\boldsymbol{\aleph}^{(f(j)\leqslant)}$ and $H$ is a maximal chain
of $\\{f(i)\leqslant\\}$ which belongs to the set

$\mathcal{H}\
=\bigl{\\{}\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\!\\!\\!H\in\mathcal{M}(\\{f(i)\leqslant\\})\mid
f(j)\in H,\,H\cap[f(i),f(j)]=B\cap[f(i),f(j)]\bigr{\\}}.$

Thus, by using (7.4) we obtain:

$\begin{split}&\left[\widehat{f}(\mathbf{a})\cdot\widehat{f}(\mathbf{b})\right]((r\cup
u,A),(r\cup v,B))\\\
&\phantom{xxxxxxxx}=\sum_{\begin{subarray}{c}w\in\boldsymbol{\aleph}^{(f(j)\leqslant)}\\\
E\in\mathcal{E}\end{subarray}}\left\\{\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\!\\!\\![\widehat{f}(\mathbf{a})]((r\cup
u,A),(r\cup v^{\prime}\cup w,C\cup H))\right.\\\
&\left.\phantom{xxxxxxxxxxxx}\cdot[\widehat{f}(\mathbf{b})]((r\cup
v^{\prime}\cup w,C\cup H),(r\cup v,B))\right\\}\\\
&\phantom{xxxxxxxx}=\sum_{\begin{subarray}{c}w\in\boldsymbol{\aleph}^{(f(j)\leqslant)}\\\
E\in\mathcal{E}\end{subarray}}\left\\{\phantom{\bigotimes}\\!\\!\\!\\!\\!\\!\\!\mathbf{a}(((r\cup
u)f,D\cup E),((r\cup v^{\prime}\cup w)f,D\cup f^{-1}(H))\right.\\\
&\left.\phantom{xxxxxxxxxxxx}\cdot\mathbf{b}(((r\cup v^{\prime}\cup w)f,D\cup
f^{-1}(H)),((r\cup v)f,D\cup F))\right\\}.\end{split}$

Finally, since $\mathcal{G}=\\{f^{-1}(H)\mid H\in\mathcal{H}\\}$, the latter
sum coincides with the sum at the second member of (7.7), therefore the
equality (7.5) holds. The case in which $i>j$ is handled in the same manner
and we conclude that $\widehat{f}$ is a $\mathbb{K}$-algebra homomorphism. ∎

We are now in a position to define the $\mathbb{K}$-algebra homomorphisms

$\mathfrak{B}(f)\colon\mathfrak{B}(I)\longrightarrow\mathfrak{B}(J)\quad\text{and}\quad\mathfrak{B}^{*}_{\boldsymbol{\aleph},\mathbb{K}}(f)\colon\mathfrak{B}^{*}_{\boldsymbol{\aleph},\mathbb{K}}(J)\longrightarrow\mathfrak{B}^{*}_{\boldsymbol{\aleph},\mathbb{K}}(I):$

we take the first as the composition of $\widehat{f}$ followed by the
inclusion into $\mathfrak{B}(J)$ and the second as the composition of the
projection of $\mathfrak{B}(J)$ onto $H(J,f(I))$, with kernel the ideal
$H(J,J\setminus f(I))$ (see 4.2 and note that $J\setminus f(I)$ is a lower
subset of $J$), followed by $\widehat{f}^{-1}$.

######  Proposition 7.5.

With the above settings, $\mathfrak{B}(-)$ and $\mathfrak{B}^{*}(-)$ are
functors from ${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$ to the
category ${\mathbf{A}\mathbf{l}\mathbf{g}}_{\mathbb{K}}$, the first covariant
and the second contravariant.

###### Proof.

For every $i\in I$, as
$\overline{\left(1_{I}\right)}_{i}=1_{\mathbb{F}\mathbb{R}_{X(i\leqslant)}(\mathbb{K})}$,
it is clear that $\widehat{\left(1_{I}\right)}_{i}=1_{H(I,i)}$, consequently
$\mathfrak{B}(1_{I})=1_{\mathfrak{B}(I)}=\mathfrak{B}^{*}(1_{I})$. Let
$f\colon I\to J$ and $g\colon J\to K$ be morphisms in
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$. A straightforward
computation shows that

$(\overline{gf})_{i}=\overline{g}_{f(i)}\cdot\overline{f_{i}},$

from which we infer easily that

$(\widehat{gf})_{i}=\widehat{g}_{f(i)}\widehat{f}_{i}\quad\text{and}\quad(\widehat{gf})^{-1}_{i}=\widehat{f}^{-1}_{i}\widehat{g}\,^{-1}_{f(i)}.$

