# Limiting Superluminal Electron and Neutrino Velocities using the 2010 Crab
Nebula Flare and the IceCube PeV Neutrino Events

Floyd W. Stecker  Astrophysics Science Division
NASA Goddard Space Flight Center
Greenbelt, MD 20771, USA<EMAIL_ADDRESS>
###### Abstract

The observation of two PeV-scale neutrino events reported by Ice Cube allows
one to place constraints on Lorentz invariance violation (LIV) in the neutrino
sector. After first arguing that at least one of the PeV IceCube events was of
extragalactic origin, I derive an upper limit for the difference between
putative superluminal neutrino and electron velocities of $\leq\sim 5.6\times
10^{-19}$ in units where $c=1$, confirming that the observed PeV neutrinos
could have reached Earth from extragalactic sources. I further derive a new
constraint on the superluminal electron velocity, obtained from the
observation of synchrotron radiation from the Crab Nebula flare of September,
2010. The inference that the $>$ 1 GeV $\gamma$-rays from synchrotron emission
in the flare were produced by electrons of energy up to $\sim 5.1$ PeV
indicates the non-occurrence of vacuum Ćerenkov radiation by these electrons.
This implies a new, strong constraint on superluminal electron velocities
$\delta_{e}\leq\sim 5\times 10^{-21}$. It immediately follows that one then
obtains an upper limit on the superluminal neutrino velocity alone of
$\delta_{\nu}\leq\sim 5.6\times 10^{-19}$, many orders of magnitude better
than the time-of-flight constraint from the SN1987A neutrino burst. However,
if the electrons are subluminal the constraint on $|\delta_{e}|\leq\sim
8\times 10^{-17}$, obtained from the Crab Nebula $\gamma$-ray spectrum, places
a weaker constraint on superluminal neutrino velocity of $\delta_{\nu}\leq\sim
8\times 10^{-17}$.

###### keywords:

Lorentz invariance; neutrino; electron

††journal: Astroparticle Physics

##  1 Introduction

Tests of Lorentz invariance violation (LIV) probe physics beyond the standard
model, more specifically to probe the structure of space-time on the Planck
scale. The Planck energy is the natural scale where it is expected that
gravity may unify with the other three fundamental forces. While it is not
possible to directly investigate space-time physics at the Planck energy of
$\sim 10^{19}\,$GeV, many lower energy effects have been predicted to arise
from the violation of Lorentz invariance. The subject of investigating LIV has
therefore generated much interest in the particle physics community.

Various astrophysical observations using X-ray, $\gamma$-ray and cosmic ray
data have been used to place limits on Lorentz violation (see e.g., Ref. [1]).
Diffuse fluxes of high energy neutrinos produced both in our galaxy and in
intergalactic space have long been predicted, but it was noted that at
energies below several hundred GeV their fluxes would be swamped by the
neutrinos produced by cosmic ray interactions in the atmosphere [2]. Until
recently there were no reported detections of cosmic-ray neutrinos. However,
the IceCube collaboration has now reported the first observation of two PeV
energy neutrinos, giving a signal $\sim 3\sigma$ above the atmospheric
background [3]. It is most likely that these neutrinos are evidence of a new
neutrino flux component above that expected from atmospheric cosmic ray
secondaries and that such a component would be of extraterrestrial origin [3,
4].

While more data are desirable, there are four indications that all or most of
these neutrinos are extragalactic in origin: (1) The IceCube collaboration has
reported 18 more events produced by neutrinos with energies above 0.1 PeV,
$\sim$ 4 $\sigma$ above the expected atmospheric background. The distribution
in arrival direction of all 20 events is consistent with isotropy; there is no
marked enhancement in the galactic plane [5, 6], although it has been argued
that a subset of these events might be of galactic origin [7, 8]. (2) The
implied peak in the energy spectrum of these neutrinos may be indicative of
photopion production followed by pion decay [9, 10, 11] such as expected in
AGN cores [12, 13] and GRBs [14, 15]. (3) The diffuse galactic neutrino flux
[2] is expected to be well below that implied by the implied Ice Cube flux.
(4) At least one of the $\sim$ PeV neutrinos came from a direction off of the
galactic plane. Even the existence of this one extragalactic neutrino event is
enough to place new constraints on LIV.

Limits on superluminal neutrino velocity $\delta_{\nu}=v_{\nu}-1\leq\sim
10^{-5}$ have been obtained directly from terrestrial time-of-flight
measurements [16, 17]. In addition, time-of-flight constraints from the
detection of a multi-MeV neutrino burst from supernova 1987A [18, 19] yielded
the constraint $\delta_{\nu}\leq 2\times 10^{-9}$ [20]. A comparison of
atmospheric neutrino spectra with theoretical spectra expected from the change
in the pion decay rate if neutrinos are superluminal has yielded the indirect
constraint $\delta_{\nu}\leq\cal{O}$(10-13) [21]. New IceCube observations,
together with new constraints on superluminal electron velocities derived from
$\gamma$-ray observations of the September, 2010 Crab Nebula flare using the
Large Area Telescope on the Fermi Gamma Ray Space Telescope, now allow one to
place stronger constraints on LIV in both the electron and the neutrino
sectors.

##  2 Neutrino Energy Loss

Calliday and Kostelecký [22] proposed an effective field theory framework for
quantifying and cataloging the empirical effects of small violations of CPT
and Lorentz invariance known as the standard model extension (SME). The SME is
based on the introduction of small Lorentz and CPT violating perturbations in
the individual free particle Lagranians. Coleman and Glashow [23] have
presented a simplified formalism, assuming rotational invariance, wherein
particle interactions that violate Lorentz invariance can be modified in terms
of the maximum attainable velocities (MAVs) of the various particles involved.
Thus superluminal particle velocities can be directly related to Lorentz
invariance violation. Cohen and Glashow [24] point out that if
$\delta_{\nu}>0$, three energy loss processes that are otherwise kinematically
forbidden, would be allowed even in vacuo, viz. (a) vacuum neutrino Ćerenkov
radiation $(\nu\to\nu\,\gamma)$, (b) ”neutrino splitting”
$(\nu\to\nu\,\nu\,\bar{\nu})$ and (c) vacuum electron-positron emission
$(\nu\to\nu\,e^{+}\,e^{-})$ (see also [14]). Of these processes, electron-
positron pair emission is the most dominant, leading to the fastest energy
loss. The MAVs of the various neutrino flavors are found to be within a factor
of less than $10^{-20}$ of each other, given the results from neutrino
oscillation experiments [23, 24].

We now define $\delta_{\nu}=v_{\nu}-1$, $\delta_{e}=v_{e}-1$, where c = 1 is
the low energy velocity of light in vacuo and the $v$’s are the MAVs of the
$\nu$’s and electrons. (N.B.: The definition of $\delta$ used here is half
that used in Refs. [23] and [24], but is consistent with that used in Ref.
[25].) For $\delta_{\nu}\geq\delta_{e}\geq 0$ and defining $\delta_{\nu
e}\equiv\delta_{\nu}-\delta_{e}$, the process $\nu\to\nu\,e^{+}\,e^{-}$ is
kinematically allowed provided that [23, 25]

$E_{\nu}\geq m_{e}\sqrt{2/\delta_{\nu e}}$ (1)

Using the results of [24], the attenuation length for this process is
determined by the differential equation for energy loss by pair production
(taking $\hbar=c=1$)

$\frac{dE}{dx}=\frac{25}{448}\frac{G_{F}^{2}E^{6}(2\delta_{\nu
e})^{3}}{192\,\pi^{3}},$ (2)

which leads to neutrinos having a terminal energy $E_{T}$ after traveling a
distance $L$ given by [24]

$E_{T}^{-5}=\frac{125}{448}\frac{G_{F}^{2}(2\delta_{\nu
e})^{3}}{192\,\pi^{3}}=9.2\times 10^{-15}\delta_{\nu e}^{3}{\rm GeV^{4}}L({\rm
f)}.$ (3)

A similar result has been obtained in Ref. [26] and has been generalized in
Ref. [27]. We note that 1 kiloparsec (kpc) = 3.085$\times 10^{34}$ f. It then
follows that for a terminal energy of $10^{6}$ GeV, a superluminal neutrino
with $\sim$1 PeV energy can have traveled a distance of

$L({\rm kpc})\leq 9.4\times 10^{-52}\delta_{\nu e}^{-3}$ (4)

Then, taking $\delta_{\nu e}=5.6\times 10^{-19}$ from equation (1) for
$E_{\nu}$ = 1 PeV, it follows from equation (4) that superluminal neutrinos
with multi-PeV or greater energies that survive to a terminal energy
$E_{T}\sim 1$ PeV cannot have propagated over a distance $\geq$ $\sim 32$ Mpc
in the presence of energy loss by pair emission. This distance is of order of
the size of the local supercluster of galaxies111Recently a new IceCube
neutrino event named ”big bird” was detected with an energy of 2.1 PeV with a
15% error [29]. The arrival direction of this event was not released. 222Very
recently, it has been shown that with fluctuations and changes in the
$e^{+}e^{-}$ radiation rate during propagation taken into account, some
neutrinos with energy $\sim$ 1 PeV can have survived over longer path lengths
than that given by equation (4). The cutoff energy in the resulting neutrino
spectrum can thus be larger than the terminal energy given by equation (3)
[30]. Assuming a cutoff energy of $1PeV\sim 2E_{T}$ [30, 31], the resulting
pathlength obtained would be $2^{5}$ times larger than that given by equation
(4) viz. $\sim$ 1 Gpc.

The neutrino event that originated clearly away from the galactic plane had a
measured cascade energy in IceCube of 1.14 PeV. The uncertainty in this
determination is 15% [28]. We can therefore use 1 PeV as a conservative value
for the its energy. In Section 1 we have given reasons as to why this
$\sim$PeV neutrino was of extragalactic origin. Since the vast majority of
candidate extragalactic candidate sources lie beyond the local supercluster,
it is thus probable that the value of $\sim 5.6\times 10^{-19}$ is a a valid
upper limit on $\delta_{\nu e}$, and most likely a conservative one.

The value of $\sim 1$ PeV for the neutrino events assumes that all of the
energy of the incoming neutrino is deposited in the Ice Cube detector. This is
the case for charged current (CC) interactions. An IceCube cascade event may
also be produced by a neutral current (NC) interaction. In that case, owing to
the small average inelasticity of the NC interaction, $<y>\ \simeq 0.26$ [32]
producing the observed cascade, the initial neutrino energy, $E_{\nu}$ can be
significantly greater than the energy deposited in the cascade. It can, in
fact, be several PeV. Thus, in the NC case, equation (1) would yield a smaller
value for the upper limit on $\delta_{\nu e}$. However, the probability for NC
events is smaller than for CC events because the NC cross section is smaller
and also because of the dependence of the expected event rate on the initial
neutrino energy spectrum.

##  3 Limits on Superluminal Electron and Neutrino Velocities

It is important to note that what we obtained in the previous section is a
limit on the difference between the neutrino and electron velocities,
$\delta_{\nu e}$, not on the neutrino velocity itself. We derived a
conservative upper limit on $\delta_{\nu e}$. However, our final goal is to
derive the more physically fundamental upper limits on the superluminal
electron and neutrino velocities separately, i.e., $\delta_{\nu}$ and
$\delta_{e}$ [33].

###  3.1 The Superluminal Electron Velocity Constraint

Previous indirect constraints on $\delta_{e}\simeq\cal{O}$(10-15)[34]. Here we
use $\gamma$-ray spectral data from the September, 2010 flare of the Crab
Nebula showing the acceleration of electrons in the flare to multi-PeV
energies and allowing the deduction of best LIV constraints in the electron
sector to date.

Synchrotron $\gamma$-rays from this strong flare were observed by the Large
Area Telescope on the Fermi Gamma Ray Space Telescope up to an energy $>$ 1
GeV [35].This has provided evidence for the acceleration of electrons in the
nebula up to PeV energies. Synchrotron emission occurs when relativistic
electrons are accelerated in magnetic fields [36]. The characteristic
$\gamma$-ray energy produced by electrons of energy $E_{e}$ in a magnetic
field of strength $B_{\perp}$ perpendicular to the motion of the electron is
given by (e.g. [37])

$E_{\gamma}=1.9\times 10^{-11}B_{\perp}E_{e}^{2}$ (5)

where the energies are in GeV and the magnetic field strength is in gauss.
Conservatively taking a maximum magnetic field strength for the inner nebula
of $2\times 10^{-3}$ G [38], we find a characteristic electron energy $E_{e}>$
5.1 PeV. The implication that electrons of this energy have not been
eliminated by the emission of vacuum Ćerenkov radiation places an upper limit
on $\delta_{e}$ given by [23, 25]

$\delta_{e}\leq\frac{1}{2(E_{e}/m_{e})^{2}}\simeq 5\times 10^{-21}.$ (6)

This new constraint is five orders of magnitude stronger than the direct
constraint given in Ref. [25] (see also [39]).

