This paper classifies all finite Wajsberg algebras of order n where n is less than or equal to 9, providing a complete enumeration for small finite cases.
Contribution
It offers a complete classification of all finite Wajsberg algebras for orders up to 9, filling a gap in the algebraic understanding of these structures.
Findings
01
Complete list of Wajsberg algebras for n<=9
02
Identification of structural properties for small orders
03
Foundation for further algebraic classification
Abstract
In this paper, we describe all finite Wajsberg algebras of order n<=9.
Equations117
(x→y)→y=(y→x)→x, for all x,y∈X,
(x→y)→y=(y→x)→x, for all x,y∈X,
x⊙y=(x′⊕y′)′,
x⊙y=(x′⊕y′)′,
x⊖y=x⊙y′=(x′⊕y)′.
x⊖y=x⊙y′=(x′⊕y)′.
x⊙y=(x∘y)
x⊙y=(x∘y)
x⊕y=x∘y,
x⊕y=x∘y,
x∘y=x′⊕y,
x∘y=x′⊕y,
d
d
x≡Iy if and only if d(x,y)∈I,x,y∈X.
x≡Iy if and only if d(x,y)∈I,x,y∈X.
x≡Iy if and only if (x∘y)∘(y∘x)′∈I.
x≡Iy if and only if (x∘y)∘(y∘x)′∈I.
x≤yifandonlyifx∘y=1.
x≤yifandonlyifx∘y=1.
\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.
\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.
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Taxonomy
TopicsAdvanced Topics in Algebra · graph theory and CDMA systems · Algebraic structures and combinatorial models
Full text
Wajsberg algebras of order n,n≤9
[TABLE]
Cristina Flaut, Šárka Hošková-Mayerová, Arsham Borumand
Saeid and
Radu Vasile
[TABLE]
Abstract. In this paper, we describe all finite Wajsberg
algebras of order n≤9.
[TABLE]
Keywords: MV-algebras, Wajsberg algebras.
**AMS Classification: **06F35, 06F99.
1. Introduction
[TABLE]
Residuated lattices were introduced by Dilworth and Ward, through the papers
[Di; 38], [WD; 39]. A residuated lattic is an algebra (X,∨,∧,⊙,→,0,1) of type (2,2,2,2,0,0) with
an order ≤ such that:
(X,∨,∧,0,1) be a bounded lattice;
(X,⊙,1) be a commutative ordered monoid;
x≤y→z if and only if y⊙x≤z, for all x,y,z∈X, that means ⊙ and → form an orderd pairs ([Pi; 07],
Definition 1.1).
A residuated lattice X is called an MV-algebra if and only if the
following supplimentary condition are satisfied:
[TABLE]
see [Tu; 99], Theorem 2.70.
MV-algebras are introduced by C. C. Chang in [CHA; 58] to give new proof
for the completeness of the Łukasiewicz axioms for infinite valued
propositional logic. These algebras appeared in the specialty literature
under some 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 [BV; 10] the authors gave an algorithm to find the number of all
non-isomorphic residuated lattices of order n, with examples for n≤12.
Knowing connections between residuated lattices, MV-algebras and Wajsbeg
algebras, starting from results obtained in [FV; 19], in this paper we give
Representation Theorem for finite Wajsberg algebras, we give a formula for
the number of nonisomorphic Wajsberg algebras types of order n, the total
number of Wajsberg algebras of order n and we describe all finite Wajsberg
algebras of order n≤9. For n∈{1,2,...,9}, we obtain the same
numbers of nonisomorphic MV-algebras as in [BV; 10], Table 8, by using other
methods, totally different from their method.
[TABLE]
2. Preliminaries
[TABLE]
Definition 2.1. ([CHA; 58]) An abelian monoid (X,θ,⊕) is called MV-algebra if and only if we have an
operation "′" such that:
i) (x′)′=x;
ii) x⊕θ′=θ′;
iii) (x′⊕y)′⊕y=(y′⊕x)′⊕x, for all x,y∈X.([Mu;
07]). We denote it by (X,⊕,′,θ).