Consequently

$\mathfrak{B}(gf)=\mathfrak{B}(g)\mathfrak{B}(f)\quad\text{and}\quad\mathfrak{B}^{*}(gf)=\mathfrak{B}^{*}(f)\mathfrak{B}^{*}(g).$

∎

######  Remark 7.6.

If $I\in{\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$, we know that
$\mathfrak{B}(I)$ is a unital $\mathbb{K}$-algebra, specifilally

$1_{\mathfrak{B}(I)}=\sum\\{\mathbf{e}_{X(I)_{m}}|m\in\mathbf{M}(I)\\}$

(see Proposition 3.8). If $f\colon I\to J$ is a morphism of
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$, while the
$\mathbb{K}$-algebra homomorphism $\mathfrak{B}^{*}(f)$ is always unital,
$\mathfrak{B}_{\boldsymbol{\aleph},K}(f)$ is unital if and only if
$\varphi(\mathbf{M}(I))=\mathbf{M}(J)$.

Given a _unital_ ring $R$, it is standard to define $\mathbf{K}_{0}(R)$ as the
Grothendieck group of the commutative monoid of isomorphism classes of all
finitely generated right $R$-modules; this makes $\mathbf{K}_{0}(-)$ a
covariant functor from the category of all unital rings and all unital ring
homomorphisms, to the category of partially ordered abelian group. Fortunately
a natural extension of $\mathbf{K}_{0}(-)$ to the category
$\mathbf{R}\mathbf{n}\mathbf{g}$ (this is a notation which seems due to Faith)
of not necessarily unital rings and not necessarily unital ring homomorphisms
is available (see Ch. 5, §7 in [28]). Specifically, given
$R\in\mathbf{R}\mathbf{n}\mathbf{g}$, one considers first the “unitization”
ring $R_{+}$ of $R$, defined as the abelian group $R\times\mathbb{Z}$ together
with the multiplication $(x,m)(y,n)\,\,\colon\\!\\!=(xy+my+nx,mn)$, so that
the assignment $x\mapsto(x,0)$ identifies $R$ as an ideal of $R_{+}$; thus, by
considering the canonical epimorphism $\pi_{R}\colon R_{+}\to
R_{+}/R\simeq\mathbb{Z}$, one defines $\mathbf{K}_{0}(R)$ as the kernel of
$\mathbf{K}_{0}(\pi_{R})$. If $\alpha\colon R\to S$ is a morphism in
$\mathbf{R}\mathbf{n}\mathbf{g}$ and $\alpha_{+}\colon R_{+}\to S_{+}$ is the
unital homomorphism defined by $(x,m)\mapsto(\alpha(x),m)$, then
$\mathbf{K}_{0}(\alpha)$ is taken as the unique homomorphism which makes
commutative the diagram

$\begin{CD}\mathbf{K}_{0}(R)@>{\subset}>{}>\mathbf{K}_{0}(R_{+})\\\
@V{\mathbf{K}_{0}(\alpha)}V{}V@V{}V{\mathbf{K}_{0}(\alpha_{+})}V\\\
\mathbf{K}_{0}(S)@>{\subset}>{}>\mathbf{K}_{0}(S_{+}).\end{CD}$

If $R$ has a multiplicative identity $e$, then $(e,0)$ is a central idempotent
of $R_{+}$ and so $R_{+}\simeq R\times\mathbb{Z}$; consequently the above
definition of $\mathbf{K}_{0}(R)$ matches with the standard one.

Suppose that $R$, $S$ are unital regular rings and $\alpha\colon R\to S$ is a
morphism in $\mathbf{R}\mathbf{n}\mathbf{g}$. By the above, for every
idempotent $e\in R$ we have:

$\displaystyle\mathbf{K}_{0}(\alpha)([eR])=\mathbf{K}_{0}(\alpha_{+})([(e,0)R_{+}])=[(e,0)R_{+}\otimes_{R_{+}}S_{+}]=[(\alpha(e),0)S_{+}]$
$\displaystyle=[\alpha(e)S\times\\{0\\}]=[\alpha(e)S].$

Since $\mathbf{K}_{0}(R)$ is generated by the set $\\{[eR]\mid e=e^{2}\in
R\\}$, we see that $\mathbf{K}_{0}(\alpha)$ is the unique homomorphism such
that $\mathbf{K}_{0}(\alpha)([eR])=[\alpha(e)S]$ for every idempotent $e\in
R$.

