# Robust zero modes in non-Hermitian systems without global symmetries

Jose D. H. Rivero  College of Staten Island, CUNY, Staten Island, New York
10314, USA  The Graduate Center, CUNY, New York, NY 10016, USA  Courtney
Fleming  College of Staten Island, CUNY, Staten Island, New York 10314, USA
The Graduate Center, CUNY, New York, NY 10016, USA  Bingkun Qi  College of
Staten Island, CUNY, Staten Island, New York 10314, USA  The Graduate Center,
CUNY, New York, NY 10016, USA  Liang Feng  Department of Materials Science and
Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA  Li Ge
College of Staten Island, CUNY, Staten Island, New York 10314, USA  The
Graduate Center, CUNY, New York, NY 10016, USA<EMAIL_ADDRESS>
###### Abstract

We present an approach to achieve zero modes in lattice models that do not
rely on any symmetry or topology of the bulk, which are robust against
disorder in the bulk of any type and strength. Such symmetry-free zero modes
(SFZMs) are formed by attaching a single site or small cluster with zero
mode(s) to the bulk, which serves as the “nucleus” that expands to the entire
lattice. We identify the requirements on the couplings between this boundary
and the bulk, which reveals that this approach is intrinsically non-Hermitian.
We then provide several examples with either an arbitrary or structured bulk,
forming spectrally embedded zero modes in the bulk continuum, midgap zero
modes, and even restoring the “zeroness” of coupling or disorder-shifted
topological corner states. Focusing on viable realizations using photonic
lattices, we show that the resulting SFZM can be observed as the single lasing
mode when optical gain is applied to the boundary.

Zero modes occupy an indisputable place of interest in physics at many
different levels: They appear as localized boundary states in topological
insulators [1, 2], Majorana fermions in superconductors and semiconductors [3,
4, 5], corner states in higher-dimensional multipole insulators [6, 7, 8],
single-mode lasers with narrow spectral lines [9, 10, 11, 12, 13], and
localized defect modes in photonic crystals [14, 15, 16, 17, 18]. Many
advantages come with the ability to excite these states in the middle of band
gaps, and the symmetry or topological properties of the underlying system can
be judiciously implemented to protect those states.

It is widely accepted that symmetries play a special role in the construction
of these zero modes in complex systems, as they introduce a privileged energy
(i.e., “zero energy”) that serves as a reference for all the energies of the
system. In condensed matter systems, this zero energy is often the Fermi
energy [19], and in coupled photonic systems, the resonant frequency of
interest in a single cavity or waveguide. Two well-known symmetries that
warrant zero modes in Hermitian systems are chiral and particle-hole
symmetries, which lead to a symmetric energy spectrum about zero [20]. The
study of non-Hermitian physics has also exploited and extended these two
symmetries in the presence of dissipation, where gain and loss can be
engineered to produce zero modes with unique properties [21]. As the spectrum
becomes complex, chiral symmetry [22, 23, 24, 25, 26], as well as pseudo-
chirality [27], still hosts paired modes with energy $E$ and $-E$; particle-
hole symmetry [9, 28, 29, 30, 31], along side anti-PT symmetry [32, 33] and
pseudo-anti-Hermiticity [34], now produces mode pairs with energy $E$ and
$-E^{*}$, where the asterisk denotes the complex conjugation as usual.

As these symmetries are global properties of the underlying system, the zero
modes they induce are easily lifted in the presence of local perturbations
that do not preserve these symmetries. For example, the zero modes of a
chiral-symmetric lattice disappear in the presence of strong onsite disorder.
While a lattice with non-Hermitian particle-hole symmetry has enhanced
protection in comparison from gain and loss disorder [9], its zero modes are
still eliminated with onsite frequency detunings. Although one may mitigate
the problem by employing topological protection [35, 36, 37, 38], the
resulting modes typically still have energy shifted away from zero, causing
uncertainties in the identification and characterization of the zero modes.

Meanwhile, a zero mode in a single site or a small cluster is relatively easy
to form, since the zero energy, for example in a photonic lattice, is chosen
as that of a resonant frequency in a single cavity as previously mentioned.
Therefore, can one utilize such a zero mode as the “nucleus” and expand it to
the entire lattice?

In this work, we elucidate this vastly different approach to realize a robust
zero mode in lattice models, which does not depend on the symmetry or
topological properties of the bulk. By attaching the aforementioned nucleus to
an arbitrary bulk and referring to the former as the boundary, we identify the
conditions needed on the bulk-boundary couplings to form such a symmetry-free
zero mode (SFZM). It is worth noting that non-Hermiticity is intrinsic in our
approach, either in the form of asymmetric couplings [39, 40, 41, 42, 43] or a
non-Hermitian boundary.

A distinct property of SFZMs is their robustness against any type of disorder
in the bulk Hamiltonian. Furthermore, they can exist in systems both with and
without a bulk energy gap at zero energy, which enables us to freely form
spectrally embedded zero modes similar to bound states in the continuum (BIC)
of the bulk [44, 45], midgap zero modes, and even restore the “zeroness” of
coupling or disorder-shifted topological corner states. Therefore, while the
mechanism we introduce to achieve a zero mode in the bulk does not rely on
global symmetries, it can be applied to systems with global symmetries and
topological protection as well. In this case, our approach provides additional
robustness to the underlying system, stabilizing its energy against symmetry-
breaking perturbations. Focusing on viable realizations using photonic
lattices, we also show that an SFZM can be observed as the single lasing mode
when optical gain is applied to the boundary. Such an approach can benefit
many applications, especially those in optics and photonics that rely on a
source of a fixed frequency for spectroscopy, metrology, and sensing.

Figure 1: Schematics of systems hosting an SFZM, where the boundary is a
single cavity in (a) and a small cluster in (b).

Below we consider the following Hamiltonian

$H=\begin{pmatrix}H_{B}&V\\\ U&H_{0}\end{pmatrix}$ (1)

consisting of the bulk ($H_{B}$), the boundary ($H_{0}$), and their couplings
$U,V$. Our boundary serves as the nucleus and hosts zero mode(s) itself
(Condition 0), i.e., $H_{0}\phi_{i}=0$. By partitioning the wave function of
the whole system as $\Psi=\psi_{B}\oplus\psi_{0}$, a zero mode of the system
satisfies

$H_{0}^{\prime}\psi_{0}\equiv(H_{0}-UH_{B}^{-1}V)\psi_{0}=0,\;\;\;\psi_{B}=-H_{B}^{-1}V\psi_{0}.$
(2)

Our goal is to achieve zero mode(s) inside a bulk without pre-existing ones,
and hence we have taken $H_{B}$ to be invertible. We also note that the second
relation in Eq. (2) indicates that the zero-mode wave function in the bulk
(i.e., $\psi_{B}$) can be obtained by propagating, via the bulk Green’s
function $G_{B}(E)=(E-H_{B})^{-1}$ at the zero energy, the couplings from the
boundary.

To identify a set of general conditions that enables an SFZM, we use the
property that a Hamiltonian always has paired left and right eigenstates of
the same energies [46, 47]. The first relation in Eq. (2) is then satisfied if
we can construct a simple left eigenstate of $H^{\prime}_{0}$ with $E=0$,
which should be independent of the bulk Hamiltonian $H_{B}$. This goal can be
achieved by requiring $\tilde{\phi}_{i}^{T}U=0$ (Condition 1), where
$\tilde{\phi}_{i}^{T}$ is the left eigenstate of a boundary zero mode, defined
by $\tilde{\phi}_{i}^{T}H_{0}=0$. This condition can be easily verified by
noting $\tilde{\phi}_{i}^{T}(H_{0}-UH_{B}^{-1}V)=0$. In other words,
$\tilde{\phi}_{i}^{T}$ and $\psi_{0}$ are the left and right eigenstates of
$H_{0}^{\prime}$ with $E=0$. In contrast, we note that the boundary zero mode
$\phi_{i}$ and the SFZM inside the boundary (i.e.,$\psi_{0}$) are not simply
related in general.

The resulting zero mode of $H$, however, has a vanished amplitude in the bulk
if $V\phi_{i}=0$: $V\phi_{i}=0$ indicates that $H_{0}^{\prime}\phi_{i}=0$,
i.e., the right eigenstate of $H_{0}^{\prime}$ with $E=0$, denoted by
$\psi_{0}$ in Eq. (2), is simply given by $\phi_{i}$. Using
$V\psi_{0}=V\phi_{i}=0$ in Eq. (2), we then find $\psi_{B}=0$. Therefore, to
achieve an SFZM that extends to the bulk, we require $V\phi_{i}\neq 0$
(Condition 2). An alternative derivation of Conditions 1 and 2 is given in
Ref. [48] using the Jordan normal form of $H_{0}$, where the case with a pre-
existing bulk zero mode is also discussed.

Conditions 1 and 2 imply that the system that hosts an SFZM is intrinsically
non-Hermitian: If the opposite were true, then $H_{0}$ must be Hermitian as
well, leading to $\tilde{\phi}_{i}^{T}=\phi_{i}^{\dagger}$. At the same time,
we have $U=V^{\dagger}$, and by taking the Hermitian conjugate of Condition 1,
we find $V\phi_{i}=0$ that violates Condition 2. Similarly, Conditions 1 and 2
also imply that $H$ is asymmetric, meaning that the left and right eigenstates
of the SFZM are distinct. This observation provides an intuitive understanding
of the SFZM: By imposing Condition 1, we achieve not just a left zero mode of
$H_{0}^{\prime}$ but also that of $H$, with the simple wave function
$\tilde{\Psi}^{T}=0\oplus\tilde{\phi}_{i}^{T}$ that vanishes in the bulk and
satisfies $\tilde{\Psi}^{T}H=0$. The corresponding and distinct right
eigenstate of $H$ is then the SFZM, which extends to the bulk under Condition
2.

As an example, we first consider the simplest case where the proposed SFZM
appears [Fig. 1(a)]. It features a single boundary site with $H_{0}=0$,
satisfying Condition 0 with a boundary zero mode given by
$\tilde{\phi}_{i}={\phi}_{i}=1$. Then Condition 1 dictates that the coupling
$U$ must be a zero row vector, i.e., no couplings from the bulk to the
boundary. The existence of a zero mode in the whole system is then apparent:
the bulk cannot act back on the boundary, and hence the frequency of the
latter’s zero mode is unchanged.

To ensure that the resulting SFZM extends into the bulk, Condition 2 requires
that the coupling $V$ is not a zero vector, which would otherwise isolate the
boundary from the bulk. We also note that the required unidirectional
couplings from the boundary to the bulk can be realized, for example, using
spiral waveguides evanescently coupled to micro-ring resonators [43]. For easy
implementation, coupling the boundary cavity to one in the bulk [e.g., the
$N$th cavity in Fig. 1(a)] is sufficient to warrant the SFZM. Importantly, the
bulk Hamiltonian $H_{B}$ can be completely arbitrary and does not affect the
existence of the SFZM [Fig. 2(a)].

Figure 2: (a) Representative complex energy spectrum for the configuration
shown in Fig. 1(a) with 30 bulk lattice sites. Both onsite energy and
couplings in the bulk are uniformly distributed in $[-g/2,g/2]$, whereas
onsite loss is uniformly distributed in $[-g/10,0]$. Black and orange dots
mark the SFZM and bulk nonzero modes. Black arrow shows the trajectory of the
SFZM when the cavity loss in the boundary is gradually compensated by an
increasing pump. (b) Spatial profiles of the SFZM and the circled bulk mode in
(a). Shaded area shows the bulk.

We note that this SFZM is the only eigenstate of the whole system that has a
finite amplitude at the boundary site. The other eigenstates are simply given
by all the $N$ eigenstates of the bulk, with a zero amplitude at the boundary
[see, for example, the dashed line in Fig. 2(b)], due to the prohibited
couplings from the bulk to the boundary. These properties facilitate the
observation of the SFZM in an active device (e.g., a laser [21]), which
exhibits mode(s) with the smallest loss or highest gain. Here by including
sufficient loss in the bulk, whether uniform or distributed, all the non-zero
modes in the whole system are in the lower half of the complex energy plane
[Fig. 2(a)]. As a result, only the SFZM is on the real axis and similar
spectrally to a BIC, especially as the density of states increases with the
bulk size.

