# Slow spin dynamics and quantum tunneling of magnetization in the dipolar
antiferromagnet DyScO3

N. D. Andriushin  Institut für Festkörper- und Materialphysik, Technische
Universität Dresden, D-01069 Dresden, Germany  S. E. Nikitin  Paul Scherrer
Institute (PSI), CH-5232 Villigen, Switzerland  G. Ehlers  Neutron
Technologies Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  A. Podlesnyak  Neutron Scattering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

We present a comprehensive study of static and dynamic magnetic properties in
the Ising-like dipolar antiferromagnet (AFM) DyScO3 by means of DC and AC
magnetization measurements supported by classical Monte-Carlo calculations.
Our AC-susceptibility data show that the magnetic dynamics exhibit a clear
crossover from an Arrhenius-like regime to quantum tunneling of magnetization
(QTM) at $T^{*}=10$ K. Below $T_{\mathrm{N}}=3.2$ K DyScO3 orders in an
antiferromagnetic $GxAy$-type magnetic structure and the magnetization
dynamics slow down to the minute timescale. The low-temperature magnetization
curves exhibit complex hysteretic behavior, which depends strongly on the
magnetic field sweep rate. We demonstrate that the low-field anomalies on the
magnetization curve are related to the metamagnetic transition, while the
hysteresis at higher fields is induced by a strong magnetocaloric effect. Our
theoretical calculations, which take into account dipolar interaction between
Dy3+ moments, reproduce essential features of the magnetic behavior of DyScO3.
We demonstrate that DyScO3 represents a rare example of inorganic compound,
which exhibits QTM at a single-ion level and magnetic order due to classical
dipolar interaction.

##  I Introduction

Timescale of spin dynamics – the time required to flip a single spin – in
conventional magnetic materials is of the order of femto- to picosecond [1, 2,
3]. Anomalous slowing down by multiple orders of magnitude down to the
millisecond range is known take place in some single-molecule magnets and is
induced by strong uniaxial anisotropy [4, 5, 6]. In that case, the strong
crystalline electric field (CEF) splits the ground-state multiplet
$\mathbf{J}$ of a magnetic ion and creates a doublet ground state, which
consists of two states with maximal projection of angular momentum, pointing
in the opposite directions, $|\psi^{0}_{\pm}\rangle=|\pm{}J\rangle$. Thus, the
direct transition between $|\psi^{0}_{+}\rangle$ and $|\psi^{0}_{-}\rangle$
states requires a change of the total momentum $\Delta{}J$ by more than one,
and thus is forbidden by selection rules of many conventional
single-(quasi)particle emission/absorption processes. Therefore, the matrix
element for this transition is low, see Fig. 1.

In this case there are two possible ways to change the spin momentum: (i) via
an activation process to the excited doublet with wavefunctions
$|\psi^{1}_{\pm}\rangle=|\pm{}J\mp 1\rangle$; or (ii) as a direct transition
between $|\psi^{0}_{+}\rangle$ and $|\psi^{0}_{-}\rangle$ via the quantum
tunneling of magnetization (QTM) process [4]. The first mechanism dominates in
the temperature range $T\gtrsim\Delta_{\mathrm{CEF}}/k_{\mathrm{B}}$
($\Delta_{\mathrm{CEF}}$ is the energy gap to the excited doublet and
$k_{\mathrm{B}}$ is the Boltzmann constant), but it becomes ineffective at
lower temperature where the QTM dominates the magnetic relaxation.

QTM is a well-known process in single-molecule magnets, however to the best of
our knowledge, among inorganic crystals there are only few well-documented
examples including Dy2Ti2O7 [7, 8, 9, 10, 11, 12] and Ca3Co2O6 [13, 14]. In
both cases, the strong CEF produces large uniaxial anisotropy, which freezes
magnetic moments below a crossover temperature $T^{*}\approx 13$ and
$T^{*}\approx\ 9$ K, for Dy2Ti2O7 and Ca3Co2O6 respectively. However, in
addition to the QTM both materials also show complex collective magnetic
behavior, classical spin-ice physics in case of Dy2Ti2O7 and a frustrated
spin-chain behavior in Ca3Co2O6, which obscure the QTM physics. Another
prominent example is a dipolar ferromagnet LiHoF4 and its diluted
modifications LiHoxY1-xF4 [15, 16, 17, 18], where electronic and nuclear spins
of Ho3+ ions are coupled because of large hyperfine interaction, which
produces complex slow dynamics at low temperatures.

In this work we focus on a classical Ising-like dipolar AFM DyScO3, whose
magnetic properties were studied in details previously [19, 20, 21, 22]. It
exhibits non-collinear AFM ordering below $T_{\mathrm{N}}=3.2$ K with the
propagation wavevector $\mathbf{k}=(001)$. Magnetization and neutron
diffraction measurements show that Dy exhibits strong uniaxial anisotropy at
low temperature and the easy-axis lies in the $ab$-plane, with a $\pm
28^{\circ}$ angle to the [010] direction. Inelastic neutron scattering (INS)
measurements show that the ground state doublet is well-isolated from the
first excited doublet located at 290 K. Point-charge model calculations
supported by magnetization measurements show that the wavefunction of the
ground state doublet consists of almost pure $|\pm 15/2\rangle$ states, making
DyScO3 a prospective material to search for the QTM effect.

In this work we performed a comprehensive study of the low-temperature
magnetic behavior in DyScO3 by means of AC and DC magnetization measurements.
We observed a clear peak in imaginary part of dynamical spin susceptibility
$\chi^{\prime\prime}$. It exhibits a crossover between an Arrhenius-like
regime at high temperatures and a temperature-independent regime below
$\lesssim\ 10$ K, which is a fingerprint of QTM behavior. The $M(B)$ curves
taken below $T_{\mathrm{N}}$ demonstrate complex hysteretic behavior. By using
classical Monte-Carlo simulations with dipolar interactions we reproduced
essential features of magnetic behavior of DyScO3: (i) the type of magnetic
ordering and the ordering temperature $T_{\mathrm{N}}$; (ii) temperature
dependences of the magnetic specific heat and magnetization; (iii) behavior of
magnetic correlation length above the $T_{\mathrm{N}}$; (iv) kink and a broad
magnetic hysteresis on the $M(B)$ curves. Our results demonstrate that the
low-temperature behavior of DyScO3 is described by a combination of CEF-
induced QTM, dipolar intersite interaction and the strong magnetocaloric
effect.

Figure 1:  Sketch of the energy diagram of $\mathbf{J}=15/2$ multiplet of Dy3+
in DyScO3, shown in $|M_{J}\rangle-E$ coordinates. Spin-orbit coupling (SOC)
produces the $\mathbf{J}=15/2$ multiplet, which is split into 8 doublets by
CEF. The ground state doublet consists of $|\pm 15/2\rangle$ wavefunctions,
following by excited $|\pm 13/2\rangle$, $|\pm 11/2\rangle$, etc. While the
temperature dependence of the spin excitation associated with $|\pm
15/2\rangle\rightarrow|\pm 13/2\rangle$ transition follows Arrhenius law, the
direct transition between $|\pm 15/2\rangle$ states is the temperature-
independent QTM process.

##  II Results and Analysis

###  II.1 Slow dynamics and magnetic order at zero field

We start the presentation of our results with the magnetization data collected
as a function of temperature with different sweep rates as shown in Fig. 2
(a). Note that the magnetic field was applied along the easy direction,
$B\parallel\ [010]$ in all measurements and calculations. All curves collected
upon warming up show a clear cusp anomaly associated with AFM ordering.
Noticeably, the position of the cusp shifts with the sweep rate $dT/dt$. One
can also see that the field-cooling (FC) curves differ considerably from those
collected upon warming up: (i) the cusp associated with the AFM transition
becomes less well-defined and is almost gone for $dT/dt\geq 1$ K/min; (ii) the
FC and warming up curves show considerable hysteresis below $T_{\mathrm{N}}$.

Figure 2:  Temperature dependences of the magnetization of DyScO3 measured
using VSM at $B=0.1$ T (a) and calculated by Monte-Carlo (b) with different
temperature sweep rates as described in Sec. II.3. Red and blue lines
represent data collected on warming and cooling respectively. Data are in
panels (a) and (b) are shifted respectively by +0.5 $\mu_{\mathrm{B}}$/f.u.
and +1 $\mu_{\mathrm{B}}$/f.u. vertically for clarity.

These results indicate the presence of a considerable magnetization relaxation
at low temperatures, which takes place on a timescale of minutes. It is
unexpected for a conventional antiferromagnet, but rather reminiscent of the
spin-glass (SG) state. At sufficiently low temperature, SG materials do not
exhibit long-range magnetic order, but instead form magnetic clusters with
short-range magnetic correlations [23, 24]. A clear fingerprint of the SG
behavior is a broad peak in the imaginary part of the AC-susceptibility,
$\chi^{\prime\prime}(f)$, which should follow an Arrhenius-like temperature
dependence over a broad temperature range [25].

To discuss DyScO3 in this context, Figures 3 and 4 display our temperature-
and frequency-dependent AC-susceptibility measurements. Figure 3 (a) shows the
temperature dependence of the real part of the AC-susceptibility,
$\chi^{\prime}(T)$, measured at different frequencies along with the static
spin susceptibility, $M/B$, measured with VSM. The curves collected at $f=1$
Hz and at static regime agree well above $T_{\mathrm{N}}$, and display a
single peak at the transition temperature. The qualitative behavior changes
when the frequency is increased. The low-temperature susceptibility measured
at $f\geq\ 10$ Hz is reduced, but returns to $M/B$ above a frequency-dependent
crossover temperature. The high-temperature tails of all curves follow the
same Curie-Weiss law.

Figure 3:  Temperature dependence of real (a) and imaginary (b) parts of the
complex longitudinal AC susceptibility of DyScO3 measured at $B=0.1$ T applied
along [010] at different frequencies. The static spin susceptibility $M/B$
measured using VSM is shown in panel (a). Crossed black lines in panel (b)
illustrate how the crossover temperature was determined.

The $\chi^{\prime\prime}(T)$ curve measured at 1 Hz shows a strong divergence
at $T_{\mathrm{N}}$, as expected for an AFM system, and a weak shoulder-like
feature at $\sim 7$ K. With increasing frequency the shape of the peak at the
ordering temperature changes significantly and becomes similar to the one
observed in $\chi^{\prime}(T)$. The second anomaly also shifts with frequency.
We quantified the positions of the second anomaly using the inflection point
as shown for 1 Hz curve in Fig. 3 (b).