Let us come back to the functors $\mathfrak{B}(-)$ and $\mathfrak{B}^{*}(-)$.
In view of all above, given a morphism $f\colon I\to J$ of
${\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$, it makes sense to
speak about the homomorphism
$\mathbf{K}_{0}(\mathfrak{B}(f))\colon\mathbf{K}_{0}(\mathfrak{B}(I))\to\mathbf{K}_{0}(\mathfrak{B}(J))$
and hence about the composite functor $\mathbf{K}_{0}(\mathfrak{B}(-))$. If
$I\in{\mathbf{P}\mathbf{o}\mathbf{s}}_{\boldsymbol{\aleph}}$, by Theorem 5.7
there is a unique isomorphism $\rho(I)\colon
G(I)\to\mathbf{K}_{0}(\mathfrak{B}(I))$ which sends each $i\in I$ to
$[\mathbf{u}\mathfrak{B}(I)]$, where $\mathbf{u}$ is an arbitrary primitive
idempotent of $H(I,i)$. We have then two diagrams

$\begin{CD}G(I)@>{\rho(I)}>{}>\mathbf{K}_{0}(\mathfrak{B}(I))\\\
@V{G(f)}V{}V@V{}V{\mathbf{K}_{0}(\mathfrak{B}(f))}V\\\
G(J)@>{\rho(J)}>{}>\mathbf{K}_{0}(\mathfrak{B}(J)),\end{CD}\qquad\qquad\qquad\begin{CD}G^{*}(I)@>{\rho(I)}>{}>\mathbf{K}_{0}(\mathfrak{B}^{*}(I))\\\
@A{G^{*}(f)}A{}A@A{}A{\mathbf{K}_{0}(\mathfrak{B}^{*}(f))}A\\\
G^{*}(J)@>{\rho(J)}>{}>\mathbf{K}_{0}(\mathfrak{B}^{*}(J)).\end{CD}$

Let $i\in I$ and let $\mathbf{u}$ be any primitive idempotent of $H(I,i)$.
Then $(\mathfrak{B}(f))(\mathbf{u})=\widehat{f}_{i}(\mathbf{u})$ is a
primitive idempotent of $H(J,f(i))$ and so we obtain:

$\displaystyle\mathbf{K}_{0}(\mathfrak{B}(f))(\rho(I)(i))=\mathbf{K}_{0}(\mathfrak{B}(f))([\mathbf{u}\mathfrak{B}(I)]$
$\displaystyle=[(\mathfrak{B}(f)(\mathbf{u}))\mathfrak{B}(J)]=[(\widehat{f}_{i}(\mathbf{u}))\mathfrak{B}(J)]=\rho(J)(f(i))=\rho(J)(G(f)(i)).$

Since $G(I)$ is free with $I$ as a basis, this shows that the first diagram
commutes.

Next, let $j\in J$ and let $\mathbf{v}$ be a primitive idempotent of $H(J,j)$.
If $j\not\in f(I)$, then $j\in\operatorname{Ker}(G^{*}(f))$ (see (7.2)) and
$H(J,j)\subset\operatorname{Ker}(\mathfrak{B}^{*}(f))$, hence both
$\rho(I)(G^{*}(f)(j))$ and $\mathbf{K}_{0}(\mathfrak{B}^{*}(f))(\rho(j))$ are
zero. If $j\in J$ and $i$ is the unique element of $I$ such that $j=f(i)$,
then:

$\displaystyle\mathbf{K}_{0}(\mathfrak{B}^{*}(f))(\rho(J)(j))=\mathbf{K}_{0}(\mathfrak{B}^{*}(f))([\mathbf{v}\mathfrak{B}^{*}(J)]$
$\displaystyle=[(\mathfrak{B}^{*}(f)(\mathbf{v}))\mathfrak{B}^{*}(I)]=[(\widehat{f}^{-1}(\mathbf{v}))\mathfrak{B}^{*}(I)]=[(\widehat{f}^{-1}_{i}(\mathbf{v}))\mathfrak{B}^{*}(I)]$
$\displaystyle=\rho(I)(i)=\rho(I)(G^{*}(f)(j)).$

Here again, since $G(J)$ is free with $J$ as a basis, this shows that the
second diagram commutes. Finally we may conclude as follows:

######  Proposition 7.7.

With the above notations and settings, the assignment $I\mapsto\rho(I)$
defines two natural equivalences

$\rho(-)\colon
G(-)\approx\mathbf{K}_{0}(-)\cdot\mathfrak{B}(-)\quad\text{and}\quad\rho^{*}(-)\colon
G^{*}(-)\approx\mathbf{K}_{0}(-)\cdot\mathfrak{B}^{*}(-).$

## References

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