This picture, of course, is simplified because we have not included the cavity
loss in the boundary cavity itself. However, even when we consider a decay
rate in this cavity comparable to the bulk modes ($\kappa/g=-0.05i$), the SFZM
can be observed directly as the only lasing mode in this system. This behavior
is shown by the black arrow in Fig. 2(a): We pump only this cavity that
reduces its effective cavity decay, i.e., $\kappa\rightarrow\kappa-\gamma$.
When the gain $\gamma$ compensates this cavity loss, i.e., $\gamma=\kappa$,
the SFZM reach its lasing threshold, i.e., $\text{Im}\left[E\right]=0$, while
the bulk modes do not move upward at all towards their respective lasing
thresholds in this process, since their wave functions do not extend into the
boundary cavity where the gain is applied.

As another example, consider a three-cavity boundary

$H_{0}=\begin{pmatrix}0&g&0\\\ g&0&g\\\
0&g&0\end{pmatrix},\,U=\begin{pmatrix}\cdots&0&k\\\ \cdots&0&q\\\
\cdots&0&k\end{pmatrix},\,V=\begin{pmatrix}\vdots&\vdots&\vdots\\\ 0&0&0\\\
k^{\prime}&q&k\end{pmatrix}.$ (3)

The boundary zero mode is given by $\tilde{\phi}_{i}={\phi}_{i}=[1,0,-1]^{T}$,
and we have required asymmetric couplings $k^{\prime}\neq k$ to meet Condition
2 (instead of unidirectional couplings as in the previous example). In this
case, the SFZM is no longer the only state with finite amplitudes in the
boundary [see Fig. 3(b)], but we can still achieve a BIC-like spectrum [see
Fig. 3(a)] by including sufficient loss in the bulk. This SFZM can then again
be observed as the single lasing mode when only the optical cavities in the
boundary are pumped to compensate for their losses, which makes the imaginary
parts of the diagonal elements in $H_{0}$ back to zero. Some of the bulk
nonzero modes, due to their finite amplitudes in the boundary cavities,
experience reduced losses in this process [see the short vertical black lines
in Fig. 3(a)], but they are still far away from their lasing thresholds (i.e.,
the real axis) [48].

Figure 3: Same as Fig. 2 but with a three-site boundary and another arbitrary
bulk. $k=g$, $k^{\prime}=-g$, and $q=2g$ are used. Figure 4: (a) Schematics of
a breathing Kagome lattice, with the non-Hermitian boundary attached. (b,c)
Band diagram without and with the boundary. $g=t=t_{b}$ and 20 rows of
cavities are used. (d) Symmetric higher-order corner state in the topological
gap at $t_{a}/t_{b}=0.4$ without the boundary. (e,f) The wave functions of the
SFZMs at $t_{a}/t_{b}=0.4,2.5$ [open and closed dots in (c)]. 10 rows of
cavities are used in (d-f) for a compact illustration.

In both examples above, we have used non-Hermitian couplings to couple an
effectively Hermitian boundary to the bulk. Below we present another example
where the bulk-boundary couplings are Hermitian but the boundary itself is
non-Hermitian instead. Consider the schematic shown in Fig. 1(b), where the
boundary consists of two cavities at zero frequency. These two cavities have
opposite gain and loss (i.e., $\pm ig$) and are connected by couplings of the
same strength ($\pm ig$). Together with the last cavity in the bulk, we denote
the Hamiltonian of this subsystem by

$H_{d}=\left(\begin{array}[]{c|c c}\omega_{N}&t&t\\\ \hline\cr t&-ig&ig\\\
t&-ig&ig\end{array}\right),$ (4)

where $\omega_{N}$ is the mode frequency in the last bulk cavity ($N$) and $t$
is its coupling to the two boundary cavities. Since the latter are not coupled
to other cavities in the bulk, the rest of $U$ and $V$ are all zero.
Conditions 0–2 are again satisfied, with the boundary zero mode given by the
left and right eigenstates $\tilde{\phi}_{i}^{T}=[1,-1]$,
${\phi}_{i}=[1,1]^{T}$. Clearly, $H_{d}$ (and $H$) has a vanished determinant
due to the repeated rows, and in turn, also an SFZM. We note that although in
this case the boundary zero mode is at the exceptional point (EP) [49] of
$H_{0}$, this condition is not required by Hermitian boundary and bulk
couplings [48]. Furthermore, even when the boundary zero mode is at an EP, the
formed SFZM is not an EP in general, as can be easily checked using $H_{d}$
given above.

Below we exemplify SFZMs in this configuration using a triangular “breathing”
Kagome lattice [50, 51], which has alternate nearest-neighbor couplings
$t_{a},t_{b}$ [see Fig. 4(a)] that enable topologically protected higher-order
corner states. Due to the lack of chiral symmetry and the finite length of
their tails, they couple and have energies slightly off $E=0$, represented by
the thick red line in the band diagram when $t_{a}/t_{b}\lesssim 0.5$ [Fig.
4(b)]. A trivial bandgap also opens for $t_{a}/t_{b}\gtrsim 1$.

With the non-Hermitian boundary attached, it is expected that some defect
modes will appear, and they are shown as the dashed lines in Fig. 4(c). Most
noticeably though, now there is a zero mode independent of the value of
$t_{a}$ (thin red line), whether it is in the topological gap on the left
[Fig. 4(e)], through the bulk band in the middle, or in the trivial gap on the
right [Fig. 4(f)]. In addition, unlike the original midgap states whose
energies are shifted from zero (e.g., $E=0.01t_{b}$ for the symmetric one in
Fig. 4(d) at $t_{a}/t_{b}=0.4$), this SFZM is exactly at $E=0$. In other
words, this process of boundary nucleation in fact restores the “zeroness” of
one topological corner state, with (essentially) the same wave function in the
bulk [52].

Figure 5: (a) Wave function of a near-zero mode without the boundary and with
disorder. Onsite and coupling disorders are uniform in the ranges
$[-0.1,0.1]t_{b}$ and $[-0.4,0.4]t_{b}$, respectively. (b) Corresponding SFZM
with the boundary at the top.

To further stress that the energy of the SFZM is robust against any disorder
in the bulk, we now introduce both onsite and coupling disorders to the bulk
and focus on the configuration with $t_{a}/t_{b}=0.4$. Before we attach the
boundary, the mode closest to $E=0$ in one disorder realization is shown in
Fig. 5(a). It resembles the corner state shown in Fig. 4(d) without disorder
because of its topological protection, with the highest amplitude now at just
one corner due to the break of the $C_{3v}$ symmetry. Its energy
$E/t_{b}=-0.0365$ is negative, partly due to mixing with a hexagon plaquette
right beneath the apex, which corresponds to one localized compact state in
the flat band at $E=-(t_{a}+t_{b})$ in the absence of disorder. Despite these
changes to its wave function, the attached boundary again restores the
zeroness of this disorder-shifted state, while maintaining its spatial profile
in the bulk [Fig. 5(b)]. The latter occurs because this disordered state is
mainly propagated by the bulk Green’s function [see Eq. (2)] from the top
corner, with or without the boundary cluster, and the energy difference in
these two cases is small.

In summary, we have introduced an approach to realize zero modes in lattice
models that do not rely on any symmetry or topology of the bulk. They
originate from a boundary with zero mode(s) that serve as the nucleus of these
SFZMs, and they are robust against disorder of any kind and strength in the
bulk Hamiltonian. While certain perturbations in the bulk can also change
bulk-boundary couplings [48] beyond the lattice model, the system is much less
error-prone with the addition of the proposed boundary cluster, given the
potentially vast number of lattice sites in the bulk versus the small numbers
of boundary sites and bulk-boundary couplings we utilize.

We have exemplified these SFZMs using both arbitrary and structured bulks.
Though not shown in the latter, these SFZMs can be observed as the single
lasing mode on an active photonic lattice. We also note that if there are $n$
boundary zero modes with distinct wave functions, then each can serve as a
nucleus and form an SFZM, leading to a maximum of $n$ SFZMs simultaneously. We
have also excluded bulks with pre-existing zero mode(s) because they may leave
the system without an SFZM [48], unlike the near-zero modes in Figs. 4 and 5.
Our finding provides a different perspective on zero modes in both mundane and
topological systems, which can also be used to restore the zero energy of
coupling or disorder-shifted topological states.

###### Acknowledgements.

This project is supported by National Science Foundation under Grants No.
PHY-1847240 and No. ECCS-1846766.

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Wajsberg algebras arising from binary block codes

Cristina FLAUT and Radu VASILE

Abstract. In this paper we presented some connections between BCK-commutative
bounded algebras, MV-algebras, Wajsberg algebras and binary block codes. Using
connections between these three algebras, we will associate to each of them a
binary block code and, in some circumstances, we will prove that the converse
is also true.

Keywords: BCK bounded commutative algebras, MV-algebras, Wajsberg algebras,
block codes.

AMS Classification: 06F35

1\. Introduction

BCK-algebras were first introduced in mathematics by Y. Imai and K. Iseki, in
1966, through the paper [II; 66], as a generalization of the concept of set
theoretic difference and propositional calculi. These algebras form an
important class of logical algebras and have many applications to various
domains of mathematics (group theory, functional analyses, sets theory, etc.).
Because of the necessity to establish certain rational logic systems as a
logical foundation for uncertain information processing, various types of
logical systems have been proposed. For this purpose, some logical algebras
appeared and have been researched.([WDH; 17]) One of these algebras are MV-
algebras, where MV is referred to ”many valued”( [GA; 90]), which were
originally introduced by Chang in [CHA; 58]. He tried to provide a new proof
for the completeness of the Łukasiewicz axioms for infinite valued
propositional logic. These algebras appeared in the specialty literature under
equivalent names: bounded commutative BCK-algebras or Wajsberg algebras, ([CT;
96]). Wajsberg algebras were introduced in 1984, by Font, Rodriguez and
Torrens, through the paper [FRT; 84] as an alternative model for the infinite
valued Łukasiewicz propositional logic.

In the following, we present some connections between BCK-commutative bounded
algebras, MV-algebras, Wajsberg algebras and binary block codes and we gave an
algorithm to find all finite partial ordered Wajsberg algebras. These new
approach allows us to find for these structures new and interesting
properties.

2\. Preliminaries

Definition 2.1. An algebra $(X,\ast,\theta)$ of type $(2,0)$ is called a BCI-
algebra if the following conditions are fulfilled:

$1)~{}((x\ast y)\ast(x\ast z))\ast(z\ast y)=\theta$, for all $x,y,z\in X;$

$2)~{}(x\ast(x\ast y))\ast y=\theta$, for all $x,y\in X;$

$3)~{}x\ast x=\theta$, for all $x\in X$;

$4)$ For all $x,y,z\in X$ such that $x\ast y=\theta,y\ast x=\theta$, it
results $x=y$.

If a BCI-algebra $X$ satisfies the following identity:

$5)$ $\theta\ast x=\theta,~{}$for all $x\in X,$ then $X$ is called a BCK-
algebra.

In a BCK-algebra we have the following order relation:

$x\leq y\text{ if and only if \ }x\ast y=\theta.$

If the algebra $\left(X,\ast,\theta\right)$ has an element $1$ such that
$x\ast 1=\theta,$ for all $x\in X~{}$(that means $x\leq 1,$ for all $x\in X)$,
then the BCK-algebra $X$ is called bounded. If $x\wedge y=y\wedge x,$ for all
$x,y\in X$, where $x\wedge y=y\ast\left(y\ast x\right)$, for all $x,y\in X$,
then $X$ is called a commutative BCK-algebra. For other details regarding BCK
algebras, the readers are referred to [AAT; 96], [Me-Ju; 94].