Figure 4:  Frequency dependence of the AC-susceptibility measured at multiple
temperatures between 2 and 20 K. Panels (a) and (b) show the real
[$\chi^{\prime}(f)$] and imaginary [$\chi^{\prime\prime}(f)$] parts of the AC-
susceptibility, respectively. Panel (c) shows the Cole-Cole plot
$\chi^{\prime\prime}(\chi^{\prime})$. All data are shown with a constant
vertical offset for clarity.

To further reveal the frequency dependence of the spin susceptibility in
DyScO3 we performed measurements of $\chi^{\prime}(f)$ and
$\chi^{\prime\prime}(f)$ at multiple temperatures, and the results are
summarized in Fig. 4. The $\chi^{\prime}(f)$ demonstrates a plateau at low
frequencies and a gradual decrease above temperature-dependent crossover
frequency. The $\chi^{\prime\prime}(f)$ curves exhibit a strong broad peak at
$T\geq 3$ K. The position of the peak shifts down with decreasing temperature,
however between $T=10$ and 4 K $\chi^{\prime\prime}(f)$ remains almost
unchanged. When cooling below $T_{\mathrm{N}}$ the peak height decreases and
shifts towards lower frequencies, which could not be followed further with our
AC setup. Figure 4(c) shows Cole-Cole plots
$\chi^{\prime\prime}(\chi^{\prime})$ at different temperatures. For a system
with a single relaxation channel (or symmetrical distribution of the
relaxation channels), the curves should follow a semi-circular trajectory.
However, the curves measured with DyScO3 are asymmetric, indicating a more
complex distribution of the relaxation times [26, 27, 28].

To characterize the timescale of the magnetization dynamics below
$T_{\mathrm{N}}$ we used a VSM magnetometer and measured magnetization
relaxation. To do this, we applied the following protocol: (i) ZFC to the base
temperature $T=1.8$ K; (ii) apply 0.01 T external field; (iii) wait for 300 s;
(iv) decrease the external field to zero with sweep rate of 0.07 T/s; (v)
collect time-dependent $M(t)$ for 3 hours; (vi) increase temperature to the
new target $T$ and repeat from the step (ii). The relaxation curves collected
at several selected temperatures above and below $T_{\mathrm{N}}$ are shown in
Fig. 5.

Conventionally, the relaxation process $M(t)$ can be described with an
exponential function, but we were not able to obtain good fit of our
experimental data using a single exponent at $T<3$ K. The measured curves were
fitted with a sum of two exponential functions:

$M(t)=M_{1}e^{-(t/\tau_{1})^{\beta_{1}}}+M_{2}e^{-(t/\tau_{2})^{\beta_{2}}}$
(1)

where, $M_{1}$ and $M_{2}$ correspond to two relaxing moments,
$\tau_{1},\tau_{2}$ are the relaxation times, $\beta_{1},\beta_{2}$ are
stretching parameters. To improve the fit quality we used $\beta_{1}$ and
$\beta_{2}$ as global parameters for fitting of $T=1.8,2.2$ and 2.6 K curves
[Fig. 5 (a)]. The fitted curves are shown by solid lines in Fig. 5. Results of
the fitting yield $\beta_{1}=0.35$ and $\beta_{2}=0.6$ and $\tau_{1}=550$ s,
$\tau_{2}=25$ s at $T=1.8$ K; the relaxation times exhibit only a minor
increase with temperature up to 2.6 K. The relaxation curve taken at $T=3$ K
can be described by a single exponent with $\beta=0.3$ and $\tau=0.2$ s [Fig.
5 (b)].

Figure 5:  Time dependence of magnetization taken after switching off 0.01 T
magnetic field. Panels (a) and (b) show low- and high-temperature data. Note
that $y$-scale is different in (a) and (b).

The temperature dependence of observed relaxation times extracted from AC
susceptibility and magnetization relaxation measurements is summarized in Fig.
6. Informed by our own INS measurements [19], we highlight three different
regimes: (i) Arrhenius regime at high-temperature, $T>10$ K; (ii) temperature-
independent relaxation between 10 K and $T_{\mathrm{N}}$; (iii) slow
relaxation in the AFM phase. The temperature dependent AC-susceptibility in
DyScO3 was also studied in Ref. [20], where the authors observed peak-like
anomalies in $\chi^{\prime\prime}(T)$ curves, which shifted with frequency.
The authors associated this peak with an Arrhenius-like relaxation process,
taking the population of the CEF level into account, and extracted
$\Delta/k_{\mathrm{B}}=229$ K. Our own later INS measurements indicated that
the first CEF level is located at higher energy, $\Delta/k_{\mathrm{B}}=290$ K
[19]. The calculated curve for a 290 K gap is shown in Fig. 6 by a red line,
and one can see good agreement with experimental points at high temperature
and a clear crossover between regimes (i) and (ii).

The behavior seen in DyScO3 strongly resembles the slowing down of the spin
dynamics in a classical spin-ice compound Dy2Ti2O7, which also shows three
regimes at different temperatures: (i) Arrhenius relaxation at $T>15$ K; (ii)
plateau at $1.5<T<15$ K; (iii) another Arrhenius regime below $1.5$ K due to
development of the spin-ice regime [7, 9, 10]. The difference in the low-
temperature regime (iii) is a consequence of the different ground states, AFM
in DyScO3 vs. spin-ice in Dy2Ti2O7.

Figure 6:  Arrhenius plot log$(f)(1/T)$, reconstructed from our results along
with data from Ref. [20]. The grey dotted line shows the transition
temperature $T_{\mathrm{N}}$. The blue and green points show the peak
positions extracted from the frequency and temperature dependencies of
imaginary part of AC susceptibility $\chi^{\prime\prime}$, and the black
squares show the high-frequency data from Ref. [20]. Red solid line shows the
calculated CEF activation curve with experimentally determined $\Delta=290$ K.
Red points show inverse relaxation constants $1/\tau$ extracted from a fit of
the magnetic relaxation curves with Eq. (1).

###  II.2 Field-induced anomalies

We next proceed with a description of the field-induced physics in DyScO3.
Authors of [19] reported magnetization curves measured at $T=2$ K along three
crystallographic directions. Interestingly, the curves measured along $[010]$
and $[100]$-axes show two consecutive hystereses at low field and just below
the saturation. These features were interpreted as two field-induced first-
order phase transitions. Motivated by those observations we measured the
magnetization of DyScO3 at several temperatures with magnetic field applied
along the $[010]$ axis. Figure 7 shows magnetization curves collected at
several temperatures below and above $T_{\mathrm{N}}$. One can see that at
$T=2$ K magnetization curves measured with high sweep rates (50, 500 and 700
Oe/s) show considerable hysteresis over the whole field range. However, we can
clearly highlight two distinct transitions: the first kink at $B\approx 0.4$ T
and the second anomaly at $B\approx 1$ T. Noticeably, when the sweep rate
decreases, only the low-field features remains visible, while magnetization at
higher fields shows simple Brillouin-like behavior. Figures 7(b-d) demonstrate
magnetization collected above $T_{\mathrm{N}}$ and one can see that the
magnetization is perfectly linear at the low-field regime in agreement for the
expectation for a paramagnet. However, the clear kink as well as the
hysteresis at 0.7–2 T are clearly seen for high field-sweep rates, while the
low-sweep curves show simple Brillouin-like behavior.

Based on these data we can associate the first transition with the field-
induced destruction of the AFM order, while the high-field hysteresis is
associated with single-ion physics, because it persists to temperatures up to
$\sim 8$ K, which is much larger than the characteristic energy scale of
magnetic interactions in DyScO3. We associate this effect with the strong
magnetocaloric effect (MCE) in DyScO3 [29] and in the next section we show
that Monte-Carlo simulations, which take into account the MCE, are capable to
reproduce this effect.

Figure 7:  Magnetization curves $M(B)$ measured at several temperatures as
indicated at each panel. The curves were measured with different sweep field
rates as shown in legend.

###  II.3 Monte-Carlo simulations

As discussed above, DyScO3 shows strong Ising-like single-ion anisotropy of
magnetic moments, meaning that Dy moments can be pointed up or down along the
CEF-dictated easy magnetization direction at each site. In addition, a
previous report associated magnetic order with the dipole-dipole interactions
between Dy moments [19]. The standard approach to describe physical properties
of an Ising system on a 3D lattice is by Monte-Carlo modeling and here we make
use of the metropolis algorithm to describe magnetic behavior of DyScO3 [30].
We considered a $10\times 10\times 10\times 4$ cluster of Ising spins, which
are coupled by dipole-dipole interaction and are in thermal contact with a
reservoir at temperature $T_{\mathrm{res}}$. Most parameters of our model,
such as interatomic distances, the field- and temperature dependence of the
thermal conductivity (approximated from the data measured on isostructural
DyAlO3), the magnetic moment of Dy3+ and the direction of easy axis were fixed
from the experimental data [19, 31]. The only free parameter is the
coefficient which converts the number of Monte-Carlo steps to the experimental
time, which we fixed by comparison of calculated and experimental
$\chi^{\prime\prime}(f)$ curves and the absolute value of the thermal
conductivity. See Sec. B for details.

Figure 8:  AFM ordering at zero field. (a) Temperature dependence of (001)
Bragg peak. Red dots represent experimental data for square of the ordered
moment [19] and solid line is the result of Monte-Carlo calculations. (b)
Calculated and measured low-temperature specific heat. The nuclear
contribution was approximated by $C(T)=\beta T^{3}$ with $\beta=0.00023$
J/mole$\cdot$K4 and added to the calculated curve. The nuclear contribution
reaches 0.047 J/mole K at $T=6$ K and therefore is barely visible at this
temperature scale. (c) Temperature dependence of the correlation length
measured by neutron diffuse scattering (red and pink dots) and calculated by
Monte Carlo (blue and light blue dots). The filled area represents the
estimated uncertainty of the calculation. (d) Simulated neutron diffraction
pattern in the $(H0L)$ scattering plane calculated at $T_{\mathrm{N}}<T=3.5$ K
showing anisotropic diffuse scattering around the primary AFM peak (001).