###  3.2 Superluminal Neutrino Velocity Constraint Assuming $\delta_{e}\geq 0$

Since eq. (6) implies that $\delta_{\nu e}\gg\delta_{e}$, we find that
$\delta_{\nu}\simeq\delta_{\nu e}\sim 5.6\times 10^{-19}$, almost ten orders
of magnitude better than the time-of-flight constraint from the SN1987A
neutrino burst [20] and more than five orders of magnitude better than the
constraint obtained from the study of atmospheric neutrino spectra [21]. Our
new constraints apply directly to the dimension-4 operators in the SME;
$c^{e}_{TT}\equiv-\delta_{e}$ and $\mathaccent 23{c}^{(4)}\equiv-\delta_{\nu}$
(see tables D6 and D19 of Ref. [40]). Such constraints have important
implications for quantum gravity models and Planck scale physics.

###  3.3 Superluminal Neutrino Velocity Constraint Assuming $\delta_{e}\leq 0$

If, however, we allow the possibility that the electron velocities may be
subluminal we get a looser, but still significant, constraint on superluminal
neutrino velocities. In this case, the in vacuo decay of photons into
electron-positron pairs is kinematically allowed for photons with energies
exceeding a maximum energy given by [25]

$E_{\gamma,\rm max}=m_{e}\,\sqrt{2/|\delta_{e}|}\;.$ (7)

The decay would take place rapidly, so that photons with energies exceeding
$E_{\gamma,\rm max}$ could not be observed either in the laboratory or as
cosmic rays. From the fact that photons have been observed with energies
$E_{\gamma}\geq$ 80 TeV from the Crab nebula [41], we deduce for this case
that $E_{\rm max}\geq 80\;$TeV, or that the magnitude of the negative value
for $|\delta_{e}|$ is less than $8\times 10^{-17}$. This value is comparable
to the constraint given in [39].

In this case then, with $\delta_{\nu e}$ less than $\sim 5.6\times 10^{-19}$
as determined from eq. (1), we find

$\delta_{\nu}=\delta_{\nu e}+|\delta_{e}|\simeq|\delta_{e}|=8\times 10^{-17}.$
(8)

## Acknowledgment

I thank Francis Halzen for discussions regarding the IceCube results. I also
thank Alice Harding and Alan Kostelecký for helpful discussions.

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# Some examples of BL-algebras using commutative rings

Cristina Flaut and Dana Piciu

###### Abstract

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy
logic.

The aim of this paper is to analize the structure of BL-algebras using
commutative rings. From computational considerations, we are very interested
in the finite case. We present new ways to generate finite BL-algebras using
commutative rings and we give summarizing statistics.

Furthermore, we investigated BL-rings, i.e., commutative rings whose the
lattice of ideals can be equiped with a structure of BL-algebra. A new
characterization for these rings and their connections to other classes of
rings are established. Also, we give examples of finite BL-rings for which
their lattice of ideals is not an MV-algebra and using these rings we
construct BL-algebras with $2^{r}+1$ elements, $r\geq 2,$ and all BL-chains
with $k$ elements, $k\geq 4.$

Keywords: commutative ring, BL-ring, ideal, residuated lattice, MV-algebra,
BL-algebra.

AMS Subject Classification 2010: 03G10, 03G25, 06A06, 06D05, 08C05, 06F35.

##  1 Introduction

The origin of residuated lattices is in Mathematical Logic. They have been
introduced by Dilworth and Ward, through the papers [7], [16]. The study of
residuated lattices is originated in 1930 in the context of theory of rings,
with the study of ring ideals. It is known that the lattice of ideals of a
commutative ring is a residuated lattice, see [1]. Several researchers ([1],
[3], [9], [14], etc) have been interested in this construction.

Two important subvarieties of residuated lattices are BL-algebras
(corresponding to Hajek’s logic, see [8]) and MV-algebras (corresponding to
Łukasiewicz many-valued logic, see [4], [5]). For instance, rings for which
the lattice of ideals is a BL-algebra are called BL-rings and are introduced
in [9].

In this paper, we obtain a description for BL-rings, using a new
characterization of BL-algebras, given in Theorem 13, i.e, residuated lattices
$L$ in which $\ [x\odot(x\rightarrow y)]\rightarrow z=(x\rightarrow
z)\vee(y\rightarrow z)$ for every $x,y,z\in L.$ Then, BL-rings are unitary and
commutative rings $A$ with the property that $K:$
$[I\otimes(J:I)]=(K:I)+(K:J),$ for every $I,J,K\in Id(A),$ see Corollary 14

Also, we show that the class of BL-rings contains other known classes of
commutative rings: rings which are principal ideal domains and some types of
finite unitary commutative rings, see Theorem 15, Corollary 16 and Corollary
17.

One of recent applications of BCK algebras is given by Coding Theory. From
computational considerations, we are interested, in this paper, to find ways
to generate finite BL-algebras using finite commutative rings, since a
solution that is computational tractable is to consider algebras with a small
number of elements. First, we give examples of finite BL-rings whose lattice
of ideals is not an MV-algebra. Using these rings we construct BL-algebras
with $2^{r}+1$ elements, $r\geq 2$ (see Proposition 28) and BL-chains with
$k\geq 4$ elements (see Proposition 29). Also, in Theorem 39, we present a way
to generate all (up to an isomorphism) finite BL-algebras with $2\leq n\leq 5$
elements, by using the ordinal product of residuated lattices and we present
summarizing statistics.

##  2 Preliminaries

###### Definition 1

([7], [16]) A _(commutative) residuated lattice_ is an algebra
$(L,\wedge,\vee,\odot,\rightarrow,0,1)$ such that:

(LR1) $\ (L,\wedge,\vee,0,1)$ is a bounded lattice;

(LR2) $\ \ (L,\odot,1)$ is a commutative ordered monoid;

(LR3) $\ z\leq x\rightarrow y$ iff $x\odot z\leq y,$ for all $x,y,z\in L.$

The property (LR3) is called _residuation_ , where $\leq$ is the partial order
of the lattice $\ (L,\wedge,\vee,0,1).$

In a residuated lattice is defined an additional operation: for $x\in L,$ we
denote $x^{\ast}=x\rightarrow 0.$

###### Example 2

([15]) Let $(\mathcal{B},\wedge,\vee,^{\prime},0,1)$ be a _Boolean algebra._
If we define for every $x,y\in\mathcal{B},x\odot y=x\wedge y$ and
$x\rightarrow y=x^{\prime}\vee y,$ then
$(\mathcal{B},\wedge,\vee,\odot,\rightarrow,0,1)$ becomes a residuated
lattice.

###### Example 3

It is known that, for a commutative unitary ring, $A,$ if we denote by
$Id\left(A\right)$ the set of all ideals, then for $I,J\in Id\left(A\right)$,
the following sets

$I+J=<I\cup J>=\\{i+j,i\in I,j\in J\\}\text{, }$ $I\otimes
J=\\{\underset{k=1}{\overset{n}{\sum}}i_{k}j_{k},\text{ }i_{k}\in I,j_{k}\in
J\\}\text{, }$ $\left(I:J\right)=\\{x\in A,x\cdot J\subseteq I\\}\text{,}$
$Ann\left(I\right)=\left(\mathbf{0}:I\right)\text{, where
}\mathbf{0}=<0>,\text{ }$

are also ideals of $A,$ called sum, product, quotient and annihilator, see
[2]. If we preserve these notations,
$(Id(A),\cap,+,\otimes\rightarrow,0=\\{0\\},1=A)$ is a residuated lattice in
which the order relation is $\subseteq$ and $I\rightarrow J=(J:I),$ for every
$I,J\in Id(A)$, see [14].

In a residuated lattice $(L,\wedge,\vee,\odot,\rightarrow,0,1)$ we consider
the following identities:

$(prel)\qquad(x\rightarrow y)\vee(y\rightarrow x)=1\qquad\text{
({prelinearity)}};$ $(div)\qquad x\odot(x\rightarrow y)=x\wedge y\qquad\text{
({divisibility)}}.$

###### Definition 4

([6], [12], [15]) A residuated lattice $L$ is called _a BL-algebra_ if $L$
verifies $(prel)+(div)$ conditions.

A _BL-chain_ is a totally ordered BL-algebra, i.e., a BL-algebra such that its
lattice order is total.

###### Definition 5

([4], [5]) An _MV-algebra_ is an algebra $\left(L,\oplus,^{\ast},0\right)$
satisfying the following axioms:

(MV1) $\left(L,\oplus,0\right)$ is an abelian monoid;

(MV2) $(x^{\ast})^{\ast}=x;$

(MV3) $x\oplus 0^{\ast}=0^{\ast};$

(MV4) $\left(x^{\ast}\oplus y\right)^{\ast}\oplus y=$ $\left(y^{\ast}\oplus
x\right)^{\ast}\oplus x$, for all $x,y\in L.$

In fact, a residuated lattice $L$ is an MV-algebra iff it satisfies the
additional condition:

$(x\rightarrow y)\rightarrow y=(y\rightarrow x)\rightarrow x,$

for every $x,y\in L,$ see [15].

###### Remark 6

([15]) If in a BL- algebra $L$, $x^{\ast\ast}=x,$ for every $x\in L$, and for
$x,y\in L$ we denote

$x\oplus y=(x^{\ast}\odot y^{\ast})^{\ast},$

then we obtain an MV-algebra $(L,\oplus,^{\ast},0).$ Conversely, if
$(L,\oplus,^{\ast},0)$ is an MV-algebra, then
$(L,\wedge,\vee,\odot,\rightarrow,0,1)$ becomes a BL-algebra, in which for
$x,y\in L:$

$x\odot y=(x^{\ast}\oplus y^{\ast})^{\ast},$ $x\rightarrow y=x^{\ast}\oplus
y,1=0^{\ast},$ $x\vee y=(x\rightarrow y)\rightarrow y=(y\rightarrow
x)\rightarrow x\text{ and }x\wedge y=(x^{\ast}\vee y^{\ast})^{\ast}.$

In fact, MV-algebras are exactly involutive BL-algebras.

###### Example 7

_(_[12]_)_ We give an example of a finite BL-algebra which is not an MV-
algebra. Let $L=\\{0,a,b,c,1\\}$ and we define on $L$ the following
operations:

$\begin{array}[]{c|ccccc}\rightarrow&0&c&a&b&1\\\ \hline\cr 0&1&1&1&1&1\\\
c&0&1&1&1&1\\\ a&0&b&1&b&1\\\ b&0&a&a&1&1\\\ 1&0&c&a&b&1\end{array},\hskip
14.22636pt\begin{array}[]{c|ccccc}\odot&0&c&a&b&1\\\ \hline\cr 0&0&0&0&0&0\\\
c&0&c&c&c&c\\\ a&0&c&a&c&a\\\ b&0&c&c&b&b\\\ 1&0&c&a&b&1\end{array}.$

We have, $0\leq c\leq a,b\leq 1,$ but $a,b$ are incomparable, hence $L$ is a
BL-algebra which is not a chain. We remark that $x^{\ast\ast}=1,$ for every
$x\in L,x\neq 0.$

##  3 BL-rings

It is known that if $A$ is a commutative unitary ring, then
$(Id(A),\cap,+,\otimes\rightarrow,0=\\{0\\},1=A)$ is a residuated lattice, see
[14].

###### Definition 8

([9]) A commutative ring whose lattice of ideals is a BL-algebra is called _a
BL-ring._

BL-rings are closed under finite direct products, arbitrary direct sums and
homomorphic images, see [9].

In the following, using the connections between BL-algebras and BL-rings, we
give a new characterizations for commutative and unitary rings for which the
lattice of ideals is a BL-algebra.

###### Proposition 9

([12]) Let $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ be a residuated lattice.
Then we have the equivalences:

(i) $L\ $satisfies $(prel)$ condition;

(ii) $\ \ (x\wedge y)\rightarrow z=(x\rightarrow z)\vee(y\rightarrow z),$ for
every $x,y,z\in L.$

###### Lemma 10

Let $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ be a residuated lattice. The
following assertions are equivalent:

(i) $L$ satisfies $(prel)$ condition;

(ii) For every $x,y,z\in L,$ if $\ x\wedge y\leq z,$ then $(x\rightarrow
z)\vee(y\rightarrow z)=1.$

Proof. $(i)\Rightarrow(ii).$ Following by Proposition 9.

$(ii)\Rightarrow(i).$ Using (ii), for $z=$ $x\wedge y$ we deduce that
$1=(x\rightarrow(x\wedge y))\vee(y\rightarrow(x\wedge y))=[(x\rightarrow
x)\wedge(x\rightarrow y)]\vee[(y\rightarrow x)\wedge(y\rightarrow y)]$
$=(x\rightarrow y)\vee(y\rightarrow x),$ so, $L$ satisfies $(prel)$ condition.

###### Lemma 11

Let $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ be a residuated lattice. The
following assertions are equivalent:

(i) $L$ satisfies $(div)$ condition;

(ii) For every $x,y,z\in L,$ if $x\odot(x\rightarrow y)\leq z,$ then $x\wedge
y\leq z.$

Proof. $(i)\Rightarrow(ii).$ Obviously.