In an MV-algebra, the following multiplications are also defined:
[TABLE]
[TABLE]
Definition 2.2.([COM; 00], Definition 4.2.1) An algebra (W,∘,,1) of type (2,1,0)is called a
Wajsberg algebra (or W-algebra) if and only if for every x,y,z∈W, we have:
i) 1∘x=x;
ii) (x∘y)∘[(y∘z)∘(x∘z)]=1;
iii) (x∘y)∘y=(y∘x)∘x;
iv) (x∘y)∘(y∘x)=1.\vskip6.0ptplus2.0ptminus2.0pt
**Remark 2.3. **([COM; 00], Lemma 4.2.2 and Theorem 4.2.5)
i) If (W,∘,,1) is a Wajsberg algebra,
defining the following multiplications
[TABLE]
and
[TABLE]
for all x,y∈W, we obtain that (W,⊕,⊙,,0,1) is an MV-algebra.
ii) If (X,⊕,⊙,′,θ,1) is an
MV-algebra, defining on X the operation
[TABLE]
it results that (X,∘,′,1) is a Wajsberg
algebra.
Definition 2.4. ([CT; 96]) Let (X,⊕,′,θ) be an MV-algebra. The nonempty subset I⊆X is called an
ideal in X if and only if the following conditions are satisfied:
i) θ∈I, where θ=1;
ii) x∈I and y≤x implies y∈I;
iii) If x,y∈I, then x⊕y∈I.
**Definition 2.5. **([COM; 00], p. 13) An ideal P of the MV-algebra (X,⊕,′,θ) is a prime ideal in X
if and only if for all x,y∈P we have (x′⊕y)′∈P or (y′⊕x)′∈P.
Definition 2.6. ([GA; 90], p. 56) Let (W,∘,,1) be a Wajsberg algebra and I⊆W be a nonempty subset. I is called an ideal in W if and only if the following
conditions are fulfilled:
i) θ∈I, where θ=1;
ii) x∈I and y≤x implies y∈I;
iii) If x,y∈I, then x∘y∈I.
Using connections between MV-algebras and Wajsberg algebras, we give below
the notion of a prime ideal in a Wajsberg algebra.
Definition 2.7. Let (W,∘,,1) be a
Wajsberg algebra and P⊆W be a nonempty subset. P is called a
primeideal in W if and only if for all x,y∈P we
have (x∘y)′∈P or (y∘x)′∈P.
Definition 2.8. ([CHA; 58]) Let (X,⊕,′,θ) be an MV-algebra. The distance function defined on the algebra A is:
[TABLE]
Definition 2.9. 1) ([COM; 00], Proposition 1.2.6) Let I
be an ideal in an MV-algebra X. We define the following binary relation on
X:
[TABLE]
We remark that I={x∈X/x≡Iθ} and the quotient set X/I become an MV-algebra.
Let (W,∘,,1) be a Wajsberg algebra and I
be an ideal in an W. Using Definition 2.8 and connections between
MV-algebras and Wajsberg algebras, we define the following binary relation
on W:
[TABLE]
We remark that I={x∈W/x≡Iθ} and the quotient set W/I become a Wajsberg algebra.
Definition 2.10. Let (X1,⊕,′,θ) and (X2,⊗,,0) be two
MV-algebras. A map f:X1→X2 is a morphism of Wajsberg
algebras if and only if:
f(θ)=0;
f(x⊕y)=f(x)⊗f(y);
f(x′)=f(x).
If f is a bijection, therefore the algebras X1 and X2 are
isomorphic. We write this X1≃X2.\vskip6.0ptplus2.0ptminus2.0pt
Definition 2.11. Let (W1,∘,,1)
and (W2,⋅,′,1) be two Wajsberg algebras. A
map f:W1→W2 is a morphism of Wajsberg algebras if and
only if:
f(0)=0;
f(x∘y)=f(x)⋅f(y);
f(x)=(f(x))′.
If f is a bijection, therefore the algebras W1 and W2 are
isomorphic. We write this W1≃W2.\vskip6.0ptplus2.0ptminus2.0pt
Definition 2.12. [FRT; 84] If (W,∘,,1) is a Wajsberg algebra, on W we define the following binary relation
[TABLE]
This relation is an order relation, called the natural order
relation on W.