We start the presentation of our calculations with the low-temperature
magnetic structure. First, our data show that the $GxAy$ is indeed the ground
state of DyScO3 and has the lowest energy among the four possible
$\mathbf{k}=0$ magnetic configurations in DyScO3, in agreement with previous
estimates for smaller clusters [19, 29]. As the next step we calculated the
static magnetic structure factor, $S(\mathbf{Q})$, and found that the AFM
order manifests itself with a strong magnetic Bragg peak at
$\mathbf{Q}=(001)$. We plot the calculated temperature dependence of (001)
peak in Fig. 8 along with the experimental measurements of the ordered moment
[19]. Fitting of calculated data near the critical temperature was performed
using the following expression:

$S=\left\\{\begin{array}[]{lr}0,&\text{for }(T>T_{c})\\\
S_{0}(1-\big{(}\frac{T}{T_{c}}\big{)}^{\beta})^{\alpha},&\text{for }(T\leq
T_{c}),\end{array}\right\\}$

and yielded the critical temperature $T_{\mathrm{N}}=2.853(1)$ K. The very
good agreement with the experimentally determined $T_{\mathrm{N}}=3.11$ K
indicates that the dipolar interaction is the primary magnetic interaction in
DyScO3. We have also modeled the magnetic diffuse scattering above the
transition temperature. The representative diffuse pattern calculated at
$T=3.5$ K is represented in Fig. 8 (d), which was obtained from averaging a
number of Monte-Carlo runs. We extracted the temperature dependence of the
correlation length [Fig. 8 (c)] and compared it with experimental results
measured at the CNCS instrument [19]. Clearly, the measured and the calculated
curves show good agreement with $\xi_{c}>\xi_{ab}$ over all temperature range.
The reason is the mutual arrangement of crystal axes and directions of Dy3+
moments: the nearest neighbor Dy moments along the $c$ axis have strong
antiferromagnetic interaction, while interaction between in-plane Dy moments
is nearly canceled out for the $GxAy$ magnetic configuration [29]. To further
characterize the magnetic transition we calculated the specific heat of DyScO3
and show it along with the experimental curve in Fig. 8 (b). Both curves show
a sharp $\lambda$-like peak at $T_{\mathrm{N}}$ due to AFM transition with
only weak shoulders above $T_{\mathrm{N}}$, which indicates weak magnetic
correlations in the paramagnetic state. The entropy change associated with
anomaly in the specific heat is in agreement with the value expected for a
Ising system: $R\ln(2S+1)$ for $S=1/2$.

To reveal the origin of the bifurcation between the $M(T)$ curves shown in
Fig. 2 (a) we performed Monte Carlo simulations of the temperature dependences
of magnetization [see Fig. 2 (b)] with different sweep rates of the reservoir
temperature, $dT_{\mathrm{res}}/dt$. We found that the calculated
magnetization curves are in qualitative agreement with experimental data.
Specifically, the magnetization curve calculated for $dT/dt=0.1$ K/min shows a
clear cusp at $T_{\mathrm{N}}$ and a weak hysteresis below the transition.
With increase sweep rate $dT/dt$ the hysteresis becomes wider and the peak for
warming up curves shifts towards higher temperatures in good qualitative
agreement with the experiment. The reason for this behavior is the poor
thermal stabilization due to both, weak thermal conductivity and slow
thermalization during the temperature sweeps, because of which the actual
temperature of the sample can differ considerably from $T_{\mathrm{res}}$.
This effect causes bifurcation between cooling and warming $M(T)$ curves and
the sweep-rate dependence of the transition temperature.

As the next step we calculated the magnetic field dependence of the
magnetization at three characteristic temperatures of the reservoir,
$T_{\mathrm{res}}=1,~{}2$ and 3.5 K and the results are summarized in Fig. 9.
We performed two versions of calculations, (i) isothermal magnetization with
$T_{\mathrm{Sample}}=T_{\mathrm{res}}$ strictly maintained, and (ii) by taking
into account the strong magnetocaloric effect [21], see details of
calculations in Sec. B. The field sweep rate for all curves in Fig. 9 was
$0.5$ K/min.

First of all, let us consider the magnetization curves calculated at $T=3.5$ K
above the ordering temperature, which are shown in Fig. 9 (a1). The isothermal
magnetization calculated at $T=3.5$ K shows featureless Brillouin-like
behavior in agreement with expectations for a paramagnet. Introduction of the
magnetocaloric effect changes qualitative behavior of the magnetization curves
and opens a broad hysteresis at $B\approx$ 0.4–1.2 T, because of the field-
induced change of $T_{\mathrm{samp}}$, which is shown in Fig. 9 (a2). Thus we
conclude that the hysteresis at intermediate fields observed in our
magnetization measurements for $dB/dt\geq\ 50$ Oe/s (Fig. 7) is caused by the
magnetocaloric effect, which however plays a minor role when the field ramp
rate is small compared to the thermalization time, meaning that
$T_{\mathrm{samp}}=T_{\mathrm{res}}$.

Below the magnetic transition temperature the magnetization curve changes
considerably. The curves calculated at $T_{\mathrm{res}}=2$ K demonstrate a
metamagnetic transition and a broad hysteresis at low fields, $B<0.3$ T, which
behaves similarly for both, the isothermal and non-isothermal curves [Fig. 9
(b1)]. Above the metamagnetic transition the isothermal curves merge and show
the featureless Brillouin-like behavior, while a narrow hysteresis emerges at
$B=0.5$–1 T for the curves calculated with magnetocaloric effect. We note that
the exact shape of the calculated curve depends considerably on the thermal
conductivity of DyScO3, which depends on both magnetic field and temperature,
$\lambda(B,T)$. This quantity was not measured in the low-temperature regime,
$T<10$ K and in our calculations we make use of the data measured in the
paramagnetic phase on the isostructural DyAlO3 [31], which we approximate down
to the temperature range of interest (see Sec. B for details). This procedure
does not provide quantitatively precise values for the thermal conductivity,
especially below $T_{\mathrm{N}}$ where thermal conductivity can exhibit
strong anomalies near the critical field [32, 33, 34]. Therefore, the exact
shape of calculated and observed magnetization curves does not match
quantitatively. However, our calculations allow us to associate the high-field
hysteresis with the magnetocaloric effect, while the low-field anomalies,
including the metamagnetic transition and the hysteresis of magnetization, are
associated with collective behavior.

Figure 9 (c1) shows magnetization calculated at $T_{\mathrm{res}}=1$ K and one
can see that isothermal and non-isothermal curves coincide over all field
scales, meaning that the magnetocaloric effect has a minor influence on the
magnetization curve at this temperature. Moreover, the curves calculated for
$B=0\rightarrow\ 2$ T demonstrate a narrow $M_{\mathrm{s}}/2$ plateau at
$B=0.4$ T. In a system with Ising-like anisotropy, the magnetic moments can
point parallel or antiparallel to the easy-axis direction, therefore below the
ordering temperature, one can expect to see the formation of SDW-like
incommensurate phases at the intermediate magnetic fields [35, 36]. Figure 9
(d) shows the calculated static spin structure factor at $T=1$ K and $B=0.4$
T, which corresponds to the plateau field range. One can see clearly formation
of the incommensurate peaks at $\mathbf{q}=(0~{}0~{}0.55)$, which can be
associated with the formation of $\uparrow\uparrow\uparrow\downarrow$ type of
order along the $c$ axis. Two other reflections correspond to the primary
magnetic reflection of the zero-field AFM phase, $\mathbf{q}=(0~{}0~{}1)$ and
ferromagnetic $\mathbf{q}=(0~{}0~{}0)$ peak. We note that a similar field-
induced incommensurate magnetic order with $\mathbf{k}=(0~{}0~{}1\pm\delta)$
was observed previously in the isostructural material YbAlO3 [37, 38]. Low-
temperature in-field neutron diffraction measurements would be required to
verify the presence of the incommensurate magnetic phases in DyScO3.

Figure 9:  (a1-c1) The magnetization curves calculated taking into account
magnetocaloric effect at various temperatures of reservoir and corresponding
system temperature dependence. The magnetization curves were calculated for
external field ramped from zero to 2 T and backward, the shaded areas show
width of the hysteresis curves $\Delta{}M(B)=M(B)\uparrow-M(B)\downarrow$.
Additional panels (a2-c2) show temperature change during simulation, grey
horizontal lines denote reservoir temperatures. Reservoir temperatures are
indicated in each panel. The contour map (d) is the calculated magnetic
structural factor for the incommensurate state around $B=0.4$ T and $T=1$ K
(red circle point at (c1)).

##  III Discussion and Conclusion

We demonstrate that DyScO3 exhibits slow dynamics of magnetization below
$T^{*}=10$ K, which is caused by strong CEF-induced uniaxial anisotropy of
Dy3+ moments. We note that similar behavior was observed previously in
multiple organic-based magnetic molecules [6, 5, 4], while such a behavior in
inorganic materials remains relatively rare. Several representative examples
are Dy2Ti2O7 and Ca3Co2O6. However, in addition to the QTM, both
aforementioned materials display complex magnetic behavior due to frustration
of magnetic interactions. The presence of the competing interactions makes it
difficult to disentangle the effects of QTM and magnetic frustration on spin
freezing in these systems, contrary to the case of DyScO3. We have
characterized the low-temperature spin dynamics of DyScO3 using AC
susceptibility and magnetization relaxation measurements and have observed
three regimes: Arrhenius-like behavior at $T>T^{*}$, a plateau between $T^{*}$
and $T_{\mathrm{N}}$, and slow relaxation of magnetization below
$T_{\mathrm{N}}$. Thus, DyScO3 represents a so-far unique example of a
classical dipolar AFM, which combines a classical magnetically-ordered ground
state with QTM.

Our Monte Carlo simulations appear to capture the essential physics of DyScO3,
including the magnetic ground state, the temperature of the AFM transition,
slow dynamics of magnetization at low temperatures and bifurcation between
$M(T)$ curves collected upon cooling and warming. In addition, by taking into
account the magnetocaloric effect we were able to describe the magnetization
curves and demonstrate that the hysteresis in the paramagnetic phase is caused
by considerable field-induced change of the sample temperature during the
measurements. Our simulations also predict the formation of an incommensurate
spin-density wave magnetic phase at low temperatures and intermediate magnetic
fields, similar to that observed in YbAlO3 [37, 38], whose existence in DyScO3
awaits experimental verification with elastic neutron scattering measurements.

We note that although our model captures the main features of magnetic
behavior in DyScO3 there are minor quantitative disagreements such as the
exact shape of the hysteresis curves shown in Figs. 7 (a-c), which could
probably be improved by including in the model exact results for the field-
and temperature-dependence of thermal conductivity. In addition, the
$\chi^{\prime\prime}(\chi^{\prime})$ curve has an asymmetric shape indicating
a complex distribution of the relaxation times, while our model implies a
single relaxation channel for simplicity.

To conclude, we have applied AC susceptibility and DC magnetization
measurements, supported by specific heat, neutron diffuse scattering and
Monte-Carlo calculations to characterize spin dynamics in DyScO3. Our results
indicate that DyScO3 represents a rare combination of single-ion QTM behavior
with classical dipolar interactions and stimulate further search of rare-earth
based condensed matter systems with QTM.