$(ii)\Rightarrow(i).$ Using (ii), for $z=$ $x\odot(x\rightarrow y)$ we deduce
that $x\wedge y\leq x\odot(x\rightarrow y).$ Since in a residuated lattice,
$x\odot(x\rightarrow y)\leq x\wedge y$ we deduce that $L$ satisfies $(div)$
condition.

Using Lemma 10 and Lemma 11 we deduce that:

###### Proposition 12

Let $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ be a residuated lattice. The
following assertions are equivalent:

(i) $L$ is a BL-algebra;

(ii) For every $x,y,z\in L,$ if $x\odot(x\rightarrow y)\leq z,$ then
$(x\rightarrow z)\vee(y\rightarrow z)=1;$

(iii) $[x\odot(x\rightarrow y)]\rightarrow z=(x\rightarrow z)\vee(y\rightarrow
z),$ for every $x,y,z\in L.$

Proof. $(i)\Rightarrow(ii).$ Let $x,y,z\in L$ such that $x\odot(x\rightarrow
y)\leq z.$ Since every BL-algebra satisfies $(div)$ condition, by Lemma 11, we
deduce that $x\wedge y\leq z.$ Since every BL-algebra satisfies $(prel)$
condition, following Lemma 10, we deduce that $1=(x\rightarrow
z)\vee(y\rightarrow z).$

$(ii)\Rightarrow(i).$ First, we prove that $L$ satisfies condition (ii) from
Lemma 10. So, let $x,y,z\in L$ such that $x\wedge y\leq z.$ Thus, $(x\wedge
y)\rightarrow z=1$ Since $x\odot(x\rightarrow y)\leq x\wedge y,$ we deduce
that $1=(x\wedge y)\rightarrow z\leq(x\odot(x\rightarrow y))\rightarrow z.$
Then, $x\odot(x\rightarrow y)\leq z.$ By hypothesis, $(x\rightarrow
z)\vee(y\rightarrow z)=1.$

To prove that $L$ verifies condition $(ii)$ from Lemma 11, let $x,y,z\in L$
such that $x\odot(x\rightarrow y)\leq z.$ By hypothesis, we deduce that,
$(x\rightarrow z)\vee(y\rightarrow z)=1.$ Since $(x\rightarrow
z)\vee(y\rightarrow z)\leq(x\wedge y)\rightarrow z$ we obtain $(x\wedge
y)\rightarrow z=1,$ that is, $x\wedge y\leq z.$

$(iii)\Rightarrow(ii).$ Obviously.

$(ii)\Rightarrow(iii).$ If we denote $t=[x\odot(x\rightarrow y)]\rightarrow
z,$ we have $1=t\rightarrow t=t\rightarrow[(x\odot(x\rightarrow y))\rightarrow
z]=[x\odot(x\rightarrow y)]\rightarrow(t\rightarrow z)$, hence,
$x\odot(x\rightarrow y)\leq t\rightarrow z.$

By hypothesis, we deduce that, $(x\rightarrow(t\rightarrow
z))\vee(y\rightarrow(t\rightarrow z))=1.$

Then $1=(t\rightarrow(x\rightarrow z))\vee(t\rightarrow(y\rightarrow z))\leq
t\rightarrow[(x\rightarrow z)\vee(y\rightarrow z)].$ Thus, $t\leq(x\rightarrow
z)\vee(y\rightarrow z).$

But $(x\rightarrow z)\vee(y\rightarrow z)\leq(x\wedge y)\rightarrow
z\leq[x\odot(x\rightarrow y)]\rightarrow z=t.$

We conclude that $t=(x\rightarrow z)\vee(y\rightarrow z),$ that is,
$[x\odot(x\rightarrow y)]\rightarrow z=(x\rightarrow z)\vee(y\rightarrow z),$
for every $x,y,z\in L.$

Using Proposition 12 we obtain a new characterisation for BL-algebras:

###### Theorem 13

A residuated lattice $L$ is a BL-algebra if and only if for every $x,y,z\in
L,$

$[x\odot(x\rightarrow y)]\rightarrow z=(x\rightarrow z)\vee(y\rightarrow z).$

Using this result, we can give a new description for BL-rings:

###### Corollary 14

Let $A$ be a commutative and unitary ring. The following assertions are
equivalent:

(i) $\ A$ is a BL-ring;

(ii) $K:$ $[I\otimes(J:I)]=(K:I)+(K:J),$ for every $I,J,K\in Id(A).$

###### Theorem 15

Let $A$ be a commutative ring which is a principal ideal domain. Then $A$ is a
BL-ring.

Proof. Since $A$ is a principal ideal domain, let $I=<a>$, $J=<b>,$ be the
principal non-zero ideals generated by $a,b\in A$ $\backslash\\{0\\}$.

If $d=$gcd$\\{a,b\\}$, then $d=a\cdot\alpha+b\cdot\beta,$ $a,b\in A$,
$a=a_{1}d$ and $b=b_{1}d$, with $1=$gcd$\\{a_{1},b_{1}\\}$. Thus, $I+J=<d>,$
$\ I\cap J=<ab/d>,$ $I\otimes J=<ab>$ and $\left(I:J\right)=$ $<a_{1}>.$

The conditions $(prel)$ is satisfied: $(I:J)+(J:I)=<a_{1}>+<b_{1}>=<1>=A$ and
$(div)$ is also satisfied: $J\otimes(I:J)=<b>\otimes<a_{1}>=<ab/d>=I\cap J$.

If $I=\\{0\\}$, since $A$ is an integral domain, we have that
$\left(\mathbf{0}:J\right)+(J:\mathbf{0})=Ann(J)+A=A$ and
$J\otimes(\mathbf{0}:J)=J\otimes Ann(J)=\mathbf{0=0}\cap J=$
$\mathbf{0}\otimes(J:\mathbf{0}),$ for every $J\in
Id\left(A\right)\backslash\\{0\\}.$

Moreover, we remark that
$\left(I:\left(I:J\right)\right)=\left(J:\left(J:I\right)\right)=I+J$, for
every non-zero ideals  $I,J\in Id(A).$ Also, since $A$ is an integral domain,
we obtain $Ann(Ann(I))=A,$ for every $I\in Id\left(A\right)\backslash\\{0\\}.$
We conclude that $Id(A)$ is a BL-algebra which is not an MV-algebra.

###### Corollary 16

A ring factor of a principal ideal domain is a BL-ring.

Proof. Using Theorem 15 since BL-rings are closed under homomorphic images,
see [9]. Moreover, we remark that, in fact, if $\mathcal{A}$ is a ring factor
of a principal ideal domain, then
$(Id(\mathcal{A}),\cap,+,\otimes\rightarrow,0=\\{0\\},1=A)$ is an MV-algebra,
see [11].

###### Corollary 17

A finite commutative unitary ring of the form
$A=\mathbb{Z}_{k_{1}}\times\mathbb{Z}_{k_{2}}\times...\times\mathbb{Z}_{k_{r}}$(direct
product of rings, equipped with componentwise operations), where
$k_{i}=p_{i}^{\alpha_{i}}$, $p_{i}$ is a prime number, with $p_{i}\neq p_{j},$
is a BL-ring.

Proof. We apply Corollary 16 using the fact that BL-rings are closed under
finite direct products, see [9].

Moreover, we remark that if $\ A$ is a finite commutative unitary ring of the
above form, then $Id\left(A\right)=$ $Id(\mathbb{Z}_{k_{1}})\times
Id(\mathbb{Z}_{k_{2}})\times...\times Id(\mathbb{Z}_{k_{r}})$ is an MV-algebra
$(Id\left(A\right),\oplus,^{\ast},0=\\{0\\})$ in which

$I\oplus J=Ann(Ann(I)\otimes Ann(J))\text{ and }I^{\ast}=Ann(I)$

for every $I,J\in Id\left(A\right)$, since, $Ann(Ann(I))=I,$ see [11].

###### Example 18

1) Following Theorem 15, the ring of integers $(\mathbb{Z},\mathbb{+},\cdot)$
is a BL-ring in which
$(Id(\mathbb{Z}),\cap,+,\otimes\rightarrow,0=\\{0\\},1=A)$ is not an MV-
algebra. Indeed, since $\mathbb{Z}$ is principal ideal domain, we have
$Ann\left(Ann\left(I\right)\right)=\mathbb{Z}$, for every $I\in
Id\left(\mathbb{Z}\right)\backslash\\{0\\}.$

2) Let $K$ be a field and $K\left[X\right]$ be the polynomial ring. For $f\in
K\left[X\right]$, the quotient ring $A=K\left[X\right]/\left(f\right)$ is a
BL-ring. Indeed, the lattice of ideals of this ring is an MV-algebra, see
[11].

##  4 Examples of BL-algebras using commutative rings

In this section we present ways to generate finite BL-algebras using finite
commutative rings.

First, we give examples of finite BL-rings whose lattice of ideals is not an
MV-algebra. Using these rings we construct BL-algebras with $2^{r}+1$
elements, $r\geq 2$ (see Proposition 28) and BL-chains with $k\geq 4$ elements
(see Proposition 29).

We recall that, in [11], we proved the following result:

###### Proposition 19

([11] If $A$ is a finite commutative unitary ring of the form
$A=\mathbb{Z}_{k_{1}}\times\mathbb{Z}_{k_{2}}\times...\times\mathbb{Z}_{k_{r}}$(direct
product of rings, equipped with componentwise operations), where
$k_{i}=p_{i}^{\alpha_{i}}$, $p_{i}$ is a prime number, with $p_{i}\neq p_{j},$
and $Id\left(A\right)$ denotes the set of all ideals of the ring $A$, then,
$\left(Id\left(A\right),\vee,\wedge,\odot,\rightarrow,0,1\right)$ is an MV-
algebra, where the order relation is $\subseteq$, $I\odot J=I\otimes J$,
$I^{\ast}=Ann(I)$, $I\rightarrow J=\left(J:I\right)$, $I\vee J=I+J$, $I\wedge
J=I\cap J$, $0=\\{0\\}$ and $1=A$. The set $Id\left(A\right)$ has
$\mathcal{N}_{A}=\overset{r}{\underset{i=1}{\prod}}\left(\alpha_{i}+1\right)$
elements.

In the following, we give examples of finite BL-rings whose lattice of ideals
is not an MV-algebra.

###### Definition 20

([2]) Let $R$ be a commutative unitary ring. The ideal $M$ of the ring $R$ is
maximal if it is maximal, with respect of the set inclusion, amongst all
proper ideals of the ring $R$. That means, there are no other ideals different
from $R$ contained $M$. The ideal $J$ of the ring $R$ is a minimal ideal if it
is a nonzero ideal which contains no other nonzero ideals. A commutative local
ring $R$ is a ring with a unique maximal ideal.

###### Example 21

(i) A field $F$ is a local ring, with $\\{0\\}$ the maximal ideal in this
ring.

(ii) In $\left(\mathbb{Z}_{8},+,\cdot\right)$, the ideal
$J=\\{\widehat{0},\widehat{4}\\}$ is a minimal ideal and the ideal
$M=\\{\widehat{0},\widehat{2},\widehat{4},\widehat{6}\\}$ is the maximal
ideal.

###### Remark 22

Let $R$ be a local ring with $M$ its maximal ideal. Then, the quotient ring
$R[X]/(X^{n})$, with $n$ a positive integer, is local. Indeed, the unique
maximal ideal of the ring $R[X]/(X^{n})$ is
$\overrightarrow{M}=\\{\overrightarrow{f}\in R[X]/(X^{n})/~{}f\in
R[X],f=a_{0}+a_{1}X+...+a_{n-1}X^{n-1}$, with $a_{0}\in M\\}$. For other
details, the reader is referred to [13].

In the following, we consider the ring $(\mathbb{Z}_{n},+,\cdot)$ with
$n=p_{1}p_{2}...p_{r},p_{1},p_{2},...,p_{r}$ being distinct prime numbers,
$r\geq 2,$ and the factor ring $R=\mathbb{Z}_{n}[X]/\left(X^{2}\right).$

###### Remark 23

(i) With the above notations, in the ring $(\mathbb{Z}_{n},+,\cdot)$, the
ideals generated by $\widehat{p}_{i},M_{p_{i}}=\left(\widehat{p}_{i}\right),$
are maximals. The ideals of $\mathbb{Z}_{n}$ are of the form
$I_{d}=\left(\widehat{d}\right)$, where $d$ is a divisor of $n$.

(ii) Each element from $\mathbb{Z}_{n}-\\{M_{p_{1}}\cup M_{p_{2}}\cup...\cup
M_{p_{r}}\\}$ is an invertible element. Indeed, if
$\widehat{x}\in\mathbb{Z}_{n}-\\{M_{p_{1}}\cup M_{p_{2}}\cup...\cup
M_{p_{r}}\\}$, we have gcd $\\{x,n\\}=1$, therefore $x$ is an invertible
element.