[TABLE]
3. Representation Theorem for finite Wajsberg algebras
[TABLE]
**Remark 3.1. **([FRT; 84], Theorem 19)
We consider** (X,≤) **a finite totally
ordered set, X={x0,x1,...,xn}, with x0 the first and xn the last element. With this order relation, the following
multiplication "∘" are defined on X:
[TABLE]
The obtained algebra (X,∘,,1) is a Wajsberg
algebra and this is the only way to define a Wajsberg algebra structure on a
finite totally ordered set such that the induced order relation on this
algebra is given by relation (2.1), with xi=xn−1.
Definition 3.2. ([FV; 19]) Let (W1,∘,,θ) and (W2,⋅,′,1)
be two finite Wajsberg algebras. On the Cartesian product of these algebras,
W=W1×W2, we define the following multiplication "∇",
[TABLE]
The algebra (W,∇,⌉,1) is also a Wajsberg
algebra, the complement of the element (x1,x2) is ⌉(x1,x2)=(x1,x2′) and 1=(θ,1).
If we consider x=(x1,x2),y=(y1,y2)∈W, then the order relation on the algebra (W,∇,⌉,1) is defined as follow:
[TABLE]
Proposition 3.3. ([COM; 00], Theorem 1.3.2) LetXbe a finite MV-algebra. Therefore,Xis a direct product
of a family of MV-algebras{Xi}i∈{1,2,...,m}if and
only if, there is a family{Ij}j∈{1,2,...,m}of
ideals in X such that
Proposition 3.4. ([CHA; 59], Lemma 1) LetXbe
an MV algebra andPa prime ideal inX. Therefore, the
quotient algebraX/Pis a totally ordered MV-algebra.□\vskip6.0ptplus2.0ptminus2.0pt
Proposition 3.5. ([CHA; 59], Lemma 3) Every MV-algebra is a
direct product of totally ordered MV-algebras.□\vskip6.0ptplus2.0ptminus2.0pt
Using connections between MV algebras and Wajsberg algebras, we obtain the
following Representation Theorem.
Theorem 3.6. 1) LetWbe a finite Wajsberg
algebra. Therefore,Wis a direct product of a family of Wajsberg
algebras{Wi}i∈{1,2,...,m}if and only if, there is
a family{Ij}j∈{1,2,...,m}of ideals inWsuch that
i) Wj≃W/Ij, for allj∈{1,2,...,m};
ii) j∈{1,2,...,m}∩Ij={0}.
([FV; 19], Theorem 4.8)Each finite Wajsberg
algebra is a direct product of totally ordered Wajsberg algebras.
Proof.
It results from Proposition 3.1 and Remark 2.3.
Indeed, if {Ij}j∈{1,2,...,m} are prime ideals in W, from
Proposition 3.2 and Remark 2.3, it results that W is a direct product of
totally ordered Wajsberg algebras. □\vskip6.0ptplus2.0ptminus2.0pt
Remark 3.7.
In [FV; 19] was developed an algorithm which generate all
finite Wajsberg algebras. These algebras are direct product of totally
ordered algebras. With Theorem 3.4, we can complete this algorithm:
i) Let n the order of the Wajsberg algebra (W,∘,,1) and
[TABLE]
be the decomposition of the number n in factors. Since this decomposition
is not unique, the decompositions with the same terms, but with other order
of them in the product, will we counted one time. The number of all such
decompositions will be denoted with πn.
ii) Since an MV-algebra is finite if and only if it is isomorphic to a
finite product of totally ordered MV algebras, using connections between
MV-algebras and Wajsberg algebras, we obtain that a Wajsberg algebra is
finite if and only if it is isomorphic to a finite product of totally
ordered Wajsberg algebras. ([HR; 99], Theorem 5.2, p. 43).
ii)([FV; 19], Theorem 4.8) There are only πn nonismorphic, as ordered sets, Wajsberg algebras with n elements. We
obtain these algebras as a finite product of totally ordered Wajsberg
algebras. We denote them with* (Win,∇i,⌉i,1i,≤in)*, where ≤inis the corresponding order relation on Win, i∈{1,2,...,πn}. It is clear that if n is prime, therefore we obtain
only totally ordered Wajsberg algebra.