###### Acknowledgements.

Acknowledgments. We thank O. Stockert and A. S. Sukhanov for stimulating
discussions. S.E.N. acknowledges financial support from the European Union
Horizon 2020 research and innovation program under Marie Sklodowska-Curie
Grant No. 884104. The work of N.D.A. was supported by the German Research
Foundation (DFG) through grant No. PE 3318/3-1. This research used resources
at the Spallation Neutron Source, a DOE Office of Science User Facility
operated by the Oak Ridge National Laboratory.

##  Appendix A Experimental details

High-quality single-crystals of DyScO3 were obtained commercially [39].
Magnetization and AC-susceptibility measurements were performed in a
temperature range 1.8 - 100 K, using an MPMS SQUID VSM instrument by Quantum
Design. For the magnetic measurements we used a sample with mass of 1.6 mg.
Magnetic field was applied along the [010] direction, which is the easy axis
of magnetization. Specific heat was measured using PPMS from Quantum Design.
The single crystal neutron scattering measurements were done at the Cold
Neutron Chopper Spectrometer (CNCS) [40, 41]. To quantify the correlation
lengths $\xi_{H}$, $\xi_{L}$ we perform a two dimensional fitting of the (0 0
1) magnetic peak using the following function [19]:

$S_{\mathrm{mag}}(\mathbf{Q})=\frac{\sinh{a/\xi_{H}}}{(\cosh{a/\xi_{H}}-\cos{\pi
q_{H}})}\frac{\sinh{c/\xi_{L}}}{(\cosh{c/\xi_{L}}-\cos{\pi q_{L}})},$ (2)

where $a$ and $c$ are the lattice parameters. The calculated magnetic
structure factor (see Sec. B.3) was fitted using the same equation.

##  Appendix B Monte-Carlo simulations

###  B.1 Magnetic Hamiltonian

At low temperature magnetic system of Dy3+ ions can be effectively described
by dipole-dipole interaction. Since the magnetic properties of DyScO3 exhibit
Ising-like behavior, it is convenient to theoretically study these with Monte
Carlo (MC) simulations. In this work we performed MC simulations of a 3D Ising
system of Dy3+ moments using the classical Metropolis single-flip algorithm
[30]. The Hamiltonian used in the calculations was taken as a combination of
dipole-dipole interaction energy and a Zeeman term due to the external field.
The Hamiltonian reads:

$\begin{array}[]{rcl}\mathcal{H}&=&-\displaystyle\frac{1}{2}\frac{\mu_{0}}{4\pi}\sum_{i,j}\left[\frac{3(\vec{m_{i}},\vec{r_{i,j}})(\vec{m_{j}},\vec{r_{i,j}})}{|\vec{r_{i,j}}|^{5}}-\frac{(\vec{m_{i}},\vec{m_{j}})}{|\vec{r_{i,j}}|^{3}}\right]\\\\[21.52771pt]
&&-\displaystyle{B}\sum_{i}\vec{m_{i}}{\;},\end{array}$ (3)

where the first term is the dipole-dipole interaction energy and the second
one is the energy of Zeeman interaction of the magnetic moments and external
field. Because the summation goes over each interaction twofold there is the
$\frac{1}{2}$ factor in front of the first term. The $\mu_{0}$ corresponds to
vacuum permeability constant, $\vec{m_{i}}$ is the magnetic moment on the $i$
site, $\vec{r_{i,j}}$ is the radius vector between the $i$ site and $j$ site
magnetic moments, the $B$ is the external field. In the calculations the
magnitude of magnetic moments was taken as $10\mu_{B}$.

The crystal structure of the DyScO3 was taken into account during the
simulations by implementing the 3D Ising system with inter-atomic distances
corresponding to the crystallographic data. This is necessary because inter-
atomic radius vectors appear in the expression for the dipolar energy. The
size of the system in the simulations was $10\times 10\times 10$ unit cells
each containing 4 spins (4000 spins in total). The dipole-dipole interaction
was calculated for up 62 interactions per one site (the cutoff distance
between neighbours was 9.5 Å) as a compromise between calculation time and
accuracy. A test simulation showed that an inclusion of further neighbors
produced minor effects on the magnetic behavior. The periodic boundary
conditions were used. Each Ising spin can be in the $s=1$ or $s=-1$ state
corresponding to the spin vector $\vec{S}=s(\pm\sin(\phi),\cos(\phi),0)$,
where the sign in front of the $\sin$ term depends on the position of site in
unit cell (see Fig. 10). $\phi$ is the angle between spin vector and the
$[010]$ or the $[0\bar{1}0]$ direction determined by CEF and is equal to 28∘
[19].

Figure 10:  Scheme of GxAy spin configuration.

###  B.2 AC-susceptibility

AC susceptibility data were used in our simulations to adjust the time-scale
transformation coefficient. For calculations of real and imaginary parts of AC
susceptibility the following expressions were used:

$\displaystyle\chi^{\prime}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}N_{\mathrm{total}}}\sum_{i=0}^{N_{\mathrm{total}}}M_{i}\sin(2{\pi}f\frac{i}{A}-\frac{\pi}{2})$
(4) $\displaystyle\chi^{\prime\prime}(f)$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}N_{\mathrm{total}}}\sum_{i=0}^{N_{\mathrm{total}}}M_{i}\cos(2{\pi}f\frac{i}{A}-\frac{\pi}{2}){\;},$
(5)

where $\chi^{\prime}$ and $\chi^{\prime\prime}$ are real and imaginary parts
of AC susceptibility, $B_{0}$ and $f$ are the magnitude and the frequency of
the alternating external field, and $M_{i}$ is the magnetization of the system
at $i$-th MC step. The alternating external field at the $i$-th MC step is
taken as $B=B_{0}\sin(2{\pi}f{i/A}-\pi/2)$. Used external field amplitude
$B_{0}$ is $0.1~{}T$. The $A$ parameter is the conversion factor between real
time and simulation time $t_{sim}=At_{real}$. In the above expressions, an
averaging was performed over numerous periods, with $N_{total}$ up to $10^{6}$
MC steps per spin. In Fig. 11 experimental and calculated AC susceptibility at
temperature 4 K were drawn on the same layer for comparison. The calculated
imaginary part of susceptibility $\chi^{\prime\prime}$ consists of one
symmetrical peak associated with a particular relaxation time. In contrast,
the experimental dependence of $\chi^{\prime\prime}$ is more broad and
asymmetric. As it was discussed in Sec. II.1, the shape of experimental
$\chi^{\prime\prime}$ curves can be interpreted as a presence of more then one
relaxation channel. The value of the $A$ parameter was adjusted for fitting
the calculated position of $\chi^{\prime\prime}$ maximum to the experimental
value and later the $A$ parameter was fixed when used in further calculations.

Figure 11:  Calculated and experimental AC susceptibility at $T=4$ K.

###  B.3 Magnetic diffraction

The magnetic structure factor was calculated as a spatial Fourier transform of
the spin-spin correlation function taking into account the polarization factor
of neutron scattering using the following expression:

$S_{mag}(\vec{q})=\sum_{\alpha,\beta}\big{[}\\{\delta_{\alpha,\beta}-\frac{q_{\alpha}q_{\beta}}{|\vec{q}|^{2}}\\}\sum_{j,j^{\prime}}S_{j,\alpha}S_{j^{\prime},\beta}\exp(i\vec{q}(\vec{R_{j}}-\vec{R_{j^{\prime}}}))\big{]},$
(6)

where $\vec{q}$ is the wave vector, $\alpha$ and $\beta$ are Cartesian
components - $x,y$ and $z$, the $j$ and $j^{\prime}$ are the variables which
go over all the sites in system, the $S_{j,\alpha}$ is the $\alpha$ component
of spin vector on $j$-th site. The term in front of the second summation is
the polarisation factor. The isotropic magnetic form factor of Dy ions, which
gives a monotonic suppression of the intensity at large $Q$, was not included
to the calculations.

###  B.4 Thermodynamic properties

Figure 12:  (a) Calculated magnetic part of the heat capacity as function of
external field and temperature. (b) The heat conductivity extrapolated in
nonzero external field region.

In order to achieve a better agreement with experimental measurements the
magnetocaloric effect was taken into account for calculations of the field
dependence of magnetization. The heat capacity used in the magnetocaloric
calculations was the combination of the lattice contribution, which was
obtained from approximation of experimental measurements, and the magnetic
contribution which was calculated from energy fluctuations:

$C_{\mathrm{mag}}(T,B)=\frac{{\langle\mathcal{H}^{2}\rangle}-{\langle\mathcal{H}\rangle}^{2}}{k_{B}T^{2}},$
(7)

where $k_{B}$ is the Boltzmann constant and the angle brackets means averaging
over numerous MC steps. The total heat capacity can be written as
${C}_{p}=C_{\mathrm{mag}}+C_{\mathrm{lat}}$, the lattice part was approximated
from experimental data above ordering temperature $T_{\mathrm{N}}$. The
magnetocaloric effect was implemented in calculations of the field dependence
of magnetization as a variable system temperature. The temperature change
comes form the magnetocaloric effect itself and also from thermal contact with
the temperature reservoir. If the external field changes from $B_{1}$ to
$B_{2}$ during time ${\Delta}t$ the system temperature change $\Delta T$ can
be written as follows:

${\Delta}T=\frac{T}{\frac{1}{2}(C_{p}(B_{1},T)+C_{p}(B_{2},T))}\frac{1}{2}\bigg{[}\frac{{\partial}M(B_{1},T)}{{\partial}T}+\\\
+\frac{{\partial}M(B_{2},T)}{{\partial}T}\bigg{]}dB-\frac{T-T_{res}}{\frac{1}{2}(C_{p}(B_{1},T)+C_{p}(B_{2},T))}\times\\\
\times\frac{1}{2}(\lambda(B_{1},T)+\lambda(B_{2},T)){\Delta}t,$ (8)

where the first term is the magnetocalorics and the second expresses thermal
contact with reservoir. $dB=B_{2}-B_{1}$ is the change of external field, $T$
is the system temperature before the correction, $T_{res}$ is the reservoir
temperature, $\lambda(B,T)$ is the thermal conductivity coefficient. The field
change $dB$ is assumed to be small enough in order to justify the replacement
of the integral from original thermodynamic formula by this simple expression.
The heat capacity $C_{p}(B,T)$ values were obtained as it was mentioned above.
The time $\Delta t$ was recalculated into Monte Carlo steps with help of
previously discussed $A$ parameter. The partial derivative
$\frac{{\partial}M(B,T)}{{\partial}T}$ was calculated beforehand from
equilibrium values of magnetization:

$\frac{{\partial}M(B,T)}{{\partial}T}=\frac{\langle M(B,T+\Delta
T)\rangle-\langle M(B,T-\Delta T)\rangle}{2\Delta T}.$ (9)

In order to numerically estimate the derivative the averaging was done over
more $10^{6}$ MC steps per spin.