###### Proposition 24

i) With the above notations, the factor ring
$R=\mathbb{Z}_{n}[X]/\left(X^{2}\right)$ has $2^{r}+1$ ideals, including
$\\{0\\}$ and $R$.

ii) For $\widehat{\gamma}\in\mathbb{Z}_{n}-\\{M_{p_{1}}\cup
M_{p_{2}}\cup...\cup M_{p_{r}}\\}$, the element $X+\widehat{\gamma}$ is an
invertible element in $R$ .

Proof. (i) Indeed, the ideals are:
$J_{p_{i}}=\left(\widehat{\alpha}X+\widehat{\alpha}_{i}\right),\widehat{\alpha}_{i}\in
M_{p\,_{i}},i\in\\{1,2,...,r\\},$ which are maximals,
$J_{d}=\left(\widehat{\beta}X+\widehat{\beta}_{d}\right),\widehat{\beta}_{d}\in
I_{d},I_{d}$ is not maximal, $\widehat{\alpha},\widehat{\beta}\in R,d\neq n$,
where $d$ is a proper divisor of $n$, the ideals $\left(X\right)$, for $d=n$
and $\left(0\right)$. Therefore, we have $\complement_{n}^{0}$ ideals, for
ideal $\left(X\right)$, $\complement_{n}^{1}$ ideals for ideals $J_{p_{i}}$,
$\complement_{n}^{2}$ ideals for ideals $J_{p_{i}p_{j}}$, $p_{i}\neq
p_{j}$,…,$\complement_{n}^{n}$ ideals for ideal $R$, for $d=1$, resulting a
total of $2^{r}+1,$ if we add ideal $\left(0\right)$.

(ii) Since $\widehat{\gamma}\in\mathbb{Z}_{n}-\\{M_{p_{1}}\cup
M_{p_{2}}\cup...\cup M_{p_{r}}\\}$, we have that $\widehat{\gamma}$ is
invertible, with $\widehat{\delta}$ its inverse. Therefore,
$\left(X+\widehat{\gamma}\right)[-\widehat{\delta}^{-2}\left(X-\widehat{\gamma}\right)]=1$.
It results that $X+\widehat{\gamma}$ is invertible, therefore
$\left(X+\widehat{\gamma}\right)=R$.

Since for any commutative unitary ring, its lattice of ideals is a residuated
lattice (see [14]), in particular, for the unitary and commutative ring
$A=\mathbb{Z}_{n}[X]/\left(X^{2}\right)$, we have that
$(Id(\mathbb{Z}_{n}/\left(X^{2}\right)),\cap,+,\otimes\rightarrow,0=\\{0\\},1=A)$
is a residuated lattice with $2^{r}+1$ elements.

###### Remark 25

As we remarked above, the ideals in the ring
$R=\mathbb{Z}_{n}[X]/\left(X^{2}\right)$ are:

(i) $\left(0\right);$

(ii) of the form
$J_{d}=\left(\widehat{\alpha}X+\widehat{\beta}_{d}\right),\widehat{\alpha}\in
R,\widehat{\beta}_{d}\in I_{d},$ where $d$ is a proper divisor of
$n=p_{1}p_{2}...p_{r},p_{1},p_{2},...,p_{r}$ being distinct prime numbers,
$r\geq 2,$ by using the notations from Remark 23. If
$I_{d}=\left(\widehat{p_{i}}\right),$ then $J_{d}$ is denoted $J_{p_{i}}$ and
it is a maximal ideal in $R=\mathbb{Z}_{n}[X]/\left(X^{2}\right)$ ;

(iii) The ring $R,$ if $d=1$;

(iv) $\left(X\right),$ if $d=n$.

###### Remark 26

We remark that for all two nonzero ideals of the above ring $R$, $I$ and $J$,
we have $\left(X\right)\subseteq I\cap J$ and the ideal $\left(X\right)$ is
the only minimal ideal of $\mathbb{Z}_{n}[X]/\left(X^{2}\right)$.

###### Remark 27

Let $D_{d}=\\{$ $p\in\\{p_{1},p_{2},...,p_{r}\\},$ such that $d=\prod
p\\},d\neq 1$.

(1) We have $J_{d_{1}}\cap J_{d_{2}}=J_{d_{1}}\otimes J_{d_{2}}=J_{d_{3}}$,
where $D_{d_{3}}=\\{p\in D_{d_{1}}\cup D_{d_{2}},d_{3}=\prod p\\}$, for
$d_{1},d_{2}$ proper divisors.

If $d_{1}=1,$ we have $R\otimes J_{d_{2}}=J_{d_{2}}=R\cap J_{d_{2}}$.

If $d_{1}=n$, $d_{2}\neq n$, we have $\left(X\right)\otimes
J_{d_{2}}=\left(X\right)\cap J_{d_{2}}=\left(X\right)$. If $d_{2}=n,$ we have
$\left(X\right)\otimes\left(X\right)=\left(0\right)$.

(2) We have $(J_{d_{1}}:J_{d_{2}})=J_{d_{3}}$, with
$D_{d_{3}}=D_{d_{1}}-D_{d_{2}}$. Indeed, $(J_{d_{1}}:J_{d_{2}})=\\{y\in
R,y\cdot J_{d_{2}}\subseteq J_{d_{1}}\\}=J_{d_{3}}$, for $d_{1},d_{2}$ proper
divisors.

If $J_{d_{1}}=\left(0\right),$ we have $(0:J_{d_{2}})=\left(0\right)$. Indeed,
if $(0:J_{d_{2}})=J\neq\left(0\right)$, it results that $J\otimes
J_{d_{2}}=\left(0\right)$. But, from the above, $J\otimes J_{d_{2}}=$ $J\cap
J_{d_{2}}$ $\neq\left(0\right)$, which is false

If $J_{d_{2}}=\left(0\right)$, we have $(J_{d_{1}}:0)=R$.

If $d_{1}=1$, we have $\left(R:J_{d_{2}}\right)=R$ and
$\left(J_{d_{2}}:R\right)=J_{d_{2}}.$

If $d_{1}=n$, $d_{2}\neq n$, we have $J_{d_{1}}=\left(X\right)$, therefore
$(J_{d_{1}}:J_{d_{2}})=J_{d_{1}}=\left(X\right)$. If $d_{1}\neq n$, $d_{2}=n$,
we have $J_{d_{2}}=\left(X\right)$, therefore $(J_{d_{1}}:J_{d_{2}})=R$. If
$d_{1}=d_{2}=n$, we have $J_{d_{1}}=J_{d_{2}}=\left(X\right)$ and
$(J_{d_{1}}:J_{d_{2}})=R$.

###### Proposition 28

(i) For $n\geq 2$, with the above notations, the residuated lattice
$(Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right)),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R),R=\mathbb{Z}_{n}[X]/\left(X^{2}\right),$
is a BL-algebra with $2^{r}+1$ elements.

(ii) By using notations from Remark 25, we have that
$(Id_{p_{i}}(\mathbb{Z}_{n}[X]/\left(X^{2}\right)),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R),$where
$Id_{p_{i}}(\mathbb{Z}_{n}[X]/\left(X^{2}\right))=\\{\left(0\right),J_{p_{i}},R\\},~{}$is
a BL-sublattice of the lattice $Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right))$,
having $3$ elements.

Proof. (i) First, we will prove the $\left(prel\right)$ condition:

$\left(I\rightarrow J\right)\vee\left(J\rightarrow
I\right)=(J:I)\vee(I:J)=\mathbb{Z}_{n}[X]/\left(X^{2}\right),$

for every $I,J\in Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right))$.

Case 1. If $d_{1}$ and $d_{2}$ are proper divisors of $n$, we have
$\left(J_{d_{1}}\rightarrow J_{d_{2}}\right)\vee\left(J_{d_{2}}\rightarrow
J_{d_{1}}\right)=(J_{d_{2}}:J_{d_{1}})\vee(J_{d_{1}}:J_{d_{2}})=J_{d_{4}}\vee
J_{d_{5}},$ where $D_{d_{4}}=D_{d_{1}}-D_{d_{2}}$ and
$D_{d_{5}}=D_{d_{2}}-D_{d_{1}}$. We remark that $D_{d_{4}}\cap
D_{d_{5}}=\emptyset,$ then gcd $\\{d_{4},d_{5}\\}=1$. From here, it results
that there are the integers $a$ and $b$ such that $ad_{4}+bd_{5}=1$. We obtain
that $J_{d_{4}}\vee J_{d_{5}}=<J_{d_{4}}\cup J_{d_{5}}>=R$, from Proposition
24, ii).

Case 2. If $d_{1}$ is a proper divisor of $n$ and $d_{2}=n$, we have
$J_{d_{2}}=\left(X\right)$. Therefore, $\left(J_{d_{1}}\rightarrow
J_{d_{2}}\right)\vee\left(J_{d_{2}}\rightarrow
J_{d_{1}}\right)=(J_{d_{2}}:J_{d_{1}})\vee(J_{d_{1}}:J_{d_{2}})=J_{d_{2}}\vee
R=R$, by using Remark 27.

Case 3. If $d_{1}$ is a proper divisor of $n$ and $J_{d_{2}}=\left(0\right)$,
we have $\left(J_{d_{1}}\rightarrow
J_{d_{2}}\right)\vee\left(J_{d_{2}}\rightarrow
J_{d_{1}}\right)=(0:J_{d_{1}})\vee(J_{d_{1}}:0)=0\vee R=R$, by using Remark
27.

Case 4. If $d_{1}$ is a proper divisor of $n$ and $J_{d_{2}}=R$, it is clear.
From here, we have that the condition $\left(prel\right)$ is satisfied.

Now, we prove condition $\left(div\right):$

$I\otimes\left(I\rightarrow J\right)=I\otimes\left(J:I\right)=I\cap J,$

for every $I,J\in Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right))$.

Case 1. If $d_{1}$ and $d_{2}$ are proper divisors of $n$, we have
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=J_{d_{1}}\otimes
J_{d_{3}}=J_{d_{4}}=J_{d_{1}}\cap J_{d_{2}},$ since
$D_{d_{3}}=D_{d_{2}}-D_{d_{1}}$ and $D_{d_{4}}=\\{p\in D_{d_{1}}\cup
D_{d_{3}},d_{4}=\prod p\\}=\\{p\in D_{d_{1}}\cup D_{d_{2}},d_{4}=\prod p\\}$.

Case 2. If $d_{1}$ is a proper divisor of $n$ and $d_{2}=n$, we have
$J_{d_{2}}=\left(X\right)$. We obtain
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=J_{d_{1}}\otimes\left(\left(X\right):J_{d_{1}}\right)=J_{d_{1}}\otimes\left(X\right)=J_{d_{1}}\cap\left(X\right)$
since $\left(X\right)\subset J_{d_{1}}$.

Case 3. If $d_{1}=n$ and $d_{2}$ is a proper divisor of $n$, we have
$J_{d_{1}}=\left(X\right)$. We obtain
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=\left(X\right)\otimes\left(J_{d_{2}}:\left(X\right)\right)=\left(X\right)\otimes
R=\left(X\right)=J_{d_{2}}\cap\left(X\right)$ since $\left(X\right)\subset
J_{d_{2}}$.

Case 4. If $d_{1}$ is a proper divisor of $n$ and $J_{d_{2}}=\left(0\right)$,
we have
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=J_{d_{1}}\otimes\left(0:J_{d_{1}}\right)=J_{d_{1}}\otimes\left(0\right)=\left(0\right)=J_{d_{1}}\cap\left(0\right),$
from Remark 27.

Case 5. If $J_{d_{1}}=\left(0\right)$ and $d_{2}$ is a proper divisor of $n$,
we have
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=0\otimes\left(J_{d_{2}}:0\right)=0.$

Case 6. If $d_{1}$ is a proper divisor of $n$ and $J_{d_{2}}=R$, we have
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=J_{d_{1}}\otimes\left(R:J_{d_{1}}\right)=J_{d_{1}}\otimes
R=J_{d_{1}}$. If $\ J_{d_{1}}=R$ and $d_{2}$ is a proper divisor of $n$, we
have
$J_{d_{1}}\otimes\left(J_{d_{2}}:J_{d_{1}}\right)=R\otimes\left(J_{d_{2}}:R\right)=R\otimes
J_{d_{2}}=J_{d_{2}}$. From here, we have that the condition $\left(div\right)$
is satisfied and the proposition is proved.

(ii) It is clear that $J_{p_{i}}\odot J_{p_{i}}=J_{p_{i}}\otimes
J_{p_{i}}=J_{p_{i}}$, and we obtain the following tables:

$\begin{tabular}[]{l|lll}$\rightarrow$&$O$&$J_{p_{i}}$&$R$\\\
\hline\cr$O$&$R$&$R$&$R$\\\ $J_{p_{i}}$&$O$&$R$&$R$\\\
$R$&$O$&$J_{p_{i}}$&$R$\end{tabular}\ \ \
\begin{tabular}[]{l|lll}$\odot$&$O$&$J_{p_{i}}$&$R$\\\
\hline\cr$O$&$O$&$O$&$O$\\\ $J_{p_{i}}$&$O$&$J_{p_{i}}$&$J_{p_{i}}$\\\
$R$&$O$&$J_{p_{i}}$&$R$\end{tabular}\ \ ,$

therefore a BL-algebra of order $3.$

###### Proposition 29

Let $n=p^{r}$ with $p$ a prime number, $p\geq 2,$ $r$ a positive integer,
$r\geq 2$. We consider the ring $R=\mathbb{Z}_{n}[X]/\left(X^{2}\right)$. The
set
$(Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right)),\cap,+,\otimes\rightarrow,0=\\{0\\},1=R)$
is a BL-chain with $r+2$ elements. In this way, for a given positive integer
$k\geq 4$, we can construct all BL-chains with $k$ elements.