iii) ([FV; 19], Remark 4.9) We denote with (Wijn,∇ij,⌉ij,1i,≤ij) the Wajsberg algebras isomorphic to Win,
considered as ordered sets, with ≤ijnthe corresponding order
relation on Wijn. Let fijn:Win→Wijn be such an isomorphism of ordered
sets. The Wajsberg structure on the algebra Wijn is
defined as follows. For x,y∈Wijn and a,b∈Win with fijn(a)=x and fijn(b)=y, we define
[TABLE]
In this way we define a Wajsberg algebra structure on Wijn
such that the induced order relation on this algebra is ∇ij. We
remark that the algebras Win and Wijn
are isomorphic as ordered sets and are not always isomorphic as Wajsberg
algebras.
iv) Using Theorem 3.4 and Definition 2.7, a finite Wajsberg algebra is a
direct product of Wajsberg algebras of the form W/Pj,j∈{1,2,...,m} with j∈{1,2,...,m}∩Pj={0} and Pj prime ideals in Wajsberg algebra W.
Proposition 3.8.Let(W1,∘,,θ)and(W2,⋅,′,1)be two Wajsberg algebras. Iff:W1→W2is
a bijective map such thatf(θ)=1andf(x∘y)=f(x)⋅f(y), thereforefandf−1are morphisms of ordered sets.
Proof. If x≤W1y, therefore x∘y=θ, we have
that f(x∘y)=f(x)⋅f(y)=f(θ)=1, therefore f(x)≤W2f(y). Conversely
is also true.□
[TABLE]
Examples of Wajsberg algebras of order n≤9.
[TABLE]
In this section, we will describe all Wajsberg algebras of order n≤9.
We will give the number of these algebras, a complete description and a way
to obtain them. For all these algebras, we also compute the prime ideals and
we provide the decompositions given in Theorem 3.6.
4.1. Wajsberg algebras of order 4.
1)Totally ordered case. Let W={O≤A≤B≤E}
be a totally ordered set. On W we define a multiplication as in relation (3.1). We have A=B and B=A.
Therefore the algebra W has the following multiplication table:
[TABLE]
This algebra has no proper ideals.
2)Partially ordered case. There is only one type of
partially ordered Wajsberg algebra with 4 elements, up to an isomorphism
of ordered sets and Wajsberg algebras. Indeed, let (W1={0,1},∘,,1) and (W2={0,e},⋅,′,e) be two finite totally ordered Wajsberg algebras. We
consider W114=W1×W2={(0,0),(0,e),(1,0),(1,e)}=
={O,A,B,E}. On W1×W2 we obtain a Wajsberg algebra
structure by defining the multiplication as in relation (3.2), (see [FV; 19]). We give this multiplication in the following table:
[TABLE]
All proper ideals are P1={O,A} and P2={O,B}. These ideals are also prime ideals. We obtain W114/P1={O,E} and W114/P2={O,E}, where O={O,A},E={B,E},O={O,B},E={A,E}. Indeed, since (O∘A)∘(A∘O)′=E∘A′=E∘A=A∈P1, it results O≡P1A, therefore O={O,A}. The same computations give us
that (E∘B)∘(B∘E)′=B∘E′=B∘O=A∈P1
and E≡P1B. We remark that (O∘B)∘(B∘O)′=E∘A′=E∘B=B∈/P1, etc. From here, we obtain that W114=W1×W2≃W114/P1×W114/P2, as in Remark 3.7, iv).
If we consider the map
[TABLE]
we obtain on W1×W2 the same Wajsberg structure, as we can see
in the below table:
[TABLE]
Therefore, there are only two nonisomorphic Wajsberg algebras of order 4,
(as W-algebras and as ordered sets). Using connections between MV-algebras
and Wajsberg algebras, we obtain that there are two nonisomorphic
MV-algebras of order 4. The same number was found in [BV; 10], Table 8, by
using other method.
4.2. Wajsberg algebras of order 6\vskip12.0ptplus4.0ptminus4.0pt
**Totally ordered case. **Let W={O≤A≤B≤C≤D≤E}
be a totally ordered set. On W we define a multiplication as in relation (3.1). We have A=D, B=C, C=B, D=A. Therefore the algebra W has the
following multiplication table:
[TABLE]
This algebra has no proper ideals.