The exact shape of the field dependence of thermodynamic properties can
significantly affect the magnetization behavior in magnetocaloric
calculations. Since the experimental data for the heat conductivity
$\lambda(B,T)$ is unknown in presence of external field and in low temperature
for DyScO3, we had to use values obtained from extrapolation process and turn
into account assumptions. The heat conductivity $\lambda(B,T)$ values in zero
field were taken from data for the isostructural DyAlO3 material. We assumed
the following when extrapolating the values to nonzero field: (i) the heat
conductivity $\lambda(B,T)$ slowly decreases in the external field [31], and
(ii) near the region of metamagnetic transition the heat conductivity
$\lambda(B,T)$ has a pronounced drop, which often occur in AFM systems [32,
33, 34]. The points of critical field at which the heat conductivity
$\lambda(B,T)$ drops were determined from anomaly on the calculated heat
capacity. The heat capacity $C_{p}(B,T)$ and the heat conductivity
$\lambda(B,T)$ as functions of external field and temperature, as these were
used in our calculations, can be seen in Fig. 12.

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Metric ideals and its Structures in AL-monoids

Tekalign Regasa[Department of Mathematics, College of Natural and Computational Science College, Addis Ababa University ,Addis Ababa,Ethiopia, email:tekalign.regasa@aau.edu.et],Girum Aklilu [Department of Mathematics,college of Natural and computational Science College, Addis Ababa University,Addis Ababa,Ethipia e-mail<EMAIL_ADDRESS> Kolluru Venkateswarlu [Department of Computer Science and System Engineering, College of Engineering, Andhra University, Visakhapatanam, AP<EMAIL_ADDRESS>
This paper introduces the concept of metric ideals in AL-monoids. We also examine the structure of AL-monoids and describe some of the properties of homomorphism and fundamentalisomorphism theorems.Additionaly we introduce and examine a direct product and subdirect products in AL-monoids.

§ INTRODUCTION

Subba Rao and Yedpalli(2018)[3] and Rao et al.(2019,2021)[1, 2] studied the concept of Representablity of Autometrized algebra. In 2023, [6] Gebre Yeshiwas et al. studied the structure of Autometrized algebra by introducing the concept of subalgebra and examining diffrent types of ideals in Autometrized algebra. In 2024, Tekalign, Girum and Kolluru venkateswarlu [12] introduces AL-monoids which is equationally definable ,Representable and generalization of DRl-semigroups as the optimal common abstraction of Boolean algebra and commutative lattice ordered groups. In this Paper, We continue in studying the metric ideals in AL-monoids, homomorphism and isomorphism Theorems in AL-monoids and some of its subirreducible product.

§ PRELIMINARIES

An Autometrized algebra A is a system $(A,+,\leq,\ast )$ where

* $(A,+)$ is a binary commutative algebra with element 0,

* $\leq$ is antisymmetric, reflexive ordering on A,

* $\ast : A\times A\rightarrow A$ is a mapping satisfying the formal properties of distance, namely,

* $a\ast b\geq 0$ for all $a,b$ in A,equality,if and only if $a=b$,

* $a\ast b=b\ast a$ for all $a,b$ in A, and

* $a\ast b\leq a\ast c +c\ast b$ for all $a,b,c$ in A.

| A system $A=(A,+,\leq,\ast )$ of arity $(2,2,2)$ is called a Lattice ordered autometrized algebra, if and only if, A satisfies the following conditions.

* $(A,+,\leq)$ is a commutative lattice ordered semi-group with $'0'$, and

* $\ast$ is a metric operation on A.i.e, $\ast$ is a mapping from $A\times A$ into A satisfying the formal properties of distance, namely,

* $a\ast b\geq 0$ for all $a,b$ in A,equality,if and only if $a=b$,

* $a\ast b=b\ast a$ for all $a,b$ in A, and

* $a\ast b\leq a\ast c +c\ast b$ for all $a,b,c$ in A.

A lattice ordered autometrized algebra $A=(A,+,\leq,\ast )$of arity $(2,2,2)$ is called representable autometrized algebra, if and only if, A satisfies the following conditions:

* $A=(A,+,\leq,\ast)$ is semiregular autometrized algebra. Which means $a\in A$ and $a\geq 0$ implies $a\ast 0=a,$ and

* for every a in A, all the mappings $x\mapsto a+x,x\mapsto a\vee x,x\mapsto a\wedge x$ and $x\mapsto a\ast x$ are contractions (i.e.,if $\theta$ denotes any one of the operations $+,\wedge,\vee$and $\ast$,then,for each a in A,$(a\theta x)\ast (a\theta y)\leq x\ast y$ for all $x,y$ in A)

$\label{2.1}$ An Autometrized lattice ordered monoid (AL-monoids, for short) is an algebra $(A,+,\vee,\wedge,\ast,0)$ of arity $(2,2,2,2,0)$ where

* $(A,+,\vee,\wedge,0)$ is a commutative lattice ordered monoid.

* $a\ast(a\wedge b)+ b  = a\vee b.$

* The mappings $x\mapsto a+x,a\vee x, a\wedge x, a\ast x $ are contractions with respect to $\ast$ (A mapping $f:A \longrightarrow A $ is called  a contraction with respect $\ast \Leftrightarrow f(x)\ast f(y) \leq x\ast y $ where $\leq $ is ordering in $A$ induced by $(A,\vee,\wedge)).$

* $[a\ast (a\vee b)]\wedge [b\ast (a\vee b)] = 0.$

§ RESULTS

Let $A=(A,+,\leq,\ast,0)$ be an AL-monoid.By a conguerence relation on A, we mean an equivalence relation, having the substituition property with respect to all operations: $+,\vee, \wedge, \ast .$ Now we introduce the following:

A non empty set $I$ of an AL-monoid $A=(A,+,\leq, \ast,0)$ is called an$ '$ideal' of $A$ if and only if $I$ saitsfies

i. $a\in I$ and $b\in I$ imply $a+b\in I.$

ii. $a\in I,b\in A,$ and $b\leq a$ imply $b\in I.$

If in particular, $A$ is an l-group, then, $S$ will be a lattice ideal. From the definition, we immidiately have the following:

Any ideal $I$ of an AL-monoid is a convex sub-AL-monoid in the sense that

* It is closed under all operations

* $a,b \in I$ and $a\wedge b\leq x\leq a\vee b \implies x\in I.$

Let $a,b \in I$ then, $a\ast 0=a$ and $a\in I.$ Similarly $b\in I.$ $a\ast b\ast 0=a\ast b\leq (0\ast a +0\ast b)\leq a\ast b,$ follows that $a\ast b\in I.$

$(a\ast b)\ast 0=(a\ast b)\vee (0\ast (a\ast b))\leq (a\ast b)\vee (b\ast a)=a\ast b$ and hence $a\ast b\in I.$ Now,let $a\wedge b \leq x \leq a\vee b.$ Then,

\begin{eqnarray*}
x\ast 0=x\vee (0\ast x)\leq a\vee b\vee (0\ast (a\wedge b))&=& a\vee b\vee (0\ast a)\vee (0\ast b)\\&=&a\vee (0\ast a)\vee b\vee (0\ast b)\\ &=& (a\ast 0)\vee (b\ast 0)\leq a\ast 0+0\ast b\\ &=&(a\ast 0 +0\ast b)\ast 0.
\end{eqnarray*}

and since $a\ast 0+0\ast b \in I,$ it follows that $x\in I.$

An equivalence $\theta$ on an AL-monoid $A=(A,+,\leq, \ast, 0)$ is called $''$ a conguerence relation" if and only if $\theta$ satisfies

$c_1$ $a\equiv b(\theta),c\equiv d(\theta)$ immply $a+c\equiv b+d(\theta)$

$c_2$ $a\equiv b(\theta),c\equiv d(\theta)$ imply $a\ast c\equiv b\ast d(\theta)$ for all $a,b,c,d$ in $A,$ and

$c_[3$ $a\equiv b(\theta)$ and $ x\ast y\leq a\ast b$ imply that $x\equiv y(\theta)$ for $a,b,x,y$ in $A$.

Let $\theta$ be an equivalence relation on AL-monoid $A=(A,+,\leq,\ast,0)$. Then, $(C_1)$ and $c_3$ imply $c_2,c_4,$ and $c_5,$ where $c_4$ and $c_5$ are given below:

$c_4$ $a\equiv b(\theta),c\equiv d(\theta)$ immply $a\vee c\equiv b\vee d(\theta)$

$c_5$ $a\equiv b(\theta), c\equiv d(\theta)$ imply $a\wedge c\equiv b\wedge d(\theta)$ for all $a,b,c,d$ in $A,$

In order to show an equivalence relation $\theta$ on $A=(A,+,\leq,0)$ is a conguerence relation, it is enough to show that $\theta$ satisfies $c_1$ and $c_3.$

The ideals of any AL-monoid is correspond one to one to its conguerence relations.

Let $I$ be an ideal of $A$. Define $a\equiv b(I)$ iff $a\ast b \in I.$ Since $0\in I \implies a\equiv b(I), a\ast b=b\ast a \implies b\equiv a(I).$ And, $a\equiv b(I), b\equiv c(I) \implies a\ast b \And b\ast c \in I$ and $a\ast c\leq a\ast b+b\ast c$ follows that $a\equiv c(I).$ Hence, $a\equiv b(I)$ is an equivalence relation.
Let $a\equiv b(I)$ and $c\equiv d(I).$ Now,since $\ast$ is isometry then,$a\ast b +c\ast d=(a +c)\ast (b\ast d).$Hence, $a+c\equiv b+d(I).$ From those we observe that $a\ast b, c\ast d \in I,$ and since $I$ is an ideal $(d\ast c)+(a\ast b)$ and $(b\ast a)+(c\ast d)$ belongs to $I.$ Now, $c+(b\ast d)+(d\ast c) +(a\ast b)\geq c+(b\ast c)+(a\ast b)\geq c+(a\ast c)=a.$ Hence follows that $a\ast c)\ast (b\ast d)\leq (d\ast c)+(a\ast b);$ and since, $a+(a\ast c)+(c\ast d)+(b\ast a)\geq d+(a\ast d)+(b\ast d)\geq d+(b\ast d)\geq b$ follows that $(b\ast d)\ast (a\ast c)\leq (c\ast d)+(b\ast a).$ Hence, $(a\ast c)\ast (b\ast d)\leq ((d\ast c)+(a\ast b))\vee ((d\ast c)+(a\ast b))\vee ((c\ast d)+(b\ast d)).$ So that, $(a\ast c)\ast (b\ast d)\in I.$ Hence, $(a\vee c)\ast (b\ast d)=(a\ast b)\vee (c\ast d)$, since $\ast$ is isometry $(a\ast b)\vee (c\ast d)\in I,$ follows that $(a\vee c)\ast (b\vee d)\in I.$ Hence, $a\vee c\equiv b\vee d(I).$

Finally, $a\wedge c)\ast (b\wedge d)=(a\ast b)\wedge (c\ast d)\implies a\wedge c\equiv b\wedge d (I).$ Hence, $I$ defines a conguerence relation on $A.$

Conversely, Let $Q$ be a conguerence relation on $A$ and let $N$ be the set of all $x\equiv 0(Q).$ Obviously,$a,b \in N \implies a+b\in N.$ Let $x\in N$ and $y\leq x.$ Then, $x\equiv 0(Q) \implies x\ast 0\equiv 0(Q)\implies y\vee x\equiv 0(Q)\implies y\equiv 0(Q)\implies y\in N$. hence $N$ is an Ideal.