Definition 2.2. ([CHA; 58]) An abelian monoid $\left(X,\theta,\oplus\right)$
is called MV-algebra if and only if we have an operation $"^{\prime}"$ such
that:

i) $(x^{\prime})^{\prime}=x;$

ii) $x\oplus\theta^{\prime}=\theta^{\prime};$

iii) $\left(x^{\prime}\oplus y\right)^{\prime}\oplus y=$
$\left(y^{\prime}\oplus x\right)^{\prime}\oplus x$, for all $x,y\in X.$([Mu;
07]). We denote it by $\left(X,\oplus,^{\prime},\theta\right).\vskip 6.0pt
plus 2.0pt minus 2.0pt$

Remark 2.3. ([Mu; 07]) In an MV-algebra the constant element $\theta^{\prime}$
is denoted with $1$, that means

$1=\theta^{\prime},$

and the following multiplications are also defined:

$x\odot y=\left(x^{\prime}\oplus y^{\prime}\right)^{\prime},$ $x\ominus
y=x\odot y^{\prime}=\left(x^{\prime}\oplus y\right)^{\prime}.$

Remark 2.4. ([COM; 00], Theorem 1.7.1, p. 30)

i) Let $\left(X,\ast,\theta,1\right)$ be a bounded commutative BCK-algebra. If
we define

$x^{\prime}=1\ast x,$ $x\oplus y=1\ast\left(\left(1\ast x\right)\ast
y\right)=\left(x^{\prime}\ast y\right)^{\prime},x,y\in X,$

we obtain that the algebra $\left(X,\oplus,^{\prime},\theta\right)$ is an MV-
algebra, with

$x\ominus y=x\ast y.$

ii) If $\left(X,\oplus,\theta,^{\prime}\right)$ is an MV-algebra, then
$\left(X,\ominus,\theta,1\right)$ is a bounded commutative BCK-algebra.

Definition 2.5.([COM; 00], Definition 4.2.1) An algebra
$\left(W,\circ,\overline{},1\right)$ of type $\left(2,1,0\right)~{}$is called
a Wajsberg algebra (or W-algebra) if and only if for every $x,y,z\in W$, we
have:

i) $1\circ x=x;$

ii) $\left(x\circ y\right)\circ\left[\left(y\circ z\right)\circ\left(x\circ
z\right)\right]=1;$

iii) $\left(x\circ y\right)\circ y=\left(y\circ x\right)\circ x;$

iv) $\left(\overline{x}\circ\overline{y}\right)\circ\left(y\circ
x\right)=1.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 2.6. ([COM; 00], Lemma 4.2.2 and Theorem 4.2.5)

i) If $\left(W,\circ,\overline{},1\right)$ is a Wajsberg algebra, defining the
following multiplications

$x\odot y=\overline{\left(x\circ\overline{y}\right)}$

and

$x\oplus y=\overline{x}\circ y,$

for all $x,y\in W$, we obtain that
$\left(W,\oplus,\odot,\overline{},0,1\right)$ is an MV-algebra.

ii) If $\left(X,\oplus,\odot,^{\prime},\theta,1\right)$ is an MV-algebra,
defining on $X$ the operation

$x\circ y=x^{\prime}\oplus y,$

it results that $\left(X,\circ,^{\prime},1\right)$ is a Wajsberg algebra.

Example 2.7. We consider the following set $X=\\{\theta,a,b,c,d,e\\}$ and the
multiplication $"\ast"$ given in the below table:

$\ast$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\theta$ | $\theta$ | $\theta$ | $\theta$ | $\theta$ | $\theta$
$a$ | $a$ | $\theta$ | $a$ | $\theta$ | $\theta$ | $\theta$
$b$ | $b$ | $b$ | $\theta$ | $b$ | $\theta$ | $\theta$
$c$ | $c$ | $a$ | $c$ | $\theta$ | $a$ | $\theta$
$d$ | $d$ | $b$ | $a$ | $b$ | $\theta$ | $\theta$
$e$ | $e$ | $d$ | $c$ | $b$ | $a$ | $\theta$

Then $\left(X,\ast,\theta,1\right)$ becomes a BCK commutative bounded algebra.
The associated MV-algebra is $\left(X,\oplus,^{\prime},\theta\right)$, with
the multiplication $\oplus$ and the operation $"^{\prime}"$ given in the below
tables:

$\oplus$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
$a$ | $a$ | $c$ | $d$ | $c$ | $e$ | $e$
$b$ | $b$ | $d$ | $b$ | $e$ | $d$ | $e$
$c$ | $c$ | $c$ | $e$ | $c$ | $e$ | $e$
$d$ | $d$ | $e$ | $d$ | $e$ | $e$ | $e$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
′ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
| $e$ | $d$ | $c$ | $b$ | $a$ | $\theta$

.

([WDH; 17], Example 3.3).

The associated Wajsberg algebra $\left(X,\circ,^{\prime},1\right)$ is:

$\circ$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$a$ | $d$ | $e$ | $d$ | $e$ | $e$ | $e$
$b$ | $c$ | $c$ | $e$ | $c$ | $e$ | $e$
$c$ | $b$ | $d$ | $b$ | $e$ | $d$ | $e$
$d$ | $a$ | $c$ | $d$ | $c$ | $e$ | $e$
$e$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$

Proposition 2.8. ([Bu; 06]) Let $\left(X,\oplus,^{\prime},\theta\right)$ be an
MV-algebra. We have that

$x\oplus x^{\prime}=1.$

Proof. Indeed, from Definition 2.2 ii) and iii), it results that
$1=\left(x^{\prime}\oplus 1\right)^{\prime}\oplus 1=\left(1^{\prime}\oplus
x\right)^{\prime}\oplus x=x^{\prime}\oplus x.\vskip 6.0pt plus 2.0pt minus
2.0pt$

Proposition 2.9. ([Mu; 07]) Let $\left(X,\oplus,^{\prime},\theta\right)$ be an
MV-algebra. For $x,y\in X$, the following statements are equivalent:

i) $x^{\prime}\oplus y=1;$

ii) $x\odot y^{\prime}=0;$

iii) $y=x\oplus\left(y\ominus x\right)=x\oplus\left(y^{\prime}\oplus
x\right)^{\prime};$

iv) There is an element $z\in X$ such that $x\oplus z=y$.

Definition 2.10. ([COM; 00]) Let $\left(X,\oplus,^{\prime},\theta\right)$ be
an MV-algebra and $x,y\in X$. On $X$, we define the following order relation:

$x\leq y\text{ if and only if }x^{\prime}\oplus y=1.$

Remark 2.11. It is clear that $x\leq y$ if and only if $x$ and $y$ satisfy one
of the equivalent conditions i)-iv) from Proposition 2.9.

Definition 2.12. [FRT; 84] If $\left(W,\circ,\overline{},1\right)$ is a
Wajsberg algebra, on $W$ we define the following binary relation

$x\leq y~{}\text{if~{}and~{}only~{}if~{}}x\circ y=1.$ (2.1.)

This relation is an order relation, called the natural order relation on $W$.

For other details regarding MV-algebras and Wajsberg algebras, the readers are
referred to [Io; 08] and [Pi; 07].

3\. Block codes associated to MV-algebras and Wajsberg algebras

In [JUN; 11], were introduced binary block-codes over finite BCK-algebras. In
a similar way, code over MV-algebras and Wajsberg algebras can be established.

Let $A$ be a nonempty set and let $\left(X,\oplus,^{\prime},\theta\right)$ be
an MV-algebra.

Definition 3.1. A mapping $f:A\rightarrow X$ is called an MV-function on $A.$
A cut function of $f$ is a map $f_{r}:A\rightarrow\\{0,1\\},r\in X$, such that

$f_{r}\left(x\right)=1\text{, if and only if \ }r^{\prime}\oplus
f\left(x\right)=1\text{,}\forall x\in A\text{.}$

A cut subset of $A$ is the following subset of $A$

$A_{r}=\\{x\in A~{}/~{}r^{\prime}\oplus f\left(x\right)=1\\}.$

Remark 3.2. If $y\in A_{r}\cup A_{s},$therefore $y\in A_{r\odot s}.$ Indeed,
for $y\in A_{r}\cup A_{s}$, we suppose that $y\in A_{s}$. From here, we have
$y\in A_{r\odot s}$, since $A_{r\odot s}=\\{x\in A~{}/~{}\left(r\odot
s\right)^{\prime}\oplus f\left(x\right)=1\\}$ and $\left(r\odot
s\right)^{\prime}\oplus f\left(y\right)=(r^{\prime}\oplus s^{\prime})\oplus
f\left(y\right)=r^{\prime}\oplus(s^{\prime}\oplus
f\left(y\right))=r^{\prime}\oplus 1=1.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 3.3. Let $f:A\rightarrow X$ be an MV-function on $A.$We define on $X$
the following binary relation

$\forall r,s\in X,r\sim s~{}~{}\text{if~{}and~{}only~{}if~{}~{}}A_{r}=A_{s}.$

This relation is an equivalence relation on $X$ and we denote with
$\widetilde{r}$ the equivalence class of an element $r\in X.\vskip 6.0pt plus
2.0pt minus 2.0pt$

Proposition 3.4. Let $f:A\rightarrow X$ be an MV-function on $A$. Therefore

$f\left(x\right)=\sup\\{r\in X~{}/~{}f_{r}\left(x\right)=1\\}.$

Proof. For a chosen element $x\in A$, we denote $f\left(x\right)=s,s\in X$.
From here we have $s^{\prime}\oplus f\left(x\right)=s^{\prime}\oplus s=1$,
therefore $f_{s}\left(x\right)=1.$ If there is an element $r\in X$ such that
$r^{\prime}\oplus f\left(x\right)=r^{\prime}\oplus s=1$, we obtain that $r\leq
s$. We consider the set $M=\\{r\in X~{}/~{}f_{r}\left(x\right)=1\\}$. Since
$f_{s}\left(x\right)=1$, it results that $s\in M$, therefore
$f\left(x\right)=s=\sup M.\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

The above proposition extends to MV-algebras results obtained in Proposition
3.4 from [JUN; 11]

Proposition 3.5. Let $f:A\rightarrow X$ be an MV-function on $A$. Therefore,
for $r,s\in X$, we have that  $r\leq s$ implies $A_{s}\subseteq A_{r}$. If
$A=X$ and $f:X\rightarrow X$ is the identity function, $f\left(x\right)=x$,
then the converse of this statement is also true.

Proof. Let $r,s\in X$ such that $r\leq s$. From Proposition 2.9., iv), it
results that $s=r\oplus t,t\in X$. For $x\in A_{s}$, we have $s^{\prime}\oplus
f\left(x\right)=1$, which is equivalent with $s\leq f\left(x\right)$, that
means $f\left(x\right)=s\oplus q,q\in X$. It results that
$f\left(x\right)=\left(r\oplus t\right)\oplus q=r\oplus(t\oplus q)$, therefore
$r\leq f\left(x\right).$ From here, we obtain that $r^{\prime}\oplus
f\left(x\right)=1$, then $x\in A_{r}$.

For the converse, if $A_{s}\subseteq A_{r}$, we have that $s\in A_{s}\subseteq
A_{r}$, therefore $s\in A_{r}$. It results that $r^{\prime}\oplus
f\left(s\right)=r^{\prime}\oplus s=1$, which implies $r\leq s$.$\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

The above proposition extends to MV-algebras results obtained in Proposition
3.6 from [JUN; 11].

Let $A$ be a set with $n$ elements. We consider $A=\\{1,2,...,n\\}$ and let
$X$ be an MV-algebra. Using above notations, to each equivalence class
$\widetilde{x},x\in X,$ will correspond the codeword
$w_{x}=f_{x}=x_{1}x_{2}...x_{n}$, with $x_{i}=j,$ if and only if
$f_{x}\left(i\right)=j,i\in A,j\in\\{0,1\\}$.We denote this code with $V_{X}$.
In this way, each MV-function $f:A\rightarrow X$ has associated a binary
block-code of length $n$ and $V_{X}=\\{f_{x},x\in X\\}$ .

Let $V$ be a binary block-code of length $n$ and $w_{x}=x_{1}x_{2}...x_{n}\in
V,$ $w_{y}=y_{1}y_{2}...y_{n}\in V$ be two codewords. On $V$ we can define the
following partial order relation:

$w_{x}\preceq w_{y}\text{ if and only if }y_{i}\leq
x_{i},i\in\\{1,2,...,n\\}.$ (3.1.)

Proposition 3.6. Let $X$ be an MV-algebra. With the above notations, relation
$A_{s}\subseteq A_{r}$ is equivalent with $f_{r}\preceq f_{s}$.

Proof. Assuming $A_{s}\subseteq A_{r}$, we have
$f_{s}\left(x\right)=f_{r}\left(x\right)=1$, for all $x\in A_{s}$. It results
$f_{s}\left(x\right)\leq f_{r}\left(x\right)$, for all $x\in X$, that means
$f_{r}\preceq f_{s}$.