Proof. The ideals in $\mathbb{Z}_{n}$ are of the form:
$\left(0\right)\subseteq\left(p^{r-1}\right)\subseteq\left(p^{r-2}\right)\subseteq...\subseteq\left(p\right)\subseteq\mathbb{Z}_{n}$.
The ideal $\left(p^{r-1}\right)$ and the ideal $\left(p\right)$ is the only
maximal ideal of $\mathbb{Z}_{n}$. The ideals in the ring $R$ are
$\left(0\right)\subseteq\left(X\right)\subseteq\left(\alpha_{r-1}X+\beta_{r-1}\right)\subseteq\left(\alpha_{r-2}X+\beta_{r-2}\right)\subseteq...\subseteq\left(\alpha_{1}X+\beta_{1}\right)\subseteq
R,$ where $\alpha_{i}\in\mathbb{Z}_{n},i\in\\{1,...,r-1\\},$
$\beta_{r-1}\in\left(p^{r-1}\right),\beta_{r-2}\in\left(p^{r-2}\right),...,\beta_{1}\in\left(p\right),$
therefore $r+2$ ideals. We denote these ideals with
$\left(0\right),\left(X\right),$ $I_{p^{r-1}},I_{p^{r-2}},...I_{p},\,R$, with
$I_{p}$ the only maximal ideal in $R$.

First, we will prove $\left(prel\right)$ condition:

$\left(I\rightarrow J\right)\vee\left(J\rightarrow
I\right)=(J:I)\vee(I:J)=\mathbb{Z}_{n}[X]/\left(X^{2}\right),$

for every $I,J\in Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right))$.

Case 1. We suppose that $I$ and $J$ are proper ideals and $I\subseteq J$. We
have $\left(I\rightarrow J\right)\vee\left(J\rightarrow
I\right)=(J:I)\vee(I:J)=R\vee I=R$.

Case 2$.I=\left(0\right)$ and $J$ a proper ideal. We have $\left(I\rightarrow
J\right)\vee\left(J\rightarrow
I\right)=(J:\left(0\right))\vee(\left(0\right):J)=R$. Therefore, the condition
$\left(prel\right)$ is satisfied.

Now, we prove $\left(div\right)$ condition:

$I\otimes\left(I\rightarrow J\right)=I\otimes\left(J:I\right)=I\cap J,$

for every $I,J\in Id(\mathbb{Z}_{n}[X]/\left(X^{2}\right))$.

Case 1. We suppose that $I$ and $J$ are proper ideals and $I\subseteq J$. We
have $I\otimes\left(I\rightarrow J\right)=I\otimes\left(J:I\right)=I\otimes
R=I=I\cap J$. If $J\subseteq I$, we have $I\otimes\left(I\rightarrow
J\right)=I\otimes\left(J:I\right)=I\otimes J=J=I\cap J$.

Case 2$.I=\left(0\right)$ and $J$ is a proper ideal.We have
$\left(0\right)\otimes\left(\left(0\right)\rightarrow
J\right)=\left(0\right)\otimes\left(J:\left(0\right)\right)=\left(0\right)=I\cap
J$. If $I\neq\left(X\right)$ is a proper ideal and $J=\left(0\right)$, we have
$I\otimes\left(I\rightarrow
J\right)=I\otimes\left(\left(0\right):I\right)=I\otimes\left(0\right)=\left(0\right)$.
If $I=\left(X\right)$ and $J=\left(0\right)$, we have
$I\otimes\left(I\rightarrow
J\right)=\left(X\right)\otimes\left(\left(0\right):\left(X\right)\right)=\left(X\right)\otimes\left(X\right)=\left(0\right)$
and $\left(0\right)\cap\left(X\right)=\left(0\right)$. From here, we have that
the condition $\left(div\right)$ is satisfied and the proposition is proved.

###### Example 30

In Proposition 28, we take $n=2\cdot 3\,$, therefore the ideals of $\
\mathbb{Z}_{6}$ are
$\left(0\right),\left(2\right),\left(3\right),\mathbb{Z}_{6}$, with
$\left(2\right)$ and $\left(3\right)$ maximal ideals. The ring
$\mathbb{Z}_{6}[X]/\left(X^{2}\right)$ has $5$ ideals:
$O=\left(0\right)\subset A=\left(X\right),B=\left(\alpha
X+\beta\right),C=\left(\gamma X+\delta\right),E=\mathbb{Z}_{6}$, with
$\alpha,\gamma\in$ $\mathbb{Z}_{4},\beta\in\left(2\right)$ and
$\delta\in\left(3\right)$. From the following tables, we have a BL-structure
on $Id(\mathbb{Z}_{6}[X]/\left(X^{2}\right))$:

$\begin{tabular}[]{l|lllll}$\rightarrow$&$O$&$A$&$B$&$C$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$&$E$\\\ $A$&$A$&$E$&$E$&$E$&$E$\\\
$B$&$O$&$A$&$E$&$C$&$E$\\\ $C$&$O$&$A$&$B$&$E$&$E$\\\
$E$&$O$&$A$&$B$&$C$&$E$\end{tabular}\ \ \
\begin{tabular}[]{l|lllll}$\odot$&$O$&$A$&$B$&$C$&$E$\\\
\hline\cr$O$&$O$&$O$&$O$&$O$&$O$\\\ $A$&$O$&$O$&$A$&$A$&$A$\\\
$B$&$O$&$A$&$B$&$A$&$B$\\\ $C$&$O$&$A$&$A$&$C$&$C$\\\
$E$&$O$&$A$&$B$&$C$&$E$\end{tabular}\ \ .$

From Proposition 28, if we consider $J_{p_{i}}=$ $B$, we have the following
BL-algebra of order $3$:

$\begin{tabular}[]{l|lll}$\rightarrow$&$O$&$B$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$\\\ $B$&$O$&$E$&$E$\\\ $E$&$O$&$B$&$E$\end{tabular}\ \
\ \begin{tabular}[]{l|lll}$\odot$&$O$&$B$&$E$\\\ \hline\cr$O$&$O$&$O$&$O$\\\
$B$&$O$&$B$&$B$\\\ $E$&$O$&$B$&$E$\end{tabular}\ \ .$

###### Example 31

In Proposition 28, we take $n=2\cdot 3\cdot 5\,$, therefore the ideals of the
ring $\mathbb{Z}_{30}$ are
$\left(0\right),\left(2\right),\left(3\right),\left(5\right),\left(6\right),\left(10\right),\left(15\right),\mathbb{Z}_{30}$,
with $\left(2\right),\left(3\right)$ and $\left(5\right)$ maximal ideals. The
ring $\mathbb{Z}_{30}[X]/\left(X^{2}\right)$ has $9$ ideals:
$O=\left(0\right)\subset
A=\left(X\right),B=\left(\alpha_{1}X+\beta_{1}\right),C=\left(\alpha_{2}X+\beta_{2}\right),$
$D=\left(\alpha_{3}X+\beta_{3}\right),E=\left(\alpha_{4}X+\beta_{4}\right),F=\left(\alpha_{5}X+\beta_{5}\right),G=\left(\alpha_{6}X+\beta_{6}\right),R=\mathbb{Z}_{30}$,
with $\alpha_{i}\in$
$\mathbb{Z}_{30},i\in\\{1,2,3,4,5,6\\},\beta_{1}\in\left(6\right)$,
$\beta_{2}\in\left(10\right),\beta_{3}\in\left(15\right),\beta_{4}\in\left(2\right),\beta_{5}\in\left(3\right)$
and $\beta_{6}\in\left(5\right)$. The ideals $E,F$ and $G$ are maximals. From
the following tables, we have a BL-structure on
$Id(\mathbb{Z}_{30}[X]/\left(X^{2}\right))$:

$\rightarrow$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $R$
---|---|---|---|---|---|---|---|---|---
$O$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$
$A$ | $A$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$ | $R$
$B$ | $O$ | $A$ | $R$ | $C$ | $D$ | $R$ | $R$ | $G$ | $R$
$C$ | $O$ | $A$ | $B$ | $R$ | $D$ | $R$ | $F$ | $R$ | $R$
$D$ | $O$ | $A$ | $B$ | $C$ | $R$ | $E$ | $R$ | $R$ | $R$
$E$ | $O$ | $A$ | $B$ | $C$ | $D$ | $R$ | $F$ | $G$ | $R$
$F$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $R$ | $G$ | $R$
$G$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $R$ | $R$
$R$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $R$
$\odot$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $R$
---|---|---|---|---|---|---|---|---|---
$O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$ | $O$
$A$ | $O$ | $O$ | $A$ | $A$ | $A$ | $A$ | $A$ | $A$ | $A$
$B$ | $O$ | $A$ | $B$ | $A$ | $A$ | $B$ | $B$ | $A$ | $B$
$C$ | $O$ | $A$ | $A$ | $C$ | $A$ | $C$ | $C$ | $A$ | $C$
$D$ | $O$ | $A$ | $A$ | $A$ | $D$ | $A$ | $D$ | $D$ | $D$
$E$ | $O$ | $A$ | $B$ | $C$ | $A$ | $E$ | $A$ | $A$ | $E$
$F$ | $O$ | $A$ | $B$ | $A$ | $D$ | $A$ | $F$ | $A$ | $F$
$G$ | $O$ | $A$ | $A$ | $C$ | $D$ | $A$ | $A$ | $G$ | $G$
$R$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $R$
.

###### Example 32

In Proposition 29, we consider $p=2,r=2$. The ideals in
$\left(\mathbb{Z}_{4},+,\cdot\right)$ are
$\left(0\right)\subset\left(2\right)\subset\mathbb{Z}_{4}$ and
$\mathbb{Z}_{4}$ is a local ring. The ring
$\mathbb{Z}_{4}[X]/\left(X^{2}\right)$ has $4$ ideals:
$O=\left(0\right)\subset A=\left(X\right)\subset B=\left(\alpha
X+\beta\right)\subset E=\mathbb{Z}_{4}$, with $\alpha,\gamma\in$
$\mathbb{Z}_{4},\beta\in\left(2\right)$. From the following tables, we have a
BL-structure for $Id(\mathbb{Z}_{4}[X]/\left(X^{2}\right))$:

$\begin{tabular}[]{l|llll}$\rightarrow$&$O$&$A$&$B$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$\\\ $A$&$A$&$E$&$E$&$E$\\\ $B$&$O$&$A$&$E$&$E$\\\
$E$&$O$&$A$&$B$&$E$\end{tabular}\ \ \
\begin{tabular}[]{l|llll}$\odot$&$O$&$A$&$B$&$E$\\\
\hline\cr$O$&$O$&$O$&$O$&$O$\\\ $A$&$O$&$O$&$A$&$A$\\\ $B$&$O$&$A$&$A$&$B$\\\
$E$&$O$&$A$&$B$&$E$\end{tabular}\ \ .$

###### Example 33

In Proposition 29, we consider $p=2,r=3$. The ideals in
$\left(\mathbb{Z}_{8},+,\cdot\right)$ are
$\left(0\right)\subset\left(4\right)\subset\left(2\right)\subset\mathbb{Z}_{8}$.
The ring $\mathbb{Z}_{8}[X]/\left(X^{2}\right)$ has $5$ ideals:
$O=\left(0\right)\subset A=\left(X\right)\subset B=\left(\alpha
X+\beta\right)\subset C=\left(\gamma X+\delta\right)\subset E=\mathbb{Z}_{8}$,
with $\alpha,\gamma\in$ $\mathbb{Z}_{8},\beta\in\left(4\right)$ and
$\delta\in\left(2\right)$. From the following tables, we have a BL-structure
for $Id(\mathbb{Z}_{8}[X]/\left(X^{2}\right))$:

$\begin{tabular}[]{l|lllll}$\rightarrow$&$O$&$A$&$B$&$C$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$&$E$\\\ $A$&$A$&$E$&$E$&$E$&$E$\\\
$B$&$O$&$A$&$E$&$E$&$E$\\\ $C$&$O$&$A$&$B$&$E$&$E$\\\
$E$&$O$&$A$&$B$&$C$&$E$\end{tabular}\ \ \
\begin{tabular}[]{l|lllll}$\odot$&$O$&$A$&$B$&$C$&$E$\\\
\hline\cr$O$&$O$&$O$&$O$&$O$&$O$\\\ $A$&$O$&$O$&$A$&$A$&$A$\\\
$B$&$O$&$A$&$A$&$A$&$B$\\\ $C$&$O$&$A$&$A$&$B$&$C$\\\
$E$&$O$&$A$&$B$&$C$&$E$\end{tabular}\ \ .$

In the following, we present a way to generate new finite BL-algebras using
the ordinal product of residuated lattices.