Now, let $Q'$ be a conguerence relation obtained by defining $a\equiv b(Q')$ iff $a\ast b\in N.$ Then, $a\equiv b(Q')$ iff $a\ast b \in N$ iff $a\ast b\equiv 0(Q)$ iff $a\vee b\equiv a\wedge b(Q)$ iff $a\equiv b(Q).$ Hence, there is one-to-one correspondence between conguerence relations on $A$ and its ideals.

The ideals of any DRl-semigroup is correspond one to one  to its conguerence relations

The ideals of any Boolean algebra is correspond one to one  to its conguerence relations

The conguerence relations on a commuative l-group are the partitions of $A$ in to the cosets of its diffrent l-ideals

1 The intersection of any non empty collection of ideals of $A$ is an ideal of $A.$

2 For any non-empty subset of $S$ of $A$, the set $\lbrace x\in A| x\leq (a_1+a_2+...+a_n)$ for $a_1,a_2,..,a_n\in S \rbrace$ is the smallest ideal containing $S$(i.e., the ideal generated by $S.$

3 The principal ideal $<a>$ generated by $a,$ that is the smallest ideal containing $a$ is $\lbrace x\in A|x\leq ma\rbrace$ for some prositive intiger $m.$

4 The join of any collection of ideals $\lbrace I_{\alpha}\rbrace_{\alpha\in \delta}$ is the ideal generated by $\bigcup_{\alpha \in \delta}I_{\alpha}$

5 If $I$ and $J$ are ideals of $A$, then, $I\vee J=\lbrace a\in A|a\ast x+y $ for some $x\in I$ and $y\in J \rbrace$ is the join of $I$ and $J.$

6 The join of any two principal ideals in $A$ is a principal ideal. In fact, $<a>\vee <b>=<a+b>$

7 An ideal $I$ of $A$ is compact if and only if $I$ is principal ideal.

The set of all ideals of $A$ is a complete algebraic lattice, ordered by a set inclusion.

The ideal lattice of any commutative DRl-semigroup, and consequently the ideal lattice of any commutative l-group, the ideal lattice of a Boolean algebra are all complete algebraic lattices.

i The set of all conguerence relations of $A,$ and

ii the lattice of ideals of $A$ is isomorphic with the lattice of conguerence relations of $A.$

§.§ Prime ideals in AL-monoids

Let $A$ be an AL-monoid. Let M be an ideal of $A$. $M$ is called Maximal ideal if and ony if whenever $J$ is an ideal such that $M\subset J \subset A,$ then either $M=J$ or $J=A.$

Radical of $A$ is the set $R(A)=\wedge \lbrace M| M$ is a maximal ideal of $A\rbrace.$

If $M$ is Maximal idealof A, then it is prime ideal.

Suppose $M$ is Maximal ideal. We know that regular ideal is prime ideal. Suppose $M=\wedge_{\alpha \in \Gamma}J_{\alpha}.$ Therefore, $M\subset J_{\alpha,\alpha \in \Gamma}.$ Since $M$ is Maximal ideal; $M=J_{\alpha},\alpha\in \Gamma.$ Thus $M$ is regular ideal. Hence, $M$ is prime ideal.

Let $A=\lbrace 0,a,b,c\rbrace$ with usual ordering. Define $'\ast'$ and $'+'$ as following table:

$\ast$     0     a     b       c

0     0    a     b    c

a     a     0     b    c

b     b     b    0     c

c     c     c     c     0

$+$     0     a     b       c

0     0    a     b    c

a     a     a     b    c

b     b     b    b     c

c     c     c     c     c

Hence, $\lbrace 0,a\rbrace$ and $\lbrace 0,a,b\rbrace$ are AL-monoid $0\ast (0\wedge a)+a=a.$ and $(0\ast (0\wedge a))\wedge (a\ast (0\wedge a))=0$. Hence it is AL-monoid. hence, $\lbrace 0,a,b\rbrace$ is a maximalideal of $A.$ But, $\lbrace 0,a\rbrace$ is prime but, not maximal.

Let $P$ be a prime ideal of $A$. Let $M_{1},M_2,...,M_K$ are maximal ideals. Assume that $\wedge_{i=1}^{k} M_{i}=\lbrace 0\rbrace.$ Then, $P=M_{i}$ for some $i.$

We know that $\wedge_{i=1}^{k} M_{i}=\lbrace 0\rbrace.\subset P.$ for some $i$ by induction on $k$. Since $M_{i}$ is Maximal; and hence, $P=M_{i}$ for some $i.$

An ideal $I$ of $A$ is called a strong ideal if

* $a\in I$ iff $a\ast I=I$. and

* $a\ast I=b\ast I $ iff $a\ast b\in I$ for $a,b\in I.$

Here $a\in I$ iff $a\ast I=I$ and $a\ast I=b\ast I$ implies $a\ast b\in I$ are not true in any ideal of $A.$ Consider the following example

Let $A=\lbrace 0,a,b,c,d,e\rbrace$ with usual ordering. Define $\ast $ and $+$ as follows.
$\lbrace 0,a,b\rbrace$ is an idealof $A.$ So, $0\ast I=I, a\ast I=\lbrace a,0\rbrace$ and $b\ast I=I$, we see that $a\in I$, but, $a\ast I\neq I.$ Then $I$ is not strong ideal.

+     0     a    b    c    d    e

0   0     a   b    c   d   e

a   a   b   b   e   d   e

b   b   b   b   e   e   e

c   c   e   e   e   e   e

d   d   e   e   e   e   e

e   e   e   e   e   e   e

$\ast $     0     a    b    c    d    e

0   0     a    b   c   d   e

a   a   0   a   c   c   c

b   b   a   0   c   c   c

c   c   c   c   0   a   a

d   d   c   c   a   0   a

e   e   c   c   a   a   e0

Every ideal of $A$ is strong ideal.

* To show, $a\in I $ iff $ a\ast I=I$. Suppose $a\in I.$ To show $a\ast I=I,$ Let $a\ast x \in a\ast I$ where $x\in I.$ Since $a,x\in I$
and $A$ is AL-monoid, $a\ast x=(a\ast x)\ast 0\leq (a\ast 0+ x\ast 0)=(a\ast 0+x\ast 0)\ast 0.$ Therefore, $a\ast x\in I.$ Then, $a\ast I\subset I.$

$(leftlongarrow)$, Let $x\in I$. So, $a\ast x\in I.$  We know that $a, a\ast x\in I.$ Clearly,$a\ast (a\ast x)\in a\ast I.a\ast (a\ast x)=(a\ast a)\ast x=0\ast x.$ $A$ is AL-monoid implies $a\ast (a\ast x)=x.$ As a result, $x\in I.$ Therefore, $I\subset a\ast I.$ Hence, $a\ast I=I.$ Now, to show $a\ast I=I$ iff $a\in I.$ Suppose $a\ast I=I.$ $x=a\ast y, y\in I.$ Therefore, $y, a\ast y\in I.$ and $(a\ast y)\ast y\in I$. Hence, $a\in I.$

* To show $a\ast I=b\ast I$iff $a\ast b\in I.$
Let $a\ast b\in I$. clearly $(a\ast b)\ast I=I$ by (1), so $(a\ast b)\ast x=y,$ for some $x,y\in I.$ Therefore, $a\ast ((a\ast b)\ast x)=a\ast y. (a\ast (a\ast b))\ast x=a\ast y\\ ((a\ast a)\ast (b\ast x)=a\ast y\\b\ast x=a\ast y.$ Hence, $b\ast I=a\ast I.$

Conversely, Suppose $a\ast I=b\ast I.$ Therefore, $a\ast x=b\ast y,$ for some $x,y\in I.$ Therefore, $a\ast (a\ast x)=a\ast (b\ast y)$ implies $x=(a\ast b)\ast y.$ Therefore, $(a\ast b)\ast I=I.$ Hence, $a\ast b\in I$ ny (1).

A srong ideal of $A$ is called a distant ideal  if there exists strong ideal $J$ of $A$ such that $I\ast J=A$ and $I\wedge J=\lbrace 0\rbrace.$ The set of all distant ideals of $A$ is denoted by $Dis(A).$ Hence, $\lbrace 0\rbrace, A\in Dis(A).$

Every Ideal of AL-monoid is Strong ideal.

AL-monoid $A$ is directly indecompasable iff its distant ideals are $\lbrace 0\rbrace$ and $A$ only.

§ SPECTRA OF AL-MONOIDS

Spec(A) is the set of all prime ideals.

Let $A$ be AL-monoid and let $P,Q\in Spec(A)$ such that P||Q, then $P$ and $Q$ in $Spec(A)$ disjoint neighbourhoods.

Let $P,Q\in Spec(A), P\nsubseteq Q$ and $Q\nsubseteq P.$ Then, there exist $0<a\in A, 0<b\in A$ such that $a\in P\ Q$ and $b\in Q\ P$. Let $a=a\ast (a\wedge b)$ and $b\ast (a\wedge b).$ Let us show that $u\neq Q$ and $v\neq p.$ let, for example, $u\in Q.$ We know that $a=(a\wedge b)+(a\ast a\wedge b)=a\wedge b+u,,$ and since $a\wedge b\in Q,$ we have $a\in Q,$ a contradiction. Hence, $P\in S(U), Q\in S(V)$ and $u\wedge v=0.$ Thus, $S(u)\wedge S(V)=S(U\wedge V)=\empty.$

If $X\subset Spec(A),$ then the topology of $X$ induced by the spectra topology of $Spec(A)$ will be called the spectral topology on $X.$

If $X\subset Spec(A)$ is a set of pairwise non-comparable prime ideals, then, the spectral topology of $X$ is $T_{2}-$ Topology.