Conversely, if $f_{r}\preceq f_{s}$, we have $f_{s}\left(x\right)\leq
f_{r}\left(x\right)$, for all $x\in X$. If $x\in A_{s}$, we obtain
$s^{\prime}\oplus f\left(x\right)=1$, which is equivalent with
$f_{s}\left(x\right)=1$. This implies that $f_{r}\left(x\right)=1$, which is
equivalent with $r^{\prime}\oplus f\left(x\right)=1$. From here, we obtain
$x\in A_{r}$.$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 3.7. Let $\left(X,\oplus,^{\prime},\theta\right)$ be a finite MV-
algebra. To algebra $X$ corresponds a block-code $V$ such that
$\left(X,\leq\right)$ is isomorphic to $\left(V_{X},\preceq\right)$ as ordered
sets.

Proof. Let $\left(X,\oplus,^{\prime},\theta\right)$ be a finite MV-algebra
and, for $A=X$, let $f:X\rightarrow X$ be the identity function which is an
MV-function. The function $f$ generates the following set $M=\\{f_{r}\ /\ r\in
X\\}$ of cuts functions. Then the set $M$ with the order $\preceq$ is the
associated block-code to the MV-algebra $X$, with the order relation defined
in $\left(3.1\right)$. Let $g:X\rightarrow M,$ $g\left(r\right)=f_{r},$ for
all $r\in X$. We will prove that this map is bijective and $r\leq s$ is
equivalent with $f_{r}\preceq f_{s}$. By definition, the map is surjective.
The map $g$ is injective. Indeed, if $r,s\in X$ with
$g\left(s\right)=g\left(r\right),$ we have $f_{r}=f_{s}$, therefore
$A_{r}=A_{s}$. It results that $1=r^{\prime}\oplus
f\left(s\right)=r^{\prime}\oplus s$ and $1=s^{\prime}\oplus
f\left(r\right)=s^{\prime}\oplus r$, therefore, $r\leq s$ and $s\leq r$. From
here, we obtain that $r=s$, therefore $g$ is a bijective map. Let $r,s\in X$
such that $r\leq s$. From Proposition 3.5, this is equivalent with
$A_{s}\subseteq A_{r}$, which is equivalent with $f_{r}\preceq f_{s}$, from
Proposition 3.6. $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Theorem 3.8. Let $A=\left(X,\ast,\theta,1\right)$ be a bounded commutative
BCK-algebra and $B=\left(X,\oplus,^{\prime},\theta\right)$ be the associated
MV-algebra. Therefore $A$ and $B$ are code equivalent, that means determine
the same binary block-code.

Proof. With the above notations, let $V_{A}$ be the code associated to the BCK
algebra $A$ and $V_{B}$ be the code associated to the MV-algebra $B$. Let
$\varphi:V_{A}\rightarrow
V_{B},\varphi\left(f_{r}\right)=f_{r^{\prime}}^{\prime}$, where $f_{r}$ is a
cut function associated to BCK-algebra as in [JUN; 11],
$f_{r^{\prime}}^{\prime}~{}$is the corresponded cut function associated to MV-
algebra and $r^{\prime}$ is the complement of the element $r\in X$. The map
$\varphi$ is injective, therefore bijective. Indeed, if
$\varphi\left(f_{r}\right)=\varphi\left(f_{s}\right)$, it results
$f_{r^{\prime}}^{\prime}=f_{s^{\prime}}^{\prime},$ therefore
$r^{\prime}=s^{\prime}$ and $r=s.\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Definition 3.9. ([CHA; 19]) Let $\left(X,\oplus,^{\prime},\theta\right)$ be an
MV-algebra. The distance function defined on the algebra $A$ is:

$\displaystyle d$ $\displaystyle:$ $\displaystyle X\times X\rightarrow
X,d\left(x,y\right)=\left(x\ominus y\right)\oplus\left(y\ominus x\right)=$
$\displaystyle=$ $\displaystyle(x\odot y^{\prime})\oplus\left(y\odot
x^{\prime}\right)=$ $\displaystyle=$ $\displaystyle\left(x^{\prime}\oplus
y\right)^{\prime}\oplus\left(y^{\prime}\oplus x\right)^{\prime}.$

In the following, inspired from the above definition, we will define a new
distance on a finite MV-algebra $X$ with $n$ elements. Let
$\left(X,\oplus,^{\prime},\theta\right)$ be an MV-algebra, $f:A\rightarrow X$
be an MV-function on $A$, $f_{r}:A\rightarrow\\{0,1\\},~{}r\in X,$ be a cut
function and $A_{r}$ be the associated cut subset. Let
$\varphi:X\rightarrow\mathcal{P}\left(A\right),\varphi\left(r\right)=A_{r}$.
Since $\left(\mathcal{P}\left(A\right),\cup,\cap,^{\prime},\emptyset,A\right)$
is an MV-algebra, let $d$ the distance on $\mathcal{P}\left(A\right)$ defined
as above. We define on $X$ the following map

$D:X\times X\rightarrow\mathbb{R}_{+},$
$D\left(r,s\right)=\left|d\left(\varphi\left(r\right),\varphi\left(s\right)\right)\right|$

Proposition 3.10.

1) $D\left(r,s\right)=0$ if and only if $r=s$.

2) $D\left(r,s\right)\leq D\left(r,t\right)+D\left(t,s\right).$

3) $D\left(r,\theta\right)=\left|A_{r}\right|$ and
$D\left(r,1\right)=\left|A_{r}^{\prime}\right|$, where $A_{r}^{\prime}$ is the
complement of the set $A_{r}^{\prime}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof.

1) Indeed, if
$D\left(r,s\right)=\left|d\left(\varphi\left(r\right),\varphi\left(s\right)\right)\right|=0$,
we have $(A_{r}^{\prime}\cap A_{s})\cup\left(A_{r}\cap
A_{s}^{\prime}\right)=\emptyset.$ From here, we have that $A_{r}^{\prime}\cap
A_{s}=A_{r}\cap A_{s}^{\prime}=\emptyset$, therefore $A_{r}=A_{s}$ which is
equivalent with $r=s.$

2) Indeed,
$D\left(r,s\right)=\left|d\left(\varphi\left(r\right),\varphi\left(s\right)\right)\right|\leq$
$\leq\left|d\left(\varphi\left(r\right),\varphi\left(t\right)\right)+d\left(\varphi\left(t\right),\varphi\left(s\right)\right)\right|\leq\left|d\left(\varphi\left(r\right),\varphi\left(t\right)\right)\right|+\left|d\left(\varphi\left(t\right),\varphi\left(s\right)\right)\right|.\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

Definition 3.11. The distance $D$ is called the Hamming distance between the
elements $r$ and $r,s\in X,$ and it is the really Hamming distance between
their associated code-words $f_{r}$ and $f_{s}$.

In a similar way as above, we can introduce binary block-codes over finite
Wajsberg algebra.

Let $A$ be a nonempty set and let $\left(W,\circ,\overline{},1\right)$ be a
Wajsberg algebra.

Definition 3.12. A mapping $f:A\rightarrow W$ is called a W-function on $A$. A
cut function of $f$ is a map $f_{r}:A\rightarrow\\{0,1\\},r\in W$, such that

$f_{r}\left(x\right)=1\text{, if and only if \ }r\circ
f\left(x\right)=1\text{,}\forall x\in A\text{.}$

A cut subset of $A$ is the following subset of $A$

$A_{r}=\\{x\in A~{}/~{}r\circ f\left(x\right)=1\\}.$

Let $f:A\rightarrow W$ be an W-function on $A.$We define on $W$ the following
binary relation

$\forall r,s\in W,r\sim s~{}~{}\text{if~{}and~{}only~{}if~{}~{}}A_{r}=A_{s}.$

This relation is an equivalence relation on $W$ and we denote with
$\widetilde{r}$ the equivalence class of an element $r\in W$.

Proposition 3.13. Let $f:A\rightarrow W$ be a W-function on $A$. Therefore,
for $r,s\in W$, we have that  $r\leq s$ implies $A_{s}\subseteq A_{r}$. If
$A=W$ and $f:W\rightarrow W$ is the identity function, $f\left(x\right)=x$,
then the converse of this statement is also true.

Proof. Assuming that $r\leq s$, we have $r\circ s=1$. Let $w\in A_{s}$,
therefore $s\circ f\left(w\right)=1$. We will prove that $r\circ
f\left(w\right)=1$. Since for all $x,y,z\in W$, from $\left(x\circ
y\right)\circ\left[\left(y\circ z\right)\circ\left(x\circ z\right)\right]=1$,
we have $\left(r\circ s\right)\circ\left[\left(s\circ
f\left(w\right)\right)\circ\left(r\circ f\left(w\right)\right)\right]=1$,
therefore $1\circ\left[1\circ\left(r\circ f\left(w\right)\right)\right]=1$.
From here, since $1\circ x=x$, it results $1=r\circ f\left(w\right)$,
therefore $A_{s}\subseteq A_{r}$.

For the converse, if $A_{s}\subseteq A_{r}$, we have that $s\in A_{r}$,
therefore $1=r\circ f\left(s\right)=r\circ s$. We obtain $r\leq s$.$\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

The above proposition extends to W-algebras results obtained in Proposition
3.6 from [JUN; 11].

Let $A$ be a set with $n$ elements. We consider $A=\\{1,2,...,n\\}$ and let
$W$ be a W-algebra. Using above notations, to each equivalence class
$\widetilde{x},x\in W$, will correspond the codeword
$w_{x}=f_{x}=x_{1}x_{2}...x_{n}$, with $x_{i}=j$, if and only if
$f_{x}\left(i\right)=j,i\in A,j\in\\{0,1\\}$.We denote this code with $V_{W}$.
In this way, each W-function $f:A\rightarrow W$ has associated a binary block-
code of length $n$ and $V_{W}=\\{f_{x},x\in W\\}$.

Proposition 3.14. Let $W$ be a W-algebra. With the above notations, relation
$A_{s}\subseteq A_{r}$ is equivalent with $f_{r}\preceq f_{s}$.

Proof. By straightforward calculation. $\Box\vskip 6.0pt plus 2.0pt minus
2.0pt$

Proposition 3.15. Let $\left(W,\circ,\overline{},1\right)$ be a finite
W-algebra. To algebra $W$ corresponds a block-code $V$ such that
$\left(W,\leq\right)$ is isomorphic to $\left(V_{W},\preceq\right)$ as ordered
sets.

Proof. By straightforward calculation. $\Box\vskip 6.0pt plus 2.0pt minus
2.0pt$

Theorem 3.16. Let $A=\left(X,\oplus,^{\prime},\theta\right)$ be an MV-algebra
and $B=\left(X,\circ,\overline{},1\right)$ be the associated Wajsberg algebra.
Therefore $A$ and $B$ are code equivalent, that means determine the same
binary block-code.

Proof. With the above notations, let $V_{A}$ be the code associated to the MV-
algebra $A$ and $V_{B}$ be the code associated to the W-algebra $B$. Let
$\varphi:V_{A}\rightarrow V_{B},\varphi\left(f_{r}\right)=f_{r}^{\prime}$,
where $f_{r}$ is a cut function associated to MV-algebra and $f_{r}^{\prime}$
is the corresponded cut function associated to W-algebra. The map $\varphi$ is
injective, therefore bijective. Indeed, if
$\varphi\left(f_{r}\right)=\varphi\left(f_{s}\right)$, it results
$f_{r}^{\prime}=f_{s}^{\prime},$ therefore $r=s$. $\Box\vskip 6.0pt plus 2.0pt
minus 2.0pt$

Theorem 3.17. Let $A=\left(X,\ast,\theta,1\right)$ be a bounded commutative
BCK-algebra, $B=\left(X,\oplus,^{\prime},\theta\right)$ be the associated MV-
algebra and $C=\left(X,\circ,\overline{},1\right)$ be the associated Wajsberg
algebra. Therefore $A,B$ and $C$ are code equivalent, that means determine the
same binary block-code.