We recall that, in [12], Iorgulescu study the influence of the conditions
$(prel)$ and $(div)$ on the ordinal product of two residuated lattices.

It is known that, if
$\mathcal{L}_{1}=(L_{1},\wedge_{1},\vee_{1},\odot_{1},\rightarrow_{1},0_{1},1_{1})$
and
$\mathcal{L}_{2}=(L_{2},\wedge_{2},\vee_{2},\odot_{2},\rightarrow_{2},0_{2},1_{2})$
are two residuated lattices such that $1_{1}=0_{2}$ and
$(M_{1}\backslash\\{1_{1}\\})\cap(M_{2}\backslash\\{0_{2}\\})=\oslash,$ then,
the ordinal product of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ is the
residuated lattice $\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}=(L_{1}\cup
L_{2},\wedge,\vee,\odot,\rightarrow,0,1)$ where

$0=0_{1}\text{ and }1=1_{2},$ $x\leq y\text{ if }(x,y\in L_{1}\text{ and
}x\leq_{1}y)\text{ or }(x,y\in L_{2}\text{ and }x\leq_{2}y)\text{ or }(x\in
L_{1}\text{ and }y\in L_{2})\text{ ,}$

$x\wedge y=\left\\{\begin{array}[]{c}x\wedge_{1}y,\text{ if }x,y\in L_{1},\\\
x\wedge_{2}y,\text{ if }x,y\in L_{2},\\\ x,\text{ if }x\in L_{1}\text{ and
}y\in L_{2}\end{array}\right.$ $x\vee
y=\left\\{\begin{array}[]{c}x\vee_{1}y,\text{ if }x,y\in L_{1},\\\
x\vee_{2}y,\text{ if }x,y\in L_{2},\\\ y,\text{ if }x\in L_{1}\text{ and }y\in
L_{2}\end{array}\right.$ $x\rightarrow y=\left\\{\begin{array}[]{c}1,\text{ if
}x\leq y,\\\ x\rightarrow_{i}y,\text{ if }x\nleq y,\text{ }x,y\in L_{i},\text{
}i=1,2,\\\ y,\text{ if }x\nleq y,\text{ }x\in L_{2},\text{ }y\in
L_{1}\backslash\\{1_{1}\\}.\end{array}\right.$ $x\odot
y=\left\\{\begin{array}[]{c}x\odot_{1}y,\text{ if }x,y\in L_{1},\\\
x\odot_{2}y,\text{ if }x,y\in L_{2},\\\ x,\text{ if }x\in
L_{1}\backslash\\{1_{1}\\}\text{ and }y\in L_{2}.\end{array}\right.$

The ordinal product is associative, but is not commutative, see [12].

###### Proposition 34

( [12], Corollary 3.5.10) Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be BL-
algebras.

(i) If $\mathcal{L}_{1}$ is a chain, then the ordinal product
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ is a BL-algebra.

(ii) If $\mathcal{L}_{1}$ is not a chain, then the ordinal product
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ is only a residuated lattice
satisfying (div) condition.

###### Remark 35

(i) An ordinal product of two BL-chains is a BL-chain. Indeed, using the
definition of implication in an ordinal product for every $x,y$ we have
$x\rightarrow y=1$ or $y\rightarrow x=1.$

(ii) An ordinal product of two MV-algebras is a BL-algebra which is not an MV-
algebra. Indeed, if $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are two MV-
algebras, using Proposition 34, the residuated lattice
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ is a BL-algebra in which, we have
$(1_{1})^{\ast\ast}=(1_{1}\rightarrow
0_{1})^{\ast}=(0_{1})^{\ast}=0_{1}\rightarrow 0_{1}=1=1_{2}\neq 1_{1},$ so,
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ is not an MV-algebra.

For a natural number $n\geq 2$ we consider the decomposition (which is not
unique) of $n$ in factors greater than $1$ and we denote by $\pi(n)$ the
number of all such decompositions. Obviously, if $n$ is prime then $\pi(n)=0.$

We recall that an MV-algebra is finite iff it is isomorphic to a finite
product of MV-chains, see [10]. Using this result, in [11] we showed that for
every natural number $n\geq 2$ there are $\pi(n)+1$ non-isomorphic MV-algebras
with $n$ elements of which only one is a chain. Moreover, all finite MV-
algebras (up to an isomorphisms) with $n$ elememts correspond to finite
commutative ring $A$ in which $\left|Id(A)\right|=n.$

In Table 1 we sum briefly describe a way to generate finite MV-algebras $M$
with $2\leq n\leq 6$ elements using commutative rings, see [11].

Table 1:

$\left|M\right|\mathbf{=n}$ | Nr of MV | Rings which generates MV
---|---|---
$n=2$ | $1$ |  $Id(\mathbb{Z}_{2})$ (chain)
$n=3$ | $1$ |  $Id(\mathbb{Z}_{4})$ (chain)
$n=4$ | $2$ |  $Id(\mathbb{Z}_{8})$ (chain) and $Id(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$
$n=5$ | $1$ |  $Id(\mathbb{Z}_{16})$ (chain)
$n=6$ | $2$ |  $Id(\mathbb{Z}_{32})$ (chain) and $Id(\mathbb{Z}_{2}\times\mathbb{Z}_{4})$
$n=7$ | $1$ |  $Id(\mathbb{Z}_{64})$ (chain)
$n=8$ | $3$ |  $Id(\mathbb{Z}_{128})$ (chain) and $Id(\mathbb{Z}_{2}\times\mathbb{Z}_{8})$ and $Id(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2})$

Using the construction of ordinal product, Proposition 34 and Remark 35, we
can generate BL-algebras (which are not MV-algebras) using commutative rings.

###### Example 36

In [11] we show that there is one MV-algebra with 3 elements (up to an
isomorphism), see Table 1. This MV-algebras is isomorphic to
$Id(\mathbb{Z}_{4})$ and is a chain. To generate a BL-chain with 3 elements
(which is not an MV-algebra) using the ordinal product we must consider only
the MV-algebra with 2 elements (which is in fact a Boolean algebra). In the
commutative ring $\ (\mathbb{Z}_{2},+,\cdot)$ the ideals are
$Id(\mathbb{Z}_{2})=\\{\\{\widehat{0}\\},$ $\mathbb{Z}_{2}\\}$. Using
Proposition 19,
$(Id\left(\mathbb{Z}_{2}\right),\cap,+,\otimes\rightarrow,0=\\{0\\},1=\mathbb{Z}_{2})$
is an MV-chain with the following operations:

$\text{
}\begin{tabular}[]{l|ll}$\rightarrow$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\\\
\hline\cr$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$&$\mathbb{Z}_{2}$\\\
$\mathbb{Z}_{2}$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\end{tabular};\text{
}\begin{tabular}[]{l|ll}$\otimes=\cap$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\\\
\hline\cr$\\{\widehat{0}\\}$&$\\{\widehat{0}\\}$&$\\{\widehat{0}\\}$\\\
$\mathbb{Z}_{2}$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\end{tabular}\text{ and
}\begin{tabular}[]{l|ll}$+$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\\\
\hline\cr$\\{\widehat{0}\\}$&$\\{\widehat{0}\\}$&$\mathbb{Z}_{2}$\\\
$\mathbb{Z}_{2}$&$\mathbb{Z}_{2}$&$\mathbb{Z}_{2}$\end{tabular}.$

Now we consider two MV-algebras isomorphic with
$Id\left(\mathbb{Z}_{2}\right)$ denoted
$\mathcal{L}_{1}=(L_{1}=\\{0,a\\},\wedge,\vee,\odot,\rightarrow,0,a)$ and
$\mathcal{L}_{2}=(L_{2}=\\{a,1\\},\wedge,\vee,\odot,\rightarrow,a,1).$ Using
Proposition 34 we can construct the BL-algebra
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}=(L_{1}\cup
L_{2}=\\{0,a.1\\},\wedge,\vee,\odot,\rightarrow,0,1)$ with $0\leq a\leq 1$ and
the following operations:

$\rightarrow$ | $0$ | $a$ | $1$
---|---|---|---
$0$ | $1$ | $1$ | $1$
$a$ | $0$ | $1$ | $1$
$1$ | $0$ | $a$ | $1$
and  $\odot$ | $0$ | $a$ | $1$
---|---|---|---
$0$ | $0$ | $0$ | $0$
$a$ | $0$ | $a$ | $a$
$1$ | $0$ | $a$ | $1$
,

obtaining the same BL-algebra of order $3$ as in Example 30.

###### Example 37

To generate the non-linearly ordered BL-algebra with 5 elements from Example
7, we consider the commutative rings $\ (\mathbb{Z}_{2},+,\cdot)$ and
$(\mathbb{Z}_{2}\times\mathbb{Z}_{2},+,\cdot).$ For
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}=\\{\left(\widehat{0},\widehat{0}\right),\left(\widehat{0},\widehat{1}\right),\left(\widehat{1},\widehat{0}\right),\left(\widehat{1},\widehat{1}\right)\\}$,
we obtain the lattice
$Id\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right)=\\{\left(\widehat{0},\widehat{0}\right),\\{\left(\widehat{0},\widehat{0}\right),\left(\widehat{0},\widehat{1}\right)\\},\\{\left(\widehat{0},\widehat{0}\right),\left(\widehat{1},\widehat{0}\right)\\},\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\}=\\{O,R,B,E\\}$,
which is an MV-algebra
$(Id\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right),\cap,+,\otimes\rightarrow,0=\\{\left(\widehat{0},\widehat{0}\right)\\},1=\mathbb{Z}_{2}\times\mathbb{Z}_{2}),$
see Proposition 19. In $Id\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right)$ we
have the following operations:

$\begin{tabular}[]{l|llll}$\rightarrow$&$O$&$R$&$B$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$\\\ $R$&$B$&$E$&$B$&$E$\\\ $B$&$R$&$R$&$E$&$E$\\\
$E$&$O$&$R$&$B$&$E$\end{tabular},\text{
}\begin{tabular}[]{l|llll}$\otimes=\cap$&$O$&$R$&$B$&$E$\\\
\hline\cr$O$&$O$&$O$&$O$&$O$\\\ $R$&$O$&$R$&$O$&$R$\\\ $B$&$O$&$O$&$B$&$B$\\\
$E$&$O$&$R$&$B$&$E$\end{tabular}\text{ and
}\begin{tabular}[]{l|llll}$+$&$O$&$R$&$B$&$E$\\\
\hline\cr$O$&$O$&$R$&$B$&$E$\\\ $R$&$R$&$R$&$E$&$E$\\\ $B$&$B$&$E$&$B$&$E$\\\
$E$&$E$&$E$&$E$&$E$\end{tabular}.$

If we consider two MV-algebras isomorphic with
$(Id\left(\mathbb{Z}_{2}\right),\cap,+,\otimes\rightarrow,0=\\{0\\},1=\mathbb{Z}_{2})$
and
$(Id\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right),\cap,+,\otimes\rightarrow,0=\\{\left(\widehat{0},\widehat{0}\right)\\},1=\mathbb{Z}_{2}\times\mathbb{Z}_{2}),$
denoted by
$\mathcal{L}_{1}=(L_{1}=\\{0,c\\},\wedge_{1},\vee_{1},\odot_{1},\rightarrow_{1},0,c)$
and
$\mathcal{L}_{2}=(L_{2}=\\{c,a,b,1\\},\wedge_{2},\vee_{2},\odot_{2},\rightarrow_{2},c,1),$
using Proposition 34 we generate the BL-algebra
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}=(L_{1}\cup
L_{2}=\\{0,c,a,b,1\\},\wedge,\vee,\odot,\rightarrow,0,1)$ from Example 7.

From Proposition 34 and Remark 35, we deduce that:

###### Proposition 38

(i) To generate a BL-algebra with $n$ elements as an ordinal product
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of two Bl-algebras $\mathcal{L}_{1}$
and $\mathcal{L}_{2}$ we have the following possibilities:

$\mathcal{L}_{1}\text{ is a BL-chain with }i\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }j\text{ elements, }$

and

$\mathcal{L}_{1}\text{ is a BL-chain with }j\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }i\text{ elements, }$

or

$\mathcal{L}_{1}\text{ is a BL-chain with }k\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }k\text{ elements, }$

for $i,j\geq 2,i+j=n+1,$ $i<j$ and $k\geq 2,$ $k=\frac{n+1}{2}$ $\in N,$

(ii) To generate a BL-chain with $n$ elements as the ordinal product
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of two Bl-algebras $\mathcal{L}_{1}$
and $\mathcal{L}_{2}$ we have the following possibilities:

$\mathcal{L}_{1}\text{ is a BL-chain with }i\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-chain with }j\text{ elements, }$

and

$\mathcal{L}_{1}\text{ is a BL-chain with }j\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-chain with }i\text{ elements, }$

or

$\mathcal{L}_{1}\text{ is a BL-chain with }k\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-chain with }k\text{ elements, }$

for $i,j\geq 2,i+j=n+1,$ $i<j$ and $k\geq 2,$ $k=\frac{n+1}{2}$ $\in N.$

We make the following notations:

$\mathcal{BL}_{n}=\text{the set of BL-algebras with }n\text{ elements}$
$\mathcal{BL}_{n}(c)=\text{the set of BL-chains with }n\text{ elements}$
$\mathcal{MV}_{n}=\text{the set of MV-algebras with }n\text{ elements}$
$\mathcal{MV}_{n}(c)=\text{the set of MV-algebras with }n\text{ elements.}$

###### Theorem 39

(i) All finite Bl-algebras (up to an isomorphism) which are not MV-algebras
with $2\leq n\leq 5$ elements can be generated using the ordinal product of
BL-algebras.