If $X\subset Spec(A)$ and $M\subset A,$ put $S_{x}(M)=S(M)\wedge x$ denoted by $m(A)$ the set of minimal and by $M(A)$ the set of all maximal prime ideals of AL-monoids.

The spectral topology of $m(A)$ is a $T_{2}-$ Topology and the sets $S_{m(A)}(a)=\lbrace P\in m(A); a\notin P\rbrace, a\in A,$ forms a basis of the space $m(A)$ is composed by closed subsets.

Obviously, the sets $S_{m}(a),$ where $a\in A,$ form a basis of spectral topoplogy of $m(A).$ Let $a\in A$ and let $P$ be a minimal prime ideal in $A,$ then $a\in P$ or $a^{1}\nsubseteq P.$ Hence, $S_{m(A)}(a)\wedge S_{m(A)}(a^{1}=\empty$ and $S_{m(A)}\vee S_{m(A)}(a^{1})=m(A).$ Therefore, since $S_{m}(a^{1})$ is open, $S_{m(A)}$ is closed.

Let, $0\neq a\in A$ and $V(a)$ is the ste of all values of $a,$ i.e., the set of all ideals maximals with respect to the property of not containinig $a.$ (For, $a=0,$ put $V(a)=\empty).$ Let $P\in S(a)$ then, the set of all ideals in $A$ containing $P$ is linearly ordered and there are ideals in $Val(a)$ that conatining $P$. Hence, there is exactly on $M_{p}\in Val(a)$ such that $P\subset M_{P}.$

Let us denote by $\mu_{a}: S(a)\rightarrow Val(a)$ the mapping such that $\mu: P\rightarrow M_{p}$.

The mapping $\mu_{a}$ is continuous.

Let $a\in A, P\in S(A)$ and let $U$ be a neighbourhood of $M_{P}$ in Val(a). We can suppose that $U=S(b)\wedge Val(a)$ for some $b\in A.$ If $Q\in Val(a)\ S(b),$ then, we can choose a neighbourhood $U_{Q}$ of $Q$ and neighbourhood $V_{Q}$ of $M_{P}$ such that $U_{Q}\wedge V_{Q}=\empty.$

It is evidient that all $U_{Q},$ where $Q$ runs over $Val(a)\ S(b).$ Since $S(a)$ is compact and $S(a)\ S(b)$ is closed in $S(a), S(a)\ S(b)$ is compact, too. Hence there exist $n\in \mathbb{N}$ and $Q_{1}, Q_{2},...,Q_{n}\in S(a)\ S(b)$ such that $S9a)\ S(b)\subset U_{Q_{1}}\vee...\vee U_{Q_{n}}.$ Let us denote $C=S(a)\ (U_{Q}\vee ... \vee Q_{n}).$ We have $V_{Q_{1}\wedge ...\wedge V_{Q_{n}}}\subset C,$ therefore $C$ is a neighbourhood of $M_{P}$ which is closed in $S(a),$ and $C\wedge Val(a)\subset U.$ Therefore, $C\subset \mu^{-1}(C\wedge Val(a))\subset \mu^{-1}(U).$ Moreover, $C,$ which is a neighbourhood of $M_{P},$ is also a neighbourhood of $P.$

$V(a)$ is a $T_{2}-$space. Further, $Val(a)$ is the image of the compact set $S(a)$ in the mapping $\mu_{a}$ which is, by above proposition, continous, hence $Val(a)$ is also compact.

The following theorem is an immidiate consequence.

The space $M(A)$ of all its maximal prime ideals is a $T_{2}-$ space. If there exists $b\in A$ such that $I(b)=A$ then, $M(A)$ is compact.

§ HOMOMORPHISM OF IDEALS IN AL-MONOIDS.

Let $A$ be an AL-monoids. Let $M$be an ideal of $A.$ Let $A/M=\lbrace a\ast M| a\in A\rbrace.$ For any $a\ast M, be\ast M \in A/M,$ define the operations:

\begin{eqnarray*}
(a\ast M)+(b\ast M)=(a+b)\ast M\\
(a\ast M)\ast (b\ast M)=(a+b)\ast M\\
(a\ast M)\leq (b\ast M) iff (a\leq b)
\end{eqnarray*}
Then, $(A/M, +,\leq,\ast,0)$ is called Quotient AL-monoid of $A$ by ideal $M.$

$(A/M,+,\leq,\ast,0)$ is Autometrized algebra from (Theorem 4.2 of [6]), we wants to show the condition $a\ast (a\wedge b)=a\vee b, f(a)\ast f(b)\leq a\ast b,(a\ast(a\vee b))\wedge (b\ast (a\vee b)=0)$....

Let $N$ be an ideal of $A.$Let $B$ be a subalgebra of $A.$ Define $B\ast N={a\in A| a=b\ast n for b\in B and n\in N}$ such that  for any $a\ast N,b\ast N\in B\ast N; (a\ast N)+(b\ast N)=(a+b)\ast N$ and $(a\ast N)\ast (b\ast N)=(a\ast b)\ast N$. Then, $B\ast N$ is a subalgebra of $A$.

(First Isomorphism Theorem)
Let $f:A\rightarrow B$ be a homomorphism then $A/kerf\cong Imf.$

[6]

Let $N$ be an ideal of $A.$ Let $B$ is a subalgebra of $A.$ Then, $B\ast N/N\cong B/(B\wedge N).$

Let $A$ be am AL-monoid then, $A$ is achain if and only if $a\wedge b=0$ implies $a=0$ or $b=0.$

Suppose $A$ is a chain and $a,b\in A.$ Then, either $a\leq b$ or $b\leq a.$ this means either $a\wedge b=a$ or $b\wedge a=b.$ Suppose $a\wedge b=0.$ Hence, either $a=0$ or $b=0.$ Conversely, suppose $a\wedge b=0$ it implies either $a=0$ or $b=0.$ To show that $A$ is achain, Let $a,b\in A.$ Since $A$ is an Al-monoid  $(a\ast (a\vee b))\wedge (b\ast (a\vee b)=0)$ implies $a\ast (a\vee b)=0$ or $b\ast (a\vee b)=0.$
as aresult, either $a=a\vee b$ or $b=a\vee b.$ Therefore, either $ b\leq a$ or $a\leq b.$ Hence $A$ is a chain.

Let $A$ be AL-monoid and M be strong ideal of $A.$ Then, $A/M$ is a chain AL-monoid iff $M$ is prime.

Suppose $A/M$ is a chain. Let $a,b\in A.$ Suppose $a\wedge b=0.$ Since $M$ is an ideal, $a\wedge b=0\in M.$ Since $M$ is strong ideal, implies that $(a\wedge b)\ast M=M$ implies $(a\ast M)\wedge (b\ast M)=M.$ Since $A/M$ is achain, either $a\ast M\leq b\ast M$ or $b\ast M\leq a\ast M.$ That implies that $(a\ast M)\wedge (b\ast M)=a\ast M$ or $(a\ast M)\wedge (b\ast M)=b\ast M.$ Therefore, $a\ast M=M$ or $b\ast M=M.$ As aresult, either $a\in M$ or $b\in M.$ Implies $M$ is a prime ideal.

Conversely, Suppose $M$ is prime ideal. To show that $A/M$ is a chain. Let $a\ast M, b\ast M\in A/M$ where $a,b\in  A$. Since $A$ is AL-monoid,$(a\ast (a\vee b))\wedge (b\ast (a\vee b))=0.$ Since $M$ is prime ideal; either $a\ast (a\vee b)\in M$ or $b\ast (a\vee b)\in M.$ Therefore, $(a\ast (a\vee b))\ast M=M$ or $(b\ast (a\vee b))=M.$ Therefore, $(a\ast (a\vee b))\ast M=M$ or $(b\ast (a\vee b))\ast M=M.) $ This implies tha $(a\ast M)\ast ((a\vee b)\ast M)=M$ or $(b\ast M)\ast ((a\vee b)\ast M)=M.$ This implies that $a\ast M=(a\vee b)\ast M$ or $b\ast M=(a\vee b)\ast M$ implies $a\ast M=(a\ast M)\vee (b\ast M)$ or $b\ast M=(a\ast M)\vee (b\ast M).$ This gives, $b\ast M\leq a\ast M$ or $a\ast M\leq b\ast M.$ Hence, $A/M$ is a chain.

Let $A$ be an AL-monoid. If $P$ is strong prime ideal, then $\lbrace I\in I(A)|P\subset I\rbrace $is a chain under inclusion.

Suppose $P \subseteq J$ and $P\subseteq K$ such that $J\nsubseteq K$ and $J\nsubseteq K.$ That is there exists $a\in J$ and $a\notin K$ and there exist $b\in K$ and $b\notin K.$ Clearly, $0\ast a\in J$ and $b\ast 0\in K.$
Now, consider $a\ast 0)\ast p$ and $(b\ast o)\ast p. Since $A/P is a chain; either $(a\ast 0)\ast p\leq (b\ast 0)\ast p.$ or $(b\ast 0)\ast p$ which implies that $a\ast 0\leq b\ast 0$ or$(b \ast 0)\\leq (a\ast 0)$ . Since $a\in J$ and $b\in K,$ implies that $b\in J$ and $a\in K.$ This is a contradiction. Hence, $\lbrace I\in I(A)|P\subseteq I\rbrace$ is a chain under inclusion.

Let $A$ and $B$ be an AL-monoids and let $f: A\longrightarrow B$ be a map. Then, $f$ is said to be a homomorphism from $A$ to $B$ if and only if

i. $f(a+b)=f(a)+f(b)$

ii. $f(a\ast b)=f(a)\ast f(b)$

iii. $f(a\vee b)=f(a)\vee f(b)$

iv. $f(a\wedge b)=f(a)\wedge f(b)$

v. $f(0)=0.$

A homomorphism $f:A\longrightarrow B$ is called

a. an epimorphism if and only if $f$ is onto.

b. a monomorphism(embedding) if and only if $f$ is 1-1.

c. an Isomorphism if and only if it is bijection.

Let $A$ be AL-monoids and $\alpha:A\longrightarrow B$ be a homomorphism. Then, $ker(\alpha)$ is a prime ideal if and only if $Im(\alpha)$ is a chain AL-monoid.

From Third Isomorphism Theorem we have $A/ker(\alpha)\cong Im(\alpha).$ Clearly, $ker(\alpha)$ is strong ideal. If $ker(\alpha)$ is prime ideal, then by above theorem $A/ker(\alpha)$ is a chain AL-monoid. Hence, $Im(\alpha)$ is a chain AL-monoid.