Proof. The result is obtained from Theorem 3.8 and Theorem 3.16. $\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

Example 3.18. Using algebras from Example 2.7, we can see that the code
associated to BCK commutative bounded algebra $\left(X,\ast,\theta,1\right)$
is
$V_{1}=\\{111111,010111,001011,000101,000011,000001\\}$, the code associated
to MV-algebra $\left(X,\oplus,^{\prime},\theta\right)$,$~{}V_{2}$, is the same
with $V_{1}$ and the code associated to Wajsberg algebra
$\left(X,\circ,\overline{},1\right)$,$~{}V_{3}$, is also the same with
$V_{1}$.

i) The BCK commutative bounded algebra and the associated code are given in
the below tables:

$\begin{tabular}[]{l|llllll}$\ast$&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\theta$&$\theta$&$\theta$&$\theta$&$\theta$&$\theta$\\\
$a$&$a$&$\theta$&$a$&$\theta$&$\theta$&$\theta$\\\
$b$&$b$&$b$&$\theta$&$b$&$\theta$&$\theta$\\\
$c$&$c$&$a$&$c$&$\theta$&$a$&$\theta$\\\
$d$&$d$&$b$&$a$&$b$&$\theta$&$\theta$\\\
$e$&$e$&$d$&$c$&$b$&$a$&$\theta$\end{tabular}~{}~{}~{}\begin{tabular}[]{l|llllll}$\ast$&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$a$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$b$&$0$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$c$&$0$&$0$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$\\\
$d$&$0$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$e$&$0$&$0$&$0$&$0$&$0$&$\mathbf{1}$\end{tabular}.$

ii) The MV-algebra algebra and the associated code are given in the below
tables:

$\oplus$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
$a$ | $a$ | $c$ | $d$ | $c$ | $e$ | $e$
$b$ | $b$ | $d$ | $b$ | $e$ | $d$ | $e$
$c$ | $c$ | $c$ | $e$ | $c$ | $e$ | $e$
$d$ | $d$ | $e$ | $d$ | $e$ | $e$ | $e$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$\oplus$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$
$a$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$b$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$c$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$d$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$e$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$

iii) The Wajsberg algebra and the associated code are given in the below
tables:

$\circ$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$a$ | $d$ | $e$ | $d$ | $e$ | $e$ | $e$
$b$ | $c$ | $c$ | $e$ | $c$ | $e$ | $e$
$c$ | $b$ | $d$ | $b$ | $e$ | $d$ | $e$
$d$ | $a$ | $c$ | $d$ | $c$ | $e$ | $e$
$e$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
$\circ$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$a$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$b$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$c$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$d$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$e$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$

We remark that the attached code for each of these algebras is as a skeleton
for these algebras( the same for all three algebras), on which we can insert
different structures, as for example: a BCK commutative bounded algebra or an
MV-algebra or a Wajsberg algebra. We consider the algebra $A\in${BCK, MV,
Wajsberg}, finite with $n$ elements. The skeleton of such an algebra is a
matrix of order $n$ in which the elements of this matrix is black or white
squares, with black square on the position $\left(i,j\right)$, if and only if
$x_{i}\leq x_{j}$ in $A$, $x_{i},x_{j}\in A$. The associated skeleton of such
a finite algebra $A\in${BCK, MV, Wajsberg} is nothing else than a
representation of the associated order relation on the algebra $A$.

As we can see in the below two tables, the skeleton is the same for the BCK
bounded commutative algebra and for the attached Wajsberg algebra. For the
attached MV algebra, the skeleton is the same but symmetric in respect to the
Ox axis. The skeleton generates the same order relation on $A$ as the attached
binary block code on $A,V_{A}$.

$\ast,\circ$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$a$ |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$b$ |  |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$c$ |  |  |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$
$d$ |  |  |  |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$e$ |  |  |  |  |  | $\mathbf{\blacksquare}$
$\oplus$ | $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ |  |  |  |  |  | $\mathbf{\blacksquare}$
$a$ |  |  |  |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$b$ |  |  |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$
$c$ |  |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$d$ |  | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$e$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
(3.2.)

4\. Some remarks regarding Wajsberg algebras

Remark 4.1. Let $\left(X,\leq\right)$ be a finite totally ordered set,
$X=\\{x_{0},x_{1},...,x_{n}\\}$, with $x_{0}$ the first element and $x_{n}$
the last element. Using this order relation, we define the following
multiplication $"\circ"$ on $X$:

$\left\\{\begin{array}[]{c}x_{i}\circ x_{j}=1\text{, if }x_{i}\leq x_{j};\\\
x_{i}\circ x_{j}=x_{n-i+j}\text{, otherwise;}\\\
x_{0}=\theta,x_{n}=1,x\circ\theta=\overline{x}.\end{array}\right.$ (4.1.)

Therefore, $\left(X,\circ,\overline{},1\right)$ is a Wajsberg algebra. We
remark that this is the only way to define a W-algebra structure on a finite
totally ordered set such that the induced order relation on this algebra is
given in $(2.1)$. We also remark that $\overline{x}_{i}=x_{n-1}$. ([FRT; 84],
Theorem 19).

Definition 4.2. Let $\left(W_{1},\circ,\overline{},1\right)$ and
$\left(W_{2},\cdot,^{\prime},1\right)$ be two finite Wajsberg algebras. We
define on the Cartesian product of these algebras, $W=W_{1}\times W_{2}$, the
following multiplication $"\nabla"$,

$\left(x_{1},x_{2}\right)\nabla\left(y_{1},y_{2}\right)=\left(x_{1}\circ
y_{1},x_{2}\cdot y_{2}\right),$ (4.2.)

The complement of the element $\left(x_{1},x_{2}\right)$ is
$\rceil\left(x_{1},x_{2}\right)=$
$\left(\overline{x}_{1},x_{2}^{\prime}\right)$ and
$\mathbf{1}=\left(1,1\right)$. Therefore, by straightforward calculation, we
obtain that $\left(W,\nabla,\rceil,\mathbf{1}\right)$ is also a Wajsberg
algebra.

Remark 4.3. If $x=\left(x_{1},x_{2}\right),y=\left(y_{1},y_{2}\right)\in W$,
then the order relation corresponded to the algebra
$\left(W,\nabla,\rceil,\mathbf{1}\right)$ is given as follow:

$x\leq_{W}y\text{ if and only if }x_{1}\leq_{W_{1}}y_{1}\text{ and
}x_{2}\leq_{W_{2}}y_{2}\text{.}$ (4.3.)

Definition 4.4. Let $\left(W_{1},\circ,\overline{},1\right)$ and
$\left(W_{2},\cdot,^{\prime},1\right)$ be two Wajsberg algebras. A map
$f:W_{1}\rightarrow W_{2}$ is a morphism of Wajsberg algebras if and only if:

1) $f\left(0\right)=0;$

2) $f\left(x\circ y\right)=f\left(x\right)\cdot f\left(y\right);$

3) $f\left(\overline{x}\right)=\left(f\left(x\right)\right)^{\prime}.\vskip
6.0pt plus 2.0pt minus 2.0pt$

Proposition 4.5. The algebras $W_{1}\times W_{2}$ and $W_{2}\times W_{1\text{
}}$are isomorphic.

Proof. Let $f:$ $W_{1}\times W_{2}\rightarrow W_{2}\times W_{1}$,
$f\left(\left(x_{1},x_{2}\right)\right)=\left(x_{2},x_{1}\right)$ is a
bijective morphism. $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 4.6. 1) ([HR; 99], Theorem 5.2, p. 43) An MV-algebra is finite if and
only if it is isomorphic to a finite product of totally ordered MV algebras.

2) If $M=\left(X,\oplus,\odot,^{\prime},\theta,1\right)$ is a totally ordered
MV-algebra, then the obtained Wasjberg algebra,
$\,W=\left(X,\circ,^{\prime},1\right)$, is also totally ordered. The converse
is also true. Indeed, since $x\circ y=x^{\prime}\oplus y$, we have that
$x\leq_{M}y$ if and only $x\leq_{W}y$.

3) Using connections between MV-algebras and Wajsberg algebras, from the
above, we have that a Wajsberg algebra is finite if and only if it is
isomorphic to a finite product of totally ordered Wajsberg algebras.

4) If an MV-algebra or a Wajsberg algebra are finite with a prime number of
elements, therefore these algebras are totally ordered.

5) If two Wajsberg algebras are isomorphic, then these algebras are also
isomorphic as ordered sets.

Definition 4.7. Let $n$ be a natural number, $n\geq 4$. We consider the
decomposition of the number $n$ in factors:

$n=q_{1}q_{2}...q_{t},q_{i}\in N,1<q_{i}<n,i\in\\{1,2,...,t\\}.$

This decomposition is not unique. We will count only one time the
decompositions with the same terms but with other order of them in the
product. We denote with $\pi_{n}$ the number of all such decompositions.

From the above, we obtain the following Theorem.

Theorem 4.8. Let $n$ be a natural number, $n\geq 2$. There are exactly
$\pi_{n}$ nonismorphic(as ordered sets) Wajsberg algebras with $n$ elements.
These algebras are obtained as a finite product of totally ordered Wajsberg
algebras.$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

We denote these algebras with
$\left(\mathcal{W}_{i}^{n},\nabla_{i},\rceil_{i},\mathbf{1}_{i},\leq_{i}^{n}\right)$,
where $\leq_{i}^{n}$is the corresponding order relation on
$\mathcal{W}_{i}^{n}$, $i\in\\{1,2,...,\pi_{n}\\}$.

Remark 4.9. We will denote with
$\left(\mathcal{W}_{ij}^{n},\nabla_{ij},\rceil_{ij},\mathbf{1}_{i},\leq_{ij}\right)$
the Wajsberg algebras isomorphic to $\mathcal{W}_{i}^{n}$, as ordered sets,
where $\leq_{ij}^{n}$is the corresponding order relation on
$\mathcal{W}_{ij}^{n}$. Let
$f_{ij}^{n}:\mathcal{W}_{i}^{n}\rightarrow\mathcal{W}_{ij}^{n}$ be such an
isomorphism of ordered sets. The Wajsberg structure on the algebra
$\mathcal{W}_{ij}^{n}$ is given as follows. Let $x,y\in$
$\mathcal{W}_{ij}^{n}$ and $a,b\in\mathcal{W}_{i}^{n}$ such that
$f_{ij}^{n}\left(a\right)=x$ and $f_{ij}^{n}\left(b\right)=y$. We define

$x\nabla_{ij}y=f_{ij}^{n}\left(a\right)\nabla_{ij}f_{ij}^{n}\left(b\right)\overset{def}{=}f_{ij}^{n}\left(a\nabla_{i}b\right)\text{.}$

We remark that this is the only way to define a Wajsberg algebra structure on
$\mathcal{W}_{ij}^{n}$ such that the induced order relation on this algebra is
$\nabla_{ij}$. Algebras $\mathcal{W}_{i}^{n}$ and $\mathcal{W}_{ij}^{n}$ are
isomorphic as ordered sets and are not always isomorphic as Wajsberg algebras.

From the above we obtain an algorithm to find all finite Wajsberg algebras of
order $n$.

Example 4.10.

i) Let $n=6$. We have that $n=2\cdot 3=3\cdot 2$, therefore $\pi_{6}=1$.

ii) Let $n=8$. We have that $n=2\cdot 4=4\cdot 2=2\cdot 2\cdot 2$. Therefore,
$\pi_{8}=2$.

iii) Let $n=12$. We have that $n=2\cdot 2\cdot 3=2\cdot 3\cdot 2=$
$=3\cdot 2\cdot 2=4\cdot 3=3\cdot 4=2\cdot 6=6\cdot 2$. Then, $\pi_{12}=3$.

Example 4.11. There is only one type of partially ordered Wajsberg algebra
with $4$ elements, up to an isomorphism. Indeed, let
$\left(W_{1}=\\{0,1\\},\circ,\overline{},1\right)$ and
$\left(W_{2}=\\{0,e\\},\cdot,^{\prime},e\right)$ be two finite totally ordered
Wajsberg algebras. We consider $W_{1}\times
W_{2}=\\{\left(0,0\right),\left(0,e\right),\left(1,0\right),\left(1,e\right)\\}=$
$=\\{O,A,B,E\\}$. On $W_{1}\times W_{2}$ we obtain a Wajsberg algebra
structure by defining the multiplication as in relation $\left(4.2\right)$. We
give this multiplication in the following table:

$\nabla$ | $O$ | $A$ | $B$ | $E$
---|---|---|---|---
$O$ | $E$ | $E$ | $E$ | $E$
$A$ | $B$ | $E$ | $B$ | $E$
$B$ | $A$ | $A$ | $E$ | $E$
$E$ | $O$ | $A$ | $B$ | $E$
.