(ii) The number of non-isomorphic BL-algebras with $n$ elements (with $2\leq
n\leq 5$) is

$\left|\mathcal{BL}_{2}\right|=\left|\mathcal{MV}_{2}\right|=\pi(2)+1,$
$\left|\mathcal{BL}_{3}\right|=\left|\mathcal{MV}_{3}\right|+\left|\mathcal{BL}_{2}\right|=\pi(3)+\pi(2)+2,$
$\left|\mathcal{BL}_{4}\right|=\left|\mathcal{MV}_{4}\right|+\left|\mathcal{BL}_{3}\right|+\left|\mathcal{BL}_{2}\right|=\pi(4)+\pi(3)+2\cdot\pi(2)+4,$
$\left|\mathcal{BL}_{5}\right|=\left|\mathcal{MV}_{5}\right|+\left|\mathcal{BL}_{4}\right|+\left|\mathcal{BL}_{3}\right|+\left|\mathcal{BL}_{2}\right|=$
$=\pi(5)+\pi(4)+2\cdot\pi(3)+4\cdot\pi(2)+8.$

Proof. From Proposition 34 and Remark 35, we remark that using the ordinal
product of two BL-algebras we can generate only BL-algebras which are not MV-
algebras.

We generate all BL-algebras with $n$ elements ($2\leq n\leq 5)$ which are not
MV-algebras.

Case $n=2.$

We have obviously an only BL-algebra (up to an isomorphism) isomorphic with

$(Id(\mathbb{Z}_{2}),\cap,+,\otimes\rightarrow,0=\\{0\\},1=\mathbb{Z}_{2}).$

In fact, this residuated lattice is a BL-chain and is the only MV-algebra with
$2$ elements. We deduce that

$\left|\mathcal{MV}_{2}\right|=\left|\mathcal{BL}_{2}\right|=\pi(2)+1=1$
$\left|\mathcal{MV}_{2}(c)\right|=\left|\mathcal{BL}_{2}(c)\right|=1.$

Case $n=3.$

Using Proposition 38, to generate a BL-algebra with $3$ elements as an ordinal
product $\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of two BL-algebras
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ we must consider:

$\mathcal{L}_{1}\text{ a BL-chain with }2\text{ elements and
}\mathcal{L}_{2}\text{ a BL-algebra with }2\text{ elements. }$

Since there is only one BL-algebra with $2$ elements and it is a chain, we
obtain the BL-algebra

$Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2}),$

which is a chain.

We deduce that

$\left|\mathcal{MV}_{3}\right|=\pi(3)+1\text{ and
}\left|\mathcal{BL}_{3}\right|=\left|\mathcal{MV}_{3}\right|+1\cdot\left|\mathcal{BL}_{2}\right|=\pi(3)+\pi(2)+2=2$
$\left|\mathcal{MV}_{3}(c)\right|=1\text{ and
}\left|\mathcal{BL}_{3}(c)\right|=\left|\mathcal{MV}_{3}(c)\right|+1=1+1=2.$

We remark that
$\left|\mathcal{BL}_{3}\right|=\left|\mathcal{MV}_{3}\right|+\left|\mathcal{BL}_{2}\right|.$

Case $n=4.$

Using Proposition 38, to generate a BL-algebra with $4$ elements as the
ordinal product $\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of two Bl-algebras
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ we must consider:

$\mathcal{L}_{1}\text{ is a BL-chain with }2\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }3\text{ elements, }$
$\mathcal{L}_{1}\text{ is a BL-chain with }3\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }2\text{ elements. }$

We obtain the following BL-algebras:

$Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4})\text{ and
}Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2}))\text{ }$

and

$Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{2})\text{ and
}(Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2}))\boxtimes
Id(\mathbb{Z}_{2}).$

Since $\boxtimes$ is associative, we obtain 3 BL-algebras which are chains,
from Remark 35.

We deduce that

$\left|\mathcal{MV}_{4}\right|=\pi(4)+1\text{ }$
$\left|\mathcal{BL}_{4}\right|=\left|\mathcal{MV}_{4}\right|+1\cdot\left|\mathcal{BL}_{3}\right|+2\cdot\left|\mathcal{BL}_{2}\right|-1=\pi(4)+\pi(3)+2\cdot\pi(2)+4=5$
$\left|\mathcal{MV}_{4}(c)\right|=1\text{ and
}\left|\mathcal{BL}_{4}(c)\right|=\left|\mathcal{MV}_{3}(c)\right|+3=1+3=4.$

We remark that
$\left|\mathcal{BL}_{4}\right|=\left|\mathcal{MV}_{4}\right|+\left|\mathcal{BL}_{3}\right|+\left|\mathcal{BL}_{2}\right|.$

Case $n=5.$

To generate a BL-algebra with $5$ elements as the ordinal product
$\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of two Bl-algebras $\mathcal{L}_{1}$
and $\mathcal{L}_{2}$ we must consider:

$\mathcal{L}_{1}\text{ is a BL-chain with }2\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }4\text{ elements, }$
$\mathcal{L}_{1}\text{ is a BL-chain with }4\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }2\text{ elements. }$
$\mathcal{L}_{1}\text{ is a BL-chain with }3\text{ elements and
}\mathcal{L}_{2}\text{ is a BL-algebra with }3\text{ elements,.}$

We obtain the following BL-algebras:

$\displaystyle Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{8}),\text{
}Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2}\times\mathbb{Z}_{2}),\text{
}Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4})),$
$\displaystyle Id(\mathbb{Z}_{2})\boxtimes[Id(\mathbb{Z}_{4})\boxtimes
Id(\mathbb{Z}_{2})]\text{ and
}Id(\mathbb{Z}_{2})\boxtimes[Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2}))]$

and

$\displaystyle Id(\mathbb{Z}_{8})\boxtimes
Id(\mathbb{Z}_{2}),[Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4})]\boxtimes
Id(\mathbb{Z}_{2}),$ $\displaystyle[Id(\mathbb{Z}_{4})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes Id(\mathbb{Z}_{2})\text{ and
}[Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2}))]\boxtimes Id(\mathbb{Z}_{2})$

and

$\displaystyle Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{4}),\text{
}[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes[Id(\mathbb{Z}_{2}))\boxtimes Id(\mathbb{Z}_{2})]$
$\displaystyle Id(\mathbb{Z}_{4})\boxtimes[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})]\text{ and }[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes Id(\mathbb{Z}_{4})$

Since $\boxtimes$ is associative,
$Id(\mathbb{Z}_{2})\boxtimes[Id(\mathbb{Z}_{4})\boxtimes
Id(\mathbb{Z}_{2})]=[Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4})]\boxtimes
Id(\mathbb{Z}_{2}),$
$Id(\mathbb{Z}_{2})\boxtimes[Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2}))]$ $=$
$[Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2}))]\boxtimes Id(\mathbb{Z}_{2})$
$=[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes[Id(\mathbb{Z}_{2}))\boxtimes
Id(\mathbb{Z}_{2})],$ $[Id(\mathbb{Z}_{4})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes
Id(\mathbb{Z}_{2})=Id(\mathbb{Z}_{4})\boxtimes[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})],$ $Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{4}))=$ $[Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{2})]\boxtimes Id(\mathbb{Z}_{4})$ and
$Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes
Id(\mathbb{Z}_{4}))=[Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2})]\boxtimes
Id(\mathbb{Z}_{4})$

We obtain 8 BL-algebras of which 7 are chains, from Remark 35.

We deduce that

$\left|\mathcal{MV}_{5}\right|=\pi(5)+1=1\text{ and
}\left|\mathcal{BL}_{5}\right|=9=\left|\mathcal{MV}_{5}\right|+\left|\mathcal{BL}_{4}\right|+\left|\mathcal{BL}_{3}\right|+\left|\mathcal{BL}_{2}\right|$
$\left|\mathcal{MV}_{5}(c)\right|=1\text{ and
}\left|\mathcal{BL}_{5}(c)\right|=8.$

Table 2 present a basic summary of the structure of BL-algebras $L$ with
$2\leq n\leq 5$ elements:

Table 2:

$\left|L\right|\mathbf{=n}$ | Nr of BL-alg | Structure
---|---|---
$n=2$ | $1$ | $\left\\{Id(\mathbb{Z}_{2})\text{ (chain, MV)}\right.$
$n=3$ | $2$ | $\left\\{\begin{array}[]{c}Id(\mathbb{Z}_{4})\text{ (chain, MV)}\\\ Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2})\text{ (chain)}\end{array}\right.$
$n=4$ | $5$ | $\left\\{\begin{array}[]{c}Id(\mathbb{Z}_{8})\text{ (chain, MV)}\\\ Id(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\text{ (MV)}\\\ Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4})\text{ (chain)}\\\ Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{2})\text{ \ (chain)}\\\ Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2}))\text{ (chain)}\end{array}\right.$
$n=5$ | $9$ | $\left\\{\begin{array}[]{c}Id(\mathbb{Z}_{16})\text{ (chain, MV)}\\\ Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{8})\text{ (chain)}\\\ Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\\\ Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{4}))\text{ (chain)}\\\ Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{2}))\text{ (chain)}\\\ Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes(Id(\mathbb{Z}_{2})\boxtimes Id(\mathbb{Z}_{2})))\text{ (chain)}\\\ Id(\mathbb{Z}_{8})\boxtimes Id(\mathbb{Z}_{2})\text{ (chain)}\\\ (Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{2}))\boxtimes Id(\mathbb{Z}_{2})\text{ (chain)}\\\ Id(\mathbb{Z}_{4})\boxtimes Id(\mathbb{Z}_{4})\text{ (chain)}\end{array}\right.$

Finally, Table 3 present a summary for the number of MV-algebras, MV-chains,
BL-algebras and BL-chains with $n\leq 5$ elements:

Table 3

| $n=2$ | $n=3$ | $n=4$ | $n=5$
---|---|---|---|---
MV-algebras | $1$ | $1$ | $2$ | $1$
MV-chains | $1$ | $1$ | $1$ | $1$
BL-algebras | $1$ | $2$ | $5$ | $9$
BL-chains | $1$ | $2$ | $4$ | $8$

.

## References

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Cristina Flaut

Faculty of Mathematics and Computer Science, Ovidius University,

Bd. Mamaia 124, 900527, Constanţa, România,

http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Dana Piciu

Faculty of Science, University of Craiova,

A.I. Cuza Street, 13, 200585, Craiova, Romania,

http://www.math.ucv.ro/dep_mate/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>


# Hubbert’s theory and photovoltaics: the nonsense race of breaking energy-
conversion records

Marcos Paulo Belançon<EMAIL_ADDRESS>Programa de Pós-graduação
em Processos Químicos e Bioquímicos
Universidade Tecnológica Federal do Paraná
CEP 85503-390, Pato Branco, Paraná, Brasil

###### Abstract

The wide public sees solar energy as the future of mankind, and media channels
quite oftenly states that our challenge is to improve efficiencies and reduce
cost. However, one may point some unconvenient truth’s about the physical
limits we are facing, that are barely discussed by the public or even by
scientists and institutions that are strongly biased towards a picture of a
sustainable oil free energy in the future. In this work we discuss some of
those physical limits of photovoltaics based on the principle of the Hubbert’s
theory for the oil peak, evidencing that much of the research is focused on
photovoltaic efficiencies and this parameter is widely overestimated: better
efficiencies oftenly are the result of complex technologies that are expensive
and not scalable. In this context, if fossil fuels proved not replaceable, it
is very likely that our socioeconomic ideas based in the past will not
withstand an energy transition to high cost and low quality sources.

##  I Improving photovoltaics

A quick search in the Web Of Science for “photovoltaic efficiencies” provides
us an insight of how much research is being carried on this subject, or is at
least related to it. This data is shown in figure 1, where in the inset we can
see the cumulative photovoltaic (PV) power in the world.

Figure 1: Number of articles in the Web Of Science database containing the
term “photovoltaic efficiencies”.

As one can see, the same exponential behavior is observed in both curves.
However, despite growth records of PV’s in the world, forecasts from the
Intergovernmental Panel on Climate Change (IPCC), the Energy Information
Administration (EIA) or from companies such as the British Petroleum (BP)
still give a picture of a future based mainly on oil and natural gas. Some
will say that the potential of solar energy is being underestimatedCreutzig
_et al._ (2017).