Conversely, Suppose $Im(\alpha)$ is a chain AL-monoid.$A/ker(alpha)$ is a chain AL-monoid. Thus $ker(\alpha)$ is a prime ideal.

If $A$ is AL-monoid, the following are equivalent:

(i) $A$ is nilradical.

(ii) there is a family ${M_{i}}_{i\in I}$ of maximalideals of $A$ with $\wedge_{i\in I}M_{i}={0},$

(iii) $A$is a subdirect prodcut of simple chain AL-monoid.

§ DIRECT PRODUCT, SUBDIRECT PRODUCT, AND REPRESENTABLITY OF AL-MONOIDS.

Let $A=\prod_{i\in A}A_{i}={a=(a(1),a(2),...)|a(i)\in A_{i}}$. Define $a=(a_{i})_{i\in I}, b=(b_{i})_{i\in I}; a+b=(a_{i}+b_{i})_{i\in I},a\ast b=(a_{i}\ast b_{i})_{i\in I}, a\leq b $ iff $a_{i}\leq b_{i},$ for all $i\in I.$ then $A=\prod_{i\in I}A_{i}$ is AL-monoid.

Let ${A_{i}}_{i\in I}$ be a family of AL-monoids. Let $A=A_{1}\times A_{2} \times ...\times A_{k}.$ Then $A$ is AL-monoid.

Let $a_{i}: A\longrightarrow A_{i}$ be a map for $i\in I.$ Define a map $\alpha: A\longrightarrow \prod_{i\in I}A_{i}$ by $\alpha(a)=(\alpha_{1}(a),\alpha_{2}(a),...).$ That is $\alpha(a)(i)=\alpha_{i}(a)$ for $i\in I.$

Let $A$ be AL-monoid and ${A_{i}}_{i\in I}$ be a family of AL-monoids. If each $\alpha_{i}:A\longrightarrow A_{i}$ is a homomorphism, then the map $\alpha:A\longrightarrow \prod_{i\in I}A_{i}$ is also a homomorphism and $ker \alpha=\wedge_{i\in I}ker(\alpha_{i}).$

Suppose $\alpha_{i}=A\longrightarrow A_{i}$ is a homomorphism. To show it is homomorphism. Let $a_{1},a_{2}\in A.$ Now,

\begin{eqnarray*} \alpha(a_{1}+a_{2})_{i}&=&\alpha_{i}(a_{1}+a_{2}\\&=& \alpha_{i}(a_{1}+a_{2})\\&=& \alpha(a_{1})(i)+\alpha(2)(i)\\&=&(\alpha(a_{1})+\alpha(a_{2})(i).
\end{eqnarray*}
Therefore, $\alpha(a_{1}+a_{2})=\alpha(a_{1})+\alpha_{2}).$

\begin{eqnarray*}\alpha(a_{1}\ast a_{2})(i)&=&\alpha_{i}(a_{1}\ast a_{2})\\&=& \alpha_{i}(a_{1}\ast \alpha_{i}(a_{2})\\&=&\alpha(a_{1}(i)\ast \alpha(a_{2})(i)\\&=&(\alpha(a_{1}\ast \alpha(a_{2})(i).\end{eqnarray*}

iii. Suppose $a\leq b.$ since $\alpha_{i}$nis homomorphism. it implies $\alpha_{i}\leq \alpha_{i}(b)\implies \alpha(a)(i)\leq \alpha(b)(i)\implies \alpha(a)\leq \alpha(b).$ Hence $\alpha$ is a homomorphism.

Now, we shal prove that $ker\alpha =\wedge_{i\in I}$

\begin{eqnarray*} ker(\alpha)&=& \lbrace a\in A|\alpha(a)=0\rbrace\\&=& a\in A|\alpha(a)(i)=0(i)\rbrace\\&=&\lbrace a\in A|\alpha_{i}(a)=0_{i}\rbrace=\lbrace a\in A|a\in ker(a_{i},\forall i\in I\rbrace\\&=& \lbrace a\in A|a\in \wedge_{i\in I}ker (\alpha_{i}\rbrace\\&=& \wedge_{i\in I}ker\alpha_{i}.\end{eqnarray*}

Let A be AL-monoid then

* A is weak chain

* If Prime strong ideal, then $ \lbrace I\in \mathbb{I(A)}|P\subset \mathbb{I}$  is a chain under inclusion.

* If $ \alpha $ is homomorphism, $ker(\alpha)$ is a aprime ideal iff $im(\alpha)$ a chain AL-monoid.

* There is a family $\lbrace p_{i}\rbrace_{i\in I}$ of prime ideals of $A$ with $\wedge_{i\in I}p_{i}=\lbrace 0\rbrace.$ iff $A$ is a subdirect product of a chain AL-monoid.

Let $A$ be AL-monoid. Then, the following are equivalent:

* $A$ is representable.

* : $A$ is subdirect product of a AL-monoid.

* there exist afamily $\lbrace p_{i}\rbrace_{i\in I}$ of prime ideals of $A$ with $\wedge_{i\in I}p_{i}={0}$

* Every subdirect irreducible order reversing homomorphic image of $A$ is a chain.

Let $A_{1},A_{2},...,A_{k}$ are AL-monoids and let $A=A_{1}\times A_{2}\times ...\times A_{k}.$ Let $\mathbb{A_{i}}$ is the set of ideals of $A_{i}$ for $i=1,2,...,k.$ If $I_{i}\in \mathbb{I(A_{i})},$ then $I=I_{1}\times ...\times I_{k}$ is an strong ideal of $A.$ conversely, if $I_{1}\times I_{2}\times...\times I_{k}$ is an ideal of $A$, then for $i=1,2,...,k ;I_{i}\prod_{i}(I)$ is an ideal of $A_{i}.$

Let $A_{1},A_{2}$ are AL-monoids. Then for $ker(\pi_{1}),ker(\pi_{2})$ are distant ideals. That is, $ker \pi_{1}, ker \pi_{2}=A_{1}\times A_{2}$ and $ker{\pi_{1}}\wedge \ker{\pi_{2}}={0}.$

i. clearly, $ker\pi_{1}\ast ker\pi_{2}\subset A_{1}\times A_{2}.$
let $(x,y)\in A_{1}\times A_{2}.$ This shows $x\in A_{1},y\in A_{2}.$ we know that $(0,y)\in ker \pi_{1}$ and $(x,0)\in ker \pi_{2};$ hence $(0,y)\ast (x,0)\in ker \pi_{1}\times ker\pi_{2}.$ Since $A$ is AL-monoid; $(0\ast x,y\ast 0)=(x,y).$ Therefore, $(x,y)\in ker(\pi_{1}\ast ker\pi_{2}).$ Whence, $A_{1}\times A_{2}\subset ker\pi_{1}\times ker \pi_{2}.$ Thus, $A_{1}\times A_{2}= ker\pi_{1}\times ker \pi_{2}.$

ii $\lbrace 0\rbrace\in ker\pi_{1},\ker\pi_{2}.$ Therefore, $\lbrace 0\rbrace \in ker\pi_{1}\wedge ker \pi_{2}.$ Conversely, let $a=(a_{1},a_{2})\in ker\pi_{1}\wedge ker\pi_{2}.$ so, $a\in ker\pi_{1}$  and $a\in ker\pi_{2}.$ So, $a\in ker\pi_{1}$ and $a\in ker\pi_{2}.$ It implies that $\pi_{1}(a)=\pi_{1}(a_{1},a_{2})=a_{1}=0$ and  $\pi_{2}(a)=\pi_{2}(a_{1},a_{2})=a_{2}=0.$

Let $A$ be AL-monoids. Let $I,J$ be distant ideals of $A.$ Then, $A\cong A/I \times A/J$

Let $I,J$ and $I\wedge J$ are strong ideals. Define a map $f:A/I \times A/J,$ by $f(a)=(a\ast I,a\ast J).$

Let $a,b\in A.$ Suppose $a=b,$ then $a\ast I=b\ast I$ and $a\ast J=b\ast J$ This implies $(a\ast I, b\ast J)=(b\ast I, b\ast J)$ implies $f(a)=f(b).$ Hence, $f$ is well-defined.

To show $f$ is Homomorphism: let $a,b\in A$

\begin{eqnarray*} f(a+b)&=&((a+b)\ast I,(a\ast b)\ast J)\\&=&((a\ast I)+(b\ast I),(a\ast J)\ast (b\ast J))\\&=&(a\ast I,a\ast J)+(b\ast I,b\ast J)\\&=&f(a)+f(b).
\end{eqnarray*}

\begin{eqnarray*}
f(a\ast b)=((a\ast b)\ast I,(a\ast b)\ast J)\\&=&((a\ast I)\ast (b\ast I),(a\ast j)\ast (b\ast J))\\&=&(a\ast I,a\ast J)\ast (b\ast I, b\ast J)\\&=& f(a)\ast f(b).
\end{eqnarray*}

iii. Suppose $a\leq b$. Therefore, $a\ast I\leq b\ast I$  and $a\ast J\leq b\ast J.$ By definition of direct product, $(a\ast I, a\ast J)\leq(b\ast I,b\ast J).$ This implies $f(a)\leq f(b).$ Hence, $f$ is homomorphism.
To show $f$ is onto.

Let $(x\ast I, y\ast J)\in A/I \ast A/J.$ Therefore, $x,y\in A=I\ast J,$ there exists $a_{1},a_{2}\in I$ and $b_{1},b_{2}\in J$ such that $x=a_{1}\ast b_{1}, y=a_{2}\ast b_{2}.$ Then,
\begin{eqnarray*}
(b_{1}\ast a_{2})\ast I&=&(b\ast I)\ast (a_{2}\ast I)\\&=& (b_{1}\ast I)\ast I\\&=& (b_{1}\ast I)\ast (0\ast I)\\&=& (b_{1}\ast 0)\ast I\\&=& b_{1}\ast I.
\end{eqnarray*}
\begin{eqnarray*}
(a_{1}\ast b_{1})\ast I&=&(a\ast I)\ast (b_{1}\ast I)\\&=& I\ast (b_{1}\ast I)\\&=& (0\ast I)\ast (b_{1}\ast I)\\&=& (0\ast b_{1})\ast I\\&=& b_{1}\ast I.
\end{eqnarray*}
Thuse $b_{1}\ast a_{2})\ast I=b_{1}\ast I.$

* We wants to show, $kerf=I\wedge J$ Thus, ${A/I}\wedge j\cong A/I \times A/J$ by first isomorphism theorem. Since $I\wedge J=\lbrace 0\rbrace ,A/ {\lbrace 0\rbrace} \cong A/I \times A/J.$ Hence $A\cong A/I\times A/ J.$

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:Visualization of common abstraction of Boolean algebra and commutative lattice ordered groups.