Example 4.12. There is only one type of partially ordered Wajsberg algebra
with $6$ elements, up to an isomorphism. Indeed, $\pi_{6}=1$. Let
$\left(W_{1}=\\{0,1\\},\circ,\overline{},1\right)$ and
$\left(W_{2}=\\{0,b,e\\},\cdot,^{\prime},e\right)$ be two finite totally
ordered Wajsberg algebras. Using relation $\left(4.1\right)$, on $W_{2}$ we
have that $b^{\prime}=b$. We consider $W_{1}\times
W_{2}=\\{\left(0,0\right),\left(0,b\right),\left(0,e\right),\left(1,0\right),\left(1,b\right),\left(1,e\right)\\}=$
$=\\{O,A,B,C,D,E\\}$. On $W_{1}\times W_{2}$ we obtain a Wajsberg algebra
structure by defining the multiplication as in relation $\left(4.2\right)$. We
give this multiplication in the following table:

$\begin{tabular}[]{l|llllll}$\nabla_{11}^{6}$&$O$&$A$&$B$&$C$&$D$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$&$E$&$E$\\\ $A$&$D$&$E$&$E$&$D$&$E$&$E$\\\
$B$&$C$&$D$&$E$&$C$&$D$&$E$\\\ $C$&$B$&$B$&$B$&$E$&$E$&$E$\\\
$D$&$A$&$B$&$B$&$D$&$E$&$E$\\\ $E$&$O$&$A$&$B$&$C$&$D$&$E$\end{tabular}\ .$
(4.4.)

We remark that $A\leq B,A\leq D,C\leq D$, and the other elements can’t be
compared in the algebra
$\mathcal{W}_{1}^{6}=\mathcal{W}_{11}^{6}=\left(W_{1}\times
W_{2},\nabla_{11}^{6}\right)$. We denote this order relation with
$\leq_{11}^{6}$.

If we consider the isomorphism

$\displaystyle f_{12}^{6}$ $\displaystyle:$ $\displaystyle\left(W_{1}\times
W_{2},\nabla_{11}^{6}\right)\rightarrow\left(W_{1}\times
W_{2},\nabla_{12}^{6}\right),$ $\displaystyle f_{12}^{6}\left(A\right)$ =
$\displaystyle
A,f_{12}^{6}\left(B\right)\text{=}C,f_{12}^{6}\left(C\right)\text{=}B,f_{12}^{6}\left(D\right)\text{=}D,f_{12}^{6}\left(O\right)\text{=}O,f_{12}^{6}\left(E\right)\text{=}E\text{,}$

we obtain on $W_{1}\times W_{2}$ a new Wajsberg algebra structure, with the
multiplication $\nabla_{12}^{6}$ given in the below table:

$\nabla_{12}^{6}$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$
---|---|---|---|---|---|---
$O$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
$A$ | $D$ | $E$ | $D$ | $E$ | $E$ | $E$
$B$ | $C$ | $C$ | $E$ | $C$ | $E$ | $E$
$C$ | $B$ | $D$ | $B$ | $E$ | $D$ | $E$
$D$ | $A$ | $C$ | $D$ | $C$ | $E$ | $E$
$E$ | $O$ | $A$ | $B$ | $C$ | $D$ | $E$
(4.5.)

These two algebras, $\mathcal{W}_{11}^{6}=\left(W_{1}\times
W_{2},\nabla_{11}^{6}\right)$ and $\mathcal{W}_{12}^{6}=\left(W_{1}\times
W_{2},\nabla_{12}^{6}\right)$, are isomorphic. We remark that $A\leq C,A\leq
D,B\leq D$ and the other elements can’t be compared in the algebra
$\mathcal{W}_{12}^{6}$. We denote this order relation with $\leq_{12}^{6}$.
This algebra is the Wajsberg algebra given in Example 3.18. The isomorphism
$f_{12}^{6}$ is in the same time isomorphism of ordered sets and isomorphism
of Wajsberg algebras.

If we consider the isomorphism

$\displaystyle f_{13}^{6}$ $\displaystyle:$ $\displaystyle\left(W_{1}\times
W_{2},\nabla_{11}^{6}\right)\rightarrow\left(W_{1}\times
W_{2},\nabla_{13}^{6}\right),$ $\displaystyle f_{13}^{6}\left(A\right)$ =
$\displaystyle
B,f_{13}^{6}\left(B\right)\text{=}D,f_{13}^{6}\left(C\right)\text{=}C,f_{13}^{6}\left(D\right)\text{=}A,f_{13}^{6}\left(O\right)\text{=}O,f_{13}^{6}\left(E\right)\text{=}E,$

we obtain on $W_{1}\times W_{2}$ a new Wajsberg algebra structure, with the
multiplication $\nabla_{13}^{6}$ given in the below table:

$\begin{tabular}[]{l|llllll}$\nabla_{13}^{6}$&$O$&$A$&$B$&$C$&$D$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$&$E$&$E$\\\ $A$&$B$&$E$&$D$&$A$&$D$&$E$\\\
$B$&$A$&$E$&$E$&$A$&$E$&$E$\\\ $C$&$D$&$E$&$D$&$E$&$D$&$E$\\\
$D$&$C$&$A$&$A$&$C$&$E$&$E$\\\ $E$&$O$&$A$&$B$&$C$&$D$&$E$\end{tabular}.$
(4.6.)

In $\mathcal{W}_{13}^{6}=\left(W_{1}\times W_{2},\nabla_{13}^{6}\right)$, we
have $B\leq A,C\leq A,B\leq D$ and the other elements can’t be compared. We
denote this order relation with $\leq_{13}^{6}$.The isomorphism $f_{13}^{6}$
is only isomorphism of ordered sets and is not isomorphism of Wajsberg
algebras.

If we consider $W_{2}\times
W_{1}=\\{\left(0,0\right),\left(0,1\right),\left(b,0\right),\left(b,1\right),\left(e,0\right),\left(e,1\right)\\}=$
$=\\{O,A,B,C,D,E\\}$, on $W_{2}\times W_{1}$ we obtain a Wajsberg algebra
structure by defining the multiplication as in relation $\left(4.2\right)$. We
give this multiplication in the following table:

$\begin{tabular}[]{l|llllll}$\nabla_{14}^{6}$&$O$&$A$&$B$&$C$&$D$&$E$\\\
\hline\cr$O$&$E$&$E$&$E$&$E$&$E$&$E$\\\ $A$&$D$&$E$&$D$&$E$&$D$&$E$\\\
$B$&$C$&$C$&$E$&$E$&$E$&$E$\\\ $C$&$B$&$C$&$D$&$E$&$D$&$E$\\\
$D$&$A$&$A$&$C$&$C$&$E$&$E$\\\ $E$&$O$&$A$&$B$&$C$&$D$&$E$\end{tabular}.$
(4.7.)

The algebras $\left(W_{1}\times W_{2},\nabla_{11}^{6}\right)$ and
$\left(W_{2}\times W_{1},\nabla_{14}^{6}\right)$ are also isomorphic, by
taking the map

$\displaystyle f_{14}^{6}$ $\displaystyle:$ $\displaystyle\left(W_{1}\times
W_{2},\nabla_{11}^{6}\right)\rightarrow\left(W_{2}\times
W_{1},\nabla_{14}^{6}\right),$ $\displaystyle f_{14}^{6}\left(A\right)$ =
$\displaystyle
B,f_{14}^{6}\left(B\right)\text{=}D,f_{14}^{6}\left(C\right)\text{=}A,f_{14}^{6}\left(D\right)\text{=}C,f_{14}^{6}\left(O\right)\text{=}O,f_{14}^{6}\left(E\right)\text{=}E.$

In $\mathcal{W}_{14}^{6}=\left(W_{2}\times W_{1},\nabla_{14}^{6}\right)$, we
have $A\leq C,B\leq C,B\leq D~{}$and the other elements can’t be compared. We
denote this order relation with $\leq_{14}^{6}$.The isomorphism $f_{14}^{6}$
is in the same time isomorphism of ordered sets and isomorphism of Wajsberg
algebras.

Example 4.13. There is only two types of partially ordered Wajsberg algebra
with $8$ elements, up to an isomorphism. Indeed, $\pi_{8}=2$. Let
$\left(W_{1}=\\{0,a,b,e\\},\circ,\overline{},1\right)$ and
$\left(W_{2}=\\{0,1\\},\cdot,^{\prime},e\right)$ be two finite totally ordered
Wajsberg algebras. Using relation $\left(4.1\right)$, on $W_{1}$ we have that
$\overline{b}=a$ and $\overline{a}=b$. We consider $W_{1}\times
W_{2}=\\{\left(0,0\right),\left(0,1\right),\left(a,0\right),\left(a,1\right),\left(b,0\right),\left(b,1\right),\left(e,0\right),\left(e,1\right)\\}=$
$=\\{O,X,Y,Z,T,U,V,E\\}$. On $W_{1}\times W_{2}$ we obtain a Wajsberg algebra
structure by defining the multiplication as in relation $\left(4.2\right)$,
namely $\mathcal{W}_{11}^{8}=\left(W_{1}\times W_{2},\nabla_{11}^{8}\right)$.
The multiplication $\nabla_{11}^{8}$ is given in the following table:

$\nabla_{11}^{8}$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
---|---|---|---|---|---|---|---|---
$O$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
$X$ | $V$ | $E$ | $V$ | $E$ | $V$ | $E$ | $V$ | $E$
$Y$ | $U$ | $U$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
$Z$ | $T$ | $U$ | $V$ | $E$ | $V$ | $E$ | $V$ | $E$
$T$ | $Z$ | $Z$ | $U$ | $U$ | $E$ | $E$ | $E$ | $E$
$U$ | $Y$ | $Z$ | $T$ | $U$ | $T$ | $E$ | $V$ | $E$
$V$ | $X$ | $X$ | $Z$ | $Z$ | $U$ | $U$ | $E$ | $E$
$E$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
(4.8.)

In $\mathcal{W}_{11}^{8}$ we have that $O\leq X\leq Z\leq U\leq E$, $O\leq
Y\leq T\leq V\leq E$, $O\leq Y\leq Z\leq U\leq E,O\leq Y\leq T\leq U\leq E$
and the other elements can’t be compared in this algebra. We denote this order
relation with $\leq_{11}^{8}$.

Now, we consider $W_{2}\times
W_{1}=\\{\left(0,0\right),\left(0,a\right),\left(0,b\right),\left(0,e\right),\left(1,0\right),\left(1,a\right),\left(1,b\right),\left(1,e\right)\\}=$
$=\\{O,X,Y,Z,T,U,V,E\\}$. On $W_{2}\times W_{1}$ we obtain a Wajsberg algebra
structure by defining the multiplication as in relation $\left(4.2\right)$,
namely $\mathcal{W}_{12}^{8}=\left(W_{2}\times W_{1},\nabla_{12}^{8}\right)$.
The multiplication $\nabla_{12}^{8}$ is given in the following table:

$\nabla_{12}^{8}$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
---|---|---|---|---|---|---|---|---
$O$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
$X$ | $V$ | $E$ | $E$ | $E$ | $V$ | $E$ | $E$ | $E$
$Y$ | $U$ | $V$ | $E$ | $E$ | $U$ | $V$ | $E$ | $E$
$Z$ | $T$ | $U$ | $V$ | $E$ | $T$ | $U$ | $V$ | $E$
$T$ | $Z$ | $X$ | $Z$ | $Z$ | $E$ | $E$ | $E$ | $E$
$U$ | $Y$ | $Z$ | $Z$ | $Z$ | $V$ | $E$ | $E$ | $E$
$V$ | $X$ | $Y$ | $Z$ | $Z$ | $U$ | $V$ | $E$ | $E$
$E$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
(4.8.)

We have that $O\leq X\leq Y\leq Z\leq E$, $O\leq X\leq Y\leq V\leq E$,
$O\leq X\leq U\leq V\leq E$, $O\leq T\leq U\leq V\leq E$. These two
structures, $\mathcal{W}_{11}^{8}$ and $\mathcal{W}_{12}^{8}$, are isomorphic.
The morphism is

$\displaystyle f_{12}^{8}$ $\displaystyle:$ $\displaystyle\left(W_{1}\times
W_{2},\nabla_{11}^{8}\right)\rightarrow\left(W_{2}\times
W_{1},\nabla_{12}^{8}\right),$ $\displaystyle f_{12}^{8}\left(X\right)$ =
$\displaystyle
T,f_{12}^{8}\left(Y\right)\text{=}X,f_{12}^{8}\left(Z\right)\text{=}U,f_{12}^{8}\left(T\right)\text{=}Y,$
$\displaystyle f_{12}^{8}\left(U\right)$ = $\displaystyle
V,f_{12}^{8}\left(V\right)\text{=}Z,f_{12}^{8}\left(O\right)\text{=}O,f_{12}^{8}\left(E\right)\text{=}E.$

The isomorphism $f_{12}^{8}$ is in the same time isomorphism of ordered sets
and isomorphism of Wajsberg algebras.