Many different dreams about our sustainable energy future are being
discussedChu and Majumdar (2012); Haegel  _et al._ (2017). Thousands of
researchers are trying to improve solar energy to electricity conversion
today; about 14 articles are being published everyday covering from
development of new PV’sLee and Ebong (2016); Albrecht and Rech (2017) until
spectral convertersHuang  _et al._ (2013) and solar thermal devicesKannan and
Vakeesan (2016); Nathan S. Lewis (2016). A handful of different PV
technologies are already very competitive in terms of efficiencyRand  _et al._
(2007); Polman  _et al._ (2016), however, as it is indicated in figure 1 our
community is worried in further improvements.

##  II The role of energy-conversion efficiencies

The unconvenient truth is that PV efficiencies may not help our energy
transition. Looking for example to the National Renewable Energy Laboratory
(NREL) efficiency chart for PV’s, one may see that the Silicon technology
widely dominant in the market today is almost 30 years old. By this way one
may interprete that, except by improvements in production techniques, Silicon
PV’s are still the same. When scientists have developed a new PV technology in
the last few decades, even though some energy conversion records were broken,
it has produced no net effect in our hability to produce solar energy. Some
technologies, such as CdTe PV‘s are competitive today, however the dependende
of scarse minerals makes impossible to scale up the production as much as we
need.

Even more interesting is to have a look in the NREL PV System Cost BenchmarkFu
_et al._ (2017) where they modeled the impact of module energy-conversion
efficiency ($\eta$) on total system costs. While authors oftenly interpret
that improve $\eta$ is “probably the key”Green (2016) to reduce costs, the
NREL model, even considering the optimistic assumption of reach $\eta=60\%$ by
the same price of today’s modules, does not project a strong reduction in the
total cost. As the data in table 2 shows, higher efficiencies may reduce costs
by only $\sim 20\%$.

Type | $\eta$ | Cost (US$/Wp) | Cost if $\eta=60\%$
---|---|---|---
Residential | 16.2% | 2.90 | 2.18
Commercial | 17.5% | 1.84 | 1.31
Utility-Scale | 17.5% | 1.09 | 0.78
Table 1: Costs and average efficiencies of different PV’s projects 222data
from Fu  _et al._ (2017), pg 45.

This data should not be miss interpreted: higher $\eta$ itself is desirable,
however, if we aim an approach towards global scale applications of PV’s, one
may not neglect the scale.

Two centuries ago, the economist Jean-Baptiste SaySay (1803) assumed that the
scale of our consumption was so small that natural resources would last
forever. This assumption, that natural resources are not produced or consumed
is the basis of his work. It is remarkable that this aproximation made by Say
has worked by so much time, however, everyone should consider that since his
time we multiplied the population several times and the energy consumption per
capita even more.

Many natural resources are now being consumed, such as oil and gas; many
others are being diluted, such as Silver, Copper and every single element
mined on earth. Mining or recycling less concentrated ores consumes more
energy, what is a consequence of the laws of thermodynamics. Modern economies
worldwide are not considering the implications that today’s reality is
imposing to us, and it does not matter if we peak up a liberalist or communist
one; the economic theories behind will always consider that the market will
find another resource once the first one is exhausted.

##  III Why scale matters?

Even though PV’s cumulative capacity growth may be interpreted as astoshing or
promising, one should take care to make a fair comparison between the facts
and our expectations. To give some perspective, in figure 2 we have the
variation of US crude oil production in each decade since the 1880’s.

Figure 2: Percentual growth of US oil production per decade.

From the begining of the oil exploration until the peak of conventional oil
production in the 1970’s the producion has grown pretty fast. Between 1900 and
1929 the production was multiplied by a factor of 15. And then, after the peak
in the 1970’s, oil imports contributed to install a chronic debt in the trade
balance of the country with the world. Even though the US had already known
non-conventional oil and gas sources, the market has choose to buy the cheaper
conventional alternative in Venezuela and Midde-east.

It is important to pay attention to some facts in this history. When the
cumulative capacity is very low it is common to have growth rates of two or
even three digits. In 1900 the US oil production has grown at rates similar to
those predicted for the photovoltaics in the coming decade. The other lesson
from the US oil peak is that consequences of such event may be huge and
difficult to predict, economically and politically.

Hubbert predicted in 1956Hubbert (1956) the decline of oil production in US
and 15 years after the history proved him to be right. Hubbert’s theory can
not be applied to solar energy, however, it may be more or less applied to all
the raw material we need to build the PV’s. By this way, we should take a look
in the real scale of supply, demand and raw material consumption for solar
energy. Figure 3 shows a plot of the added capacity (AC) in the last few
years, from which we may found that it has grown more or less linearly.

Figure 3: PV’s world production capacity.

The challenge that PV’s are facing is not only in reducing cost or improving
efficiencies; neither is increase production capacity. The challenge is if AC
can increase faster than linearly, or at least what slope is sustainable from
the perspective of resources and cost. In the last few year the prices were
pressured down mainly by overproduction in China, what put many companies in
the west in a difficult situation.

On the perspective of available resources, the World Silver Survey 2017, from
The Silver Institute, shows that in 2011 and in 2016 the PV’s industry has
consumed the same amouth of Silver, of about 75 million ounces, while the AC
speed up by $\sim 3$. This achievement is remarkable, however, as it shown in
International Technology Roadmap For Photovoltaics 2017, it is not so clear
that industry will continue to reduce Silver consumption per cell so fast as
in the past. There are not much room for improvements when the technology is
already mature, and Silver production is a concernSverdrup  _et al._ (2014);
Grandell and Thorenz (2014).

The world PV’s installed capacity is around 300 GWp today, which corresponds
to an average of 60 GW of electricity. This is in the range of the the energy
equivalent to the US oil production in the time of the first world war, which
was less than 1 million barrels a day. These data discussed here supports the
idea that even running far from the scale of mankind’s needs, PV’s are already
facing finite resources constraintsTao  _et al._ (2011).

After China dominated the PV’s market, many companies in US, Europe and Japan
lost value or even declared insolvency. To keep AC increasing while
maintaining or reducing the prices the PV’s industry needs to keep sharing
more or less the same 75 million ounces of Silver they consumed last year. In
other words, it is necessary to reduce the silver contentGrandell and Thorenz
(2014). From the materials perspective Copper may be used to replace Silver,
however it does not have the same conductivity and chemical resistance. The
assumption that we will use less materials, with lower quality but engineered
in such way that will result in more efficient devices is too much optimistic
and not scientific.

Back to the scale needed, even a conservative projectionFraunhofer ISE (2015)
considers an AC of PV’s per year three times higher than today. While Silicon
technology needs Silver, others are based on elements even more difficult to
obtain, such as Cadmium, Tellurium, Indium, Gallium, Selenium or some
combination of them. A few of those elements are byproduct mineralBleiwas
(2010); Graedel (2011); Grandell  _et al._ (2016) and by this way the
production of such CdTe, CIS or CIGS PV’s is very unlikely to reach the
Silicon PV’s production scaleTao  _et al._ (2011). If our best shot to produce
solar electricity will be with the 30 years old single junction Silicon
technology, we may reduce Silver consumption per cell further, what will
enable AC to be increased by the “conservative” (or realistic) factor of three
in the next decades.

##  IV Production capacity in the future

To clarify this picture of how far we are from the “Hubbert’s limits” the
figure 4 show some very simple scenarios for PC of PV’s. At left in the figure
we can see the same data from figure 3, for the AC, as well the linear fit
represented by the blue line. The slope for this curve is about 7 GWp/year.
This is how much we are speeding up the addition of PV‘s in the world in the
last 10 years, on average. If the slope surpasses the growing demand of PV‘s,
we should have and excess suply that in will in turn make some companies to
crash: a common market driven adjustment.

The three other lines in the figure 4 project what should be the slope if we
aim to reach $PC=500$ GWp in 2030, 2040 or 2050. This 500 GWp of PC should be
enough to reach in a few decades about 12.5 TWp of installed capacity; which
means something like 2.5 TW on average. This value is still a fraction of how
much energy we consume worldwide but it is useful to provide a picture of the
point discussed here.

Figure 4: PV’s world production capacity and slopes needed to reach 500 GWp in
the next decades.

If PC keeps increasing at the present rate, we are not going to reach the 500
GWp before 2060. Even to reach this level in 2050 the slope needs to be
increased to about 12 GWp/year. This means for example that, to avoid Silver
constraints the industry should reduce its consumption per Wp in about 80% in
33 years. Even though this kind of reduction was achieved in the last ten
years, it is not clear that such reduction can be achieved again without
undesirable performance effects. And this is only one example.

The point here is that the question that should receive more attention from
the community is if are all the pieces in the production chain of
photovoltaics “free” to grow? Every time one constraint appear, the prices may
not decrease, what can reduce demand and investment, compromising the PC
growth. Our biggest challenge is than the scale of our needs and a quite short
deadline.

##  V Scaling up production instead of improving efficiencies

In a recent report from the HSBC Global ResearchFustier  _et al._ (2017),
Fustier et al made estimates that it is very likely that today’s oil field
will decline in production by more than 40 million barrels a day until 2040.
To replace it, the non-conventional fossil fuels are already been extracted,
mainly in Shale plays in USKerr (2010); Hughes (2013) and oil sands in Canada.

Since 2008 the shale rush made the US Oil production increase from $\sim 5$ to
$\sim 9$ million barrels a day, deviating from the Hubbert prediction made 70
years ago. 4 million barrels more per day is energy equivalent to much more
than what PV’s of the world can deliver today. However, this non-conventional
oil source has its own problems: the decline production of individual Shale
wells may be as fast as 50% in the first year. So, the Hubbert’s peak for this
non-conventional source is very likely to come faster than for conventional
sources. By this way, the oil decline is very likely do impact the cost of our
energy at the same time it may introduce a kind of deadline to perform a
transition to renewables.

I wish PV’s could be the alternative to replace conventional oil and gas.
However, considering the prediction made by Fustier, to accomplish that we
should multiply our installed PV’s by a factor of about 40 until 2040, which
should require a PC of 500 GWp on average from now on; which is clearly not
feasible. Anyway, if such oil decline is going to happen, to replace it
scientists working in the field should change their research focus.

Across the globe we are working in “photovoltaic efficiencies” as mentioned
earlier, and it is very unlikely that such contributions will ever have some
effect in our everyday lives. We should than focus on expanding the limits of
the cumulative capacity, by increasing life span of PV’s, reducing and
eventually replacing scarse materials such as Silver and developing techniques
that could make recycling PV’s easier.

All of that could make possible to achieve a higher slope in PC growth;
increase photovoltaic efficiences does not help pretty much towards this
direction because we are already near the limits of single junction solar cell
efficiencies. And we may add that it could be helpful for scientists and the
public in general to think more broadly about the energy transition. If
replace fossil fuels prove not to be possible in the coming decades, we may
need to look to other options.

##  VI Other option?

The mankind’s appetite for energy is huge and our institutions are also
addicted to it. The IPCC has more specialists in economics than in ecology,
and the prevalence of men and europeans is clearvan den Bergh (2017). When
scientists try to follow the biased view of this kind of institution to pursue
grants and publications, science as a whole is in serious danger. The IPCC
does not put all option in the table (and in the reports), because only
perpetual growth of gross domestic product (GDP) are allowed to be present in
the debate. A possible limit to the ideology of unlimited growthKallis (2011);
van den Bergh (2017) is not being considered, even though it could solve many
aspects of our situation.

Once we are not near of solving our sustainability problemRaftery  _et al._
(2017), all the options should be present in the scientific debate promoted by
institution such as the IPCC. That should include many aspects of the
feasibility in the long term of our civilization as it is today. There are
socioeconomic options that aim reduce consumption of resources and increase
prosperityJackson (2009), however, the institutions did not consider it
seriously.

##  VII Conclusion

If we choose to replace conventional by non-conventional fossil fuels, the
overall cost of energy will rise. On the other hand, if we choose renewables,
such as PV‘s, to reach the scale of our needs it is very likely that the
overall cost of energy will rise, also.

While scientists are blindly wasting time following what may be proved as
useless goals, they are also wasting intellectual resources that could be used
to fix our civilization instead of, at maximum, create a new PV company to
produce a few more GW’s that may solve a tiny part of the problem. The tragedy
of pursuing continuous growth of the energy supply is that the manutention
costs of our civilization are growing togetherTainter and Patzek (2012); and
after the Hubbert’s peak of the key resource we will face deplection while our
manutention cost keeps rising. There are no high tech solution that will avoid
this dilemma.

In the 1980’s France has builtDussud  _et al._ (2016) an impressive fleet of
nuclear reactors, and even though Uranium production did not peaked, it did
not avoid the problem that now the country needs to replace the nuclear
reactors to keep the supply of energy. There are other hundreds of nuclear
reactors facing the same critical point around the world.

So, the big question is: can we replace fossil fuels by PV‘s or other
renewables without replace our socioeconomic status quo? Some scientists will
answer that yes, of course. But these answers are about faith and not
supported by scientific evidence.

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