If we take $W_{2}\times W_{2}\times
W_{2}=\\{\left(0,0,0\right),\left(0,0,1\right),\left(0,1,0\right),\left(0,1,1\right),\left(1,0,0\right),$
$\left(1,0,1\right),\left(1,1,0\right),\left(1,1,1\right)\\}=\\{O,X,Y,Z,T,U,V,E\\}$,
on $W_{2}\times W_{2}\times W_{2}$ we obtain a Wajsberg algebra structure by
defining the multiplication as in relation $\left(4.2\right)$, namely
$\mathcal{W}_{21}^{8}=\left(W_{2}\times W_{2}\times
W_{2},\nabla_{21}^{8}\right)$. The multiplication $\nabla_{21}^{8}$ is given
in the following table:

$\nabla_{21}^{8}$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
---|---|---|---|---|---|---|---|---
$O$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$ | $E$
$X$ | $V$ | $E$ | $V$ | $E$ | $V$ | $E$ | $V$ | $E$
$Y$ | $U$ | $U$ | $E$ | $E$ | $U$ | $U$ | $E$ | $E$
$Z$ | $T$ | $U$ | $V$ | $E$ | $T$ | $U$ | $V$ | $E$
$T$ | $Z$ | $Z$ | $Z$ | $Z$ | $E$ | $E$ | $E$ | $E$
$U$ | $Y$ | $Z$ | $Y$ | $Z$ | $V$ | $E$ | $V$ | $E$
$V$ | $X$ | $X$ | $Z$ | $Z$ | $U$ | $U$ | $E$ | $E$
$E$ | $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
(4.9.)

We have that $X\leq Z,X\leq U,Y\leq Z,T\leq V,Y\leq V,T\leq U$. These two
structures, $\mathcal{W}_{11}^{8}$ and $\mathcal{W}_{21}^{8}$, are not
isomorphic as ordered sets, therefore are not isomorphic as Wajsberg algebras.

Remark 4.14. 1) There is only one type of partially ordered Wajsberg algebra
with $9$ elements, up to an isomorphism as ordered sets.

2) There is only one type of partially ordered Wajsberg algebra with $10$
elements, up to an isomorphism, up to an isomorphism as ordered sets..

3) There is only three types of partially ordered Wajsberg algebra with $12$
elements, up to an isomorphism, up to an isomorphism as ordered sets..

5\. Special algebras arising from block codes

In [FL; 15], was developed an algorithm which provide conditions to attach a
BCK algebra to a given block code. In the following, we will use those ideas
to obtain a similar algorithm in the case of MV-algebras and Wajsberg
algebras. The difference is that, in the first case, the BCK algebra arising
from a block code is a non-commutative, a non-implicative, but a positive
implicative BCK algebra ([FL; 17]). In the second case, of MV-algebras and
Wajsberg algebras, we must obtain from a given block-code a BCK commutative
bounded algebra. Our idea is to obtain a Wajsberg finite algebra associated to
a given block code and from here the desired MV-algebra and the desired
bounded commutative BCK-algebra.

Let $V$ be a binary block-code with $n+1$ codewords of length $n+1$,
lexicographically ordered, $V=\\{w_{0},w_{1},...,w_{n}\\}$ and
$w_{x}=x_{1}x_{2}...x_{n}x_{n+1}\in V,$ $w_{y}=y_{1}y_{2}...y_{n}y_{n+1}\in V$
be two codewords. On $V$ we can define the following partial order relation:

$w_{x}\preceq w_{y}\text{ if and only if }y_{i}\leq
x_{i},i\in\\{0,1,2,...,n\\}.$ (5.1.)

We consider the matrix
$M_{V}=\left(m_{i,j}\right)_{i,j\in\\{1,2,...,n+1\\}}\in\mathcal{M}_{n+1}(\\{0,1\\})$
with the rows consisting of the codewords of $V.$ This matrix is called the
matrix associated to the block-code $V.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Theorem 5.1. With the above notations, if the matrix $M_{V}$ has the first
line and the last column of the form
$\underset{n+1-\text{time}}{\underbrace{11...1}}$, the last line of the form
$\underset{n-\text{time}}{\underbrace{00...00}1}$, the first column of the
form $\underset{n-\text{time}}{1\underbrace{00...00}}$, $m_{ii}=1$, for all
$i\in\\{1,2,...,n+1\\}$, and if the order relation given by $\left(5.1\right)$
coincide with one of the order relations $\leq_{ij}^{n+1}$given by the Remark
4.9, then there are a set $A$ with $n+1$ elements, a Wajsberg algebra
$\mathcal{W}_{ij}^{n+1}$ and a W-function
$f:A\rightarrow\mathcal{W}_{ij}^{n+1}$ such that $f$ determines $V.\vskip
6.0pt plus 2.0pt minus 2.0pt$

Proof.  We consider on $V$ the lexicographic order, denoted by $\leq_{lex}$.
It results that $(V,\leq_{lex})$ is a totally ordered set. Let
$V=\\{w_{0},w_{1},...,w_{n}\\},$ with
$w_{n}\geq_{lex}w_{n-1}\geq_{lex}...\geq_{lex}w_{0}$. We denote
$w_{n}=\underset{n+1-\text{time}}{\underbrace{11...1}}$ and
$w_{0}=\underset{n-\text{time}}{\underbrace{00...0}1}$. We remark that
$w_{0}=\mathbf{1}$ is the ”one” element and $w_{n}=\theta$ is the ”zero”
element in $\left(V,\preceq\right)$, considered with order relation
$\left(5.1\right)$. If this order relation coincides with one of the order
relation $\leq_{ij}^{n+1}$, given in Remark 4.9, it results that on
$\left(V,\preceq\right)$ we can obtain a Wajsberg algebra structure, which is
isomorphic to a Wajsberg algebra $\mathcal{W}_{ij}^{n+1}$, with attached order
relation $\leq_{ij}^{n+1}$. If we consider $A=V$ and the identity map
$f:A\rightarrow V,f\left(w\right)=w$ as a W-function, the decomposition of $f$
provides a family of maps
$V_{\mathcal{W}_{ij}^{n+1}}=\\{f_{r}:A\rightarrow\\{0,1\\}~{}/~{}$
$f_{r}\left(x\right)=1$ if and only if $r\ast f\left(x\right)=w_{0},\forall
x\in A,r\in X\\}.$ This family is the binary block-code $V$ relative to the
order relation $\preceq.\vskip 6.0pt plus 2.0pt minus 2.0pt\Box$

The above Theorem extends to W-algebras results obtained in Theorem 3.2 from
[FL; 15].

Proposition 5.2.  Let
$A=\left(a_{i,j}\right)_{\begin{subarray}{c}i\in\\{1,2,...,n\\}\\\
j\in\\{1,2,...,m\\}\end{subarray}}\in\mathcal{M}_{n,m}(\\{0,1\\})$ be a
matrix. Starting from this matrix, we can find a matrix
$B=\left(b_{i,j}\right)_{i,j\in\\{1,2,...,q\\}}\in\mathcal{M}_{q}(\\{0,1\\})$,
$q\geq max\\{m,n\\}$, such that $B$ has the first line and the last column of
the form $\underset{q-\text{time}}{\underbrace{11...1}}$, the last line of the
form $\underset{q-1-\text{time}}{\underbrace{00...00}1}$, the first column of
the form $\underset{q-1-\text{time}}{1\underbrace{00...00}}~{}$with
$b_{ii}=1,\forall i\in\\{1,2,...,q\\}$ and $A$ becomes a submatrix of the
matrix $B.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof. Obviously$.\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Theorem 5.3. With the above notations, let  $V$ be a binary block-code with
$n$ codewords of length $m,n\neq m$. Let $q$ be a natural number, $q\geq
max\\{m,n\\}$, and
$B=\left(b_{i,j}\right)_{i,j\in\\{1,2,...,q\\}}\in\mathcal{M}_{q}(\\{0,1\\})$
be a matrix such that the matrix $M_{V}$ is a submatrix of the matrix $B$. If
$B$ is the matrix attached to a code $C$ which satisfies the conditions from
Theorem 5.1, therefore there is a set $A$ with $m$ elements, a Wajsberg
algebra $\mathcal{W}_{ij}^{n+1}$ and a W-function
$f:A\rightarrow\mathcal{W}_{ij}^{n+1}$ such that the obtained block-code
$C_{nm}$ contains the block-code $V$ as a subset.

Proof. Let $V$ be a binary block-code, $V=\\{w_{1,}w_{2},...,w_{n}\\}$, with
codewords of length $m$. Let $M\in\mathcal{M}_{n,m}(\\{0,1\\})$ be its
associated matrix. Using Proposition 5.2, we can extend the matrix $M$ to a
square matrix $M^{\prime}\in\mathcal{M}_{q}(\\{0,1\\})$ and we can apply
Theorem 5.1 for the matrix $M^{\prime}$. Assuming that the initial columns of
the matrix $M$ have in the new matrix $M^{\prime}$ positions
$i_{j_{1}},i_{j_{2}},...,i_{j_{m}}\in\\{1,2,...,q\\},$ let
$A=\\{x_{j_{1}},x_{j_{2}},...,x_{j_{m}}\\}\subseteq C.$ The W-function
$f:A\rightarrow C,f\left(x_{j_{i}}\right)=$ $x_{j_{i}},$
$i\in\\{1,2,...,m\\}$, determines the binary block-code $C_{nm}$ such that
$V\subseteq C_{nm}.\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

The above Theorem extends to W-algebras results obtained in Theorem 3.9 from
[FL; 15].

Remark 5.4. In the above theorem, the associated Wajsberg algebra is not
unique, as we can see in the Example 6.10.

Theorem 5.5. If a block-code $V$ satisfies the conditions from Theorem 5.1,
there is an MV-algebra $\mathcal{X}$ and a commutative bounded BCK-algebra
$\mathcal{Y}$ associated to this code.

Proof. We use Theorem 5.1, Remark 2.4 and Remark 2.6.

6\. Examples

Example 6.1. We consider the code
$V_{1}=\\{111111,011011,001001,000111,000011,000001\\}=$
$=\\{\theta,a,b,c,d,e\\}$, with elements lexicographically ordered. The
associated matrix is

$\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$a$&$0$&$\mathbf{1}$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$b$&$0$&$0$&$\mathbf{1}$&$0$&$0$&$\mathbf{1}$\\\
$c$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$d$&$0$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$e$&$0$&$0$&$0$&$0$&$0$&$\mathbf{1}$\end{tabular}.$

Using order given by the relation $\left(5.1\right)$, we have that $a\preceq
b,c\preceq d,a\preceq d$ and the other elements can’t be compared. Therefore
this order relation is $\leq_{11}^{6}$ and the associated Wajsberg algebra is
algebra $\mathcal{W}_{11}^{6}$, given by relation $\left(4.4\right)$.

Example 6.2. We consider the code
$V_{2}=\\{111111,010111,001011,000101,000011,000001\\}=$
$=\\{\theta,a,b,c,d,e\\}$, with elements lexicographically ordered. The
associated matrix is

$~{}\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$a$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$b$&$0$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$c$&$0$&$0$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$\\\
$d$&$0$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$e$&$0$&$0$&$0$&$0$&$0$&$\mathbf{1}$\end{tabular}.$

Using order given by the relation $\left(5.1\right),$ we have that $a\preceq
c,a\preceq d,b\preceq d$ and the other elements can’t be compared. Therefore
this order relation is $\leq_{12}^{6}$ and the associated Wajsberg algebra is
$\mathcal{W}_{12}^{6}$, given by relation $\left(4.5\right)$. Algebra
$\mathcal{W}_{12}^{6}$ is the Wajsberg algebra given in the Example 3.18. From
here, we can see the MV-algebra and the BCK commutative bounded algebra
associated to the block-code $V_{2}$.

Example 6.3. We consider the code
$V_{3}=\\{111111,010001,011011,010101,000011,000001\\}=$
$=\\{\theta,a,b,c,d,e\\}$, with elements lexicographically ordered. The
associated matrix is

$~{}\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$a$&$0$&$\mathbf{1}$&$0$&$0$&$0$&$\mathbf{1}$\\\
$b$&$0$&$\mathbf{1}$&$\mathbf{1}$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$c$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$&$0$&$\mathbf{1}$\\\
$d$&$0$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{1}$\\\
$e$&$0$&$0$&$0$&$0$&$0$&$\mathbf{1}$\end{tabular}.$

Using order given by the relation $\left(5.1\right),$ we have that $b\preceq
a,c\preceq a,b\preceq d$ and the other elements can’t be compared. Therefore
this order relation is $\leq_{13}^{6}$ and the associated Wajsberg algebra is
$\mathcal{W}_{13}^{6}$, given by relation $\left(4.6\right)$.

The skeleton of this algebra is

$\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$a$&&$\mathbf{\blacksquare}$&&&&$\mathbf{\blacksquare}$\\\
$b$&&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$c$&&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$\\\
$d$&&&&&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$e$&&&&&&$\mathbf{\blacksquare}$\end{tabular}.$ (6.1.)

Example 6.4. We consider the code
$V_{4}=\\{111111,010101,001111,000101,000011,000001\\}=$
$=\\{\theta,a,b,c,d,e\\}$, with elements lexicographically ordered. Using
order given by the relation $\left(5.1\right),$ we have that $a\preceq
c,b\preceq c,b\preceq d$ and the other elements can’t be compared. Therefore
this order relation is $\leq_{14}^{6}$ and the associated Wajsberg algebra is
$\mathcal{W}_{14}^{6}$, given by relation $\left(4.7\right)$.

The skeleton of this algebra is

$\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$a$&&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$\\\
$b$&&&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$c$&&&&$\mathbf{\blacksquare}$&&$\mathbf{\blacksquare}$\\\
$d$&&&&&$\mathbf{\blacksquare}$&$\mathbf{\blacksquare}$\\\
$e$&&&&&&$\mathbf{\blacksquare}$\end{tabular}.$ (6.2.)

Example 6.5. We consider the code $V_{5}=\\{011,101,010,001,000\\}$. Using
Theorem 5.3, we have the matrix $B\,\ $

$B=~{}\begin{tabular}[]{|llllll}$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$1$&$1$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$&$1$\\\
$0$&$1$&$\mathbf{0}$&$\mathbf{1}$&$\mathbf{1}$&$1$\\\
$0$&$0$&$\mathbf{1}$&$\mathbf{0}$&$\mathbf{1}$&$1$\\\
$0$&$0$&$\mathbf{0}$&$\mathbf{1}$&$\mathbf{0}$&$1$\\\
$0$&$0$&$\mathbf{0}$&$\mathbf{0}$&$\mathbf{1}$&$1$\\\
$0$&$0$&$\mathbf{0}$&$\mathbf{0}$&$\mathbf{0}$&$1$\end{tabular}\text{,}$

the set $A=\\{b,c,d\\}$, the Wajsberg algebra $\mathcal{W}_{12}^{6}$ and the
W-function
$f:A\rightarrow\mathcal{W}_{12}^{6},f\left(b\right)=b,f\left(c\right)=c,f\left(d\right)=d$.
We have the code $C_{53}=\\{111,011,101,010,001,000\\}$. The code
$V_{5}=\\{011,101,010,001,000\\}$ is a subcode of the code $C_{53}$.

Example 6.6. We consider the code
$V_{6}=\\{111111,011101,001101,000111,000011,000001\\}=$
$=\\{\theta,a,b,c,d,e\\}$, with elements lexicographically ordered. The
associated matrix is

| $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$a$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$b$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$c$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$d$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$e$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$

and the skeleton is

| $\theta$ | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---|---
$\theta$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$a$ |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$
$b$ |  |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ |  | $\mathbf{\blacksquare}$
$c$ |  |  |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$d$ |  |  |  |  | $\mathbf{\blacksquare}$ | $\mathbf{\blacksquare}$
$e$ |  |  |  |  |  | $\mathbf{\blacksquare}$

Using order given by the relation $\left(5.1\right),$ we have that $a\preceq
b,c\preceq d$ and the other elements can’t be compared. If we consider the
binary relation given by the skeleton: $i\leq_{s}j$ if and only if in the
position $\left(i,j\right)$ we have a black square, this relation is not an
order relation. Indeed, we do not have the transitivity: we have
$a\leq_{s}b,b\leq_{s}c,c\leq_{s}d$ but $a$ and $d$ can’t be compared,
therefore the sets $\left(V_{6},\leq_{s}\right)$ and
$\left(V_{6},\preceq\right)$ are not isomorphic as ordered sets, as in
Proposition 3.15. Even if the associated matrix to this code is on the form
asked in Theorem 5.1, to this code we can’t associate a Wajsberg algebra,
since the order relation $\preceq$ is not on the form $\leq_{ij}^{6}$, given
by Theorem 4.8 and Remark 4.9.

We must remark that, from [FL; 15], Theorem 3.2, to the code $V_{6}$ we can
attached a BCK-algebra, namely

$B=\begin{tabular}[]{l|llllll}&$\theta$&$a$&$b$&$c$&$d$&$e$\\\
\hline\cr$\theta$&$\theta$&$\theta$&$\theta$&$\theta$&$\theta$&$\theta$\\\
$a$&$a$&$\theta$&$\theta$&$a$&$a$&$\theta$\\\
$b$&$b$&$b$&$\theta$&$b$&$b$&$\theta$\\\
$c$&$c$&$c$&$c$&$\theta$&$\theta$&$\theta$\\\
$d$&$d$&$d$&$d$&$d$&$\theta$&$\theta$\\\
$e$&$e$&$e$&$e$&$e$&$e$&$\theta$\end{tabular},$

but the block code associated to the BCK-algebra $B$ is
$V_{B}=\\{111111,011001,001001,000111,000011,000001\\}$ and it is different
from $V_{6}$. Therefore, the above mentioned Theorem needs to be understood as
follows:

i) In some circumstances, we can associate a BCK-algebra to a binary block
code.

ii) The BCK-algebra $B$ associated to a block code $V$ generates the same code
$V$ if and only if the order relation $\leq_{s}$, generated by the skeleton
associated to the code $V$, is the same with the order relation $\preceq$,
defined on the obtained BCK-algebra $B$.

Example 6.7. We consider the code
$V_{7}=\\{11111111,01010101,00111111,00010101,$
$00001111,00000101,00000011,00000001\\}=$
$=\\{O,X,Y,Z,T,U,V,E\\}$, with elements lexicographically ordered. The
associated matrix is

| $O$ | $X$ | $Y$ | $Z$ | $T$ | $U$ | $V$ | $E$
---|---|---|---|---|---|---|---|---
$O$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$X$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$Y$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$Z$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$T$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$U$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$V$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$ | $\mathbf{1}$
$E$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathbf{1}$

Using the order given by the relation $\left(5.1\right)$, we have that
$O\preceq X\preceq Z\preceq U\preceq E$, $O\preceq Y\preceq T\preceq V\preceq
E$, $O\preceq Y\preceq Z\preceq U\preceq E,O\preceq Y\preceq T\preceq U\preceq
E$ and the other elements can’t be compared. Therefore this order relation is
$\leq_{11}^{8}$and the associated Wajsberg algebra is $\mathcal{W}_{11}^{8}$,
given by relation $\left(4.7\right)$.

Example 6.8. We consider the code
$V_{8}=\\{11111111,01110111,00110011,00010001,$
$00001111,00000111,00000011,00000001\\}=$
$=\\{O,X,Y,Z,T,U,V,E\\}$, with elements lexicographically ordered. Using order
given by the relation $\left(5.1\right)$, we have that $O\leq X\leq Y\leq
Z\leq E$, $O\leq X\leq Y\leq V\leq E$,
$O\leq X\leq U\leq V\leq E$, $O\leq T\leq U\leq V\leq E$, and the other
elements can’t be compared. Therefore this order relation is $\leq_{12}^{8}$
and the associated Wajsberg algebra is $\mathcal{W}_{12}^{8}$, given by
relation $\left(4.8\right)$.

Example 6.9. We consider the code
$V_{9}=\\{11111111,01010101,0010001,00010001,$
$00001111,00000101,00000011,00000001\\}=$
$=\\{O,X,Y,Z,T,U,V,E\\}$, with elements lexicographically ordered. Using order
given by the relation $\left(5.1\right)$, we have that $O\leq X\leq Y\leq
Z\leq E$, $O\leq X\leq Y\leq V\leq E$,
$O\leq X\leq U\leq V\leq E$, $O\leq T\leq U\leq V\leq E$, and the other
elements can’t be compared. Therefore this order relation is
$\leq_{21}^{8}$and the associated Wajsberg algebra is $\mathcal{W}_{21}^{8}$,
given by the relation $\left(4.9\right)$.

Example 6.10. We consider the code $V_{10}=\\{011,101\\}$. Using Theorem 5.3,
we have the matrix $B$

$B=\begin{tabular}[]{|llllllll}$O$&$X$&$Y$&$Z$&$T$&$U$&$V$&$E$\\\
$1$&$1$&$1$&$1$&$1$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$0$&$1$&$0$&$1$&$0$&$\mathbf{1}$&$\mathbf{0}$&$\mathbf{1}$\\\
$0$&$0$&$1$&$1$&$0$&$\mathbf{0}$&$\mathbf{1}$&$\mathbf{1}$\\\
$0$&$0$&$0$&$1$&$0$&$\mathbf{1}$&$\mathbf{0}$&$\mathbf{1}$\\\
$0$&$0$&$0$&$0$&$1$&$\mathbf{1}$&$\mathbf{1}$&$\mathbf{1}$\\\
$0$&$0$&$0$&$0$&$0$&$\mathbf{1}$&$\mathbf{0}$&$\mathbf{1}$\\\
$0$&$0$&$0$&$0$&$0$&$\mathbf{0}$&$\mathbf{1}$&$\mathbf{1}$\\\
$0$&$0$&$0$&$0$&$0$&$\mathbf{0}$&$\mathbf{0}$&$\mathbf{1}$\end{tabular},$

the set $A=\\{U,V,E\\}$, the Wajsberg algebra $\mathcal{W}_{11}^{8}$ and the
W-function
$f:A\rightarrow\mathcal{W}_{11}^{8},f\left(U\right)=U,f\left(V\right)=V,f\left(E\right)=E$.
We have the code $C_{23}=\\{111,101,011,111,001\\}$. The code
$V_{10}=\\{011,101\\}$ is a subcode of the code $C_{23}$. We remark that the
associated Wajsberg is not unique. Indeed, we can consider the set
$A=\\{b,c,d\\}$, the Wajsberg algebra $\mathcal{W}_{12}^{6}$ and the code
$C_{23}^{\prime}=\\{111,011,101,010,001,000\\}$, as in Example 6.5. The code
$V_{10}$ is a subcode of the code $C_{23}^{\prime}$.

Conclusions

In this paper, we presented some connections between BCK-commutative bounded
algebras, MV-algebras, Wajsberg algebras and binary block codes. By studying
these connections were identified two types of approaches. First of them is if
these algebras can generate good codes? At a first glance, the answer can be
no. Indeed, the codes generates by the BCK-commutative bounded algebras, MV-
algebras and Wajsberg algebras have, in general, the minimum Hamming distance
equal with $1$, that means are not so good codes. Therefore, we turned our
attention to the second approach: if we use the attached codes, we can find
new and interesting properties for these algebras? The answer is yes and that
is exactly what we did in this paper.

We found an algorithm to generate all finite partially ordered Wajsberg
algebras and, from here, all finite partially ordered MV algebras and all
finite partially ordered BCK commutative bounded algebras. In Section 6, we
gave examples of the above mentioned algebras associated to a binary block
codes. We remarked that to a two isomorphic algebras correspond different
codes, but we can find different algebras which can generate the same code.

Even if, for the moment, the answer is no, we will not give up the first
approach and we hope that in a future research to achieve important results in
this direction.

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Cristina Flaut,

Faculty of Mathematics and Computer Science,

Ovidius University of Constanţa, România

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Radu Vasile,

PhD student at Doctoral School of Mathematics,

Ovidius University of Constanţa, România

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