2) **Partially ordered case. **There is only one type of
partially ordered Wajsberg algebra with 6 elements, up to an isomorphism
of ordered sets. Indeed, π6=1. Let (W1={0,1},∘,,1) and (W2={0,b,e},⋅,′,e) be two finite totally ordered Wajsberg algebras. Using relation
(3.1), on W2 we have that b′=b. We consider W1×W2={(0,0),(0,b),(0,e),(1,0),(1,b),(1,e)}=
={O,A,B,C,D,E}. On W1×W2 we obtain a Wajsberg algebra
structure by defining the multiplication as in relation (3.2). We give this multiplication in the following table:
[TABLE]
(see relation (4.4) from [FV; 19], Example 4.12).
We remark that A≤B,A≤D,C≤D and the other elements can’t be
compared in the algebra W16=W116=(W1×W2,∇116). We denote this order relation
with ≤116. All proper ideals are P1={O,A,B},P2={O,C}
and are prime ideals. We obtain W116/P1={O,E} and W116/P2={O,A,E}, where O={O,A,B},E={C,D,E},O={O,C},A={A,D},E={B,E}. Indeed, we have (E∘B)∘(B∘E)′=B∘E′=B∘O=C,
(A∘D)∘(D∘A)′=E∘B′=C and so on. From here, it results that W116≃W116/P1×W116/P2, as in Remark 3.7, iv).
If we consider the isomorphism
[TABLE]
we obtain on W1×W2 a new Wajsberg algebra structure, with the
multiplication ∇126 given in relation (4.5)
from [FV;19], Example 4.12. We denote this algebra with W126=(W1×W2,∇126). Using the
above isomorphism, we get that: -algebras W116=(W1×W2,∇116) and W126=(W1×W2,∇126) are isomorphic as Wajsberg
algebras;
A≤C,A≤D,B≤D and the other elements can’t be compared in the
algebra W126. We denote this order relation with ≤126;
-all proper ideals P1={O,A,C}, P2={O,B} are prime ideals;
W126/P1={O,E} and W126/P2={O,A,E}, where O={O,A,C},E={B,D,E},O={O,B},A={A,D},E={C,E};
-W126≃W126/P1×W126/P2, as in Remark 3.7, iv).
If we consider the map
[TABLE]
we obtain on W1×W2 a new Wajsberg algebra structure, with the
multiplication ∇136 given in relation (4.6)
from [FV;19], Example 4.12.
We denote this algebra with W136=(W1×W2,∇136). Using the above map and Proposition 3.8
from above, we get that:
-algebras W116=(W1×W2,∇116) and W136=(W1×W2,∇136) are isomorphic only as ordered sets;
-B≤A,C≤A,B≤D and the other elements can’t be compared in the
algebra W136. We denote this order relation with ≤136.
-all proper ideals P1={O,B,D}, P2={O,C} are prime ideals;
W136/P1={O,E} and W136/P2={O,B,E}, where O={O,B,D},E={C,A,E},O={O,C},A={A,D},E={C,E};
-W136≃W136/P1×W136/P2, as in Remark 3.7, iv).
If we consider W2×W1={(0,0),(0,1),(b,0),(b,1),(e,0),(e,1)}=
={O,A,B,C,D,E}, on W2×W1 we obtain a Wajsberg algebra
structure by defining the multiplication as in relation (3.2). We give this multiplication in the following table:
[TABLE]
(see relation (4.7) from [FV; 19], Example 4.12)
The algebras (W1×W2,∇116) and (W2×W1,∇146) are isomorphic as Wajsberg
algebras, by taking the map
[TABLE]
In W146=(W2×W1,∇146),
we have A≤C,B≤C,B≤Dand the other elements can’t be compared.
We denote this order relation with ≤146. All proper ideals P1={O,B,D}, P2={O,A} are prime ideals.
We also have:
-W146/P1={O,E} and W146/P2={O,B,E}, where O={O,B,D},E={C,A,E},O={O,A},B={B,C},E={D,E};
-W136≃W136/P1×W136/P2, as in Remark 3.7, iv).
If we take the map
[TABLE]
[TABLE]
we obtain the Wajsberg algebra W156=(W2×W1,∇156). In this algebra, we have B≤C,B≤A,D≤Cand the other elements can’t be compared. We denote this order
relation with ≤156. The algebras W116=(W1×W2,∇116) and W156=(W2×W1,∇156) are isomorphic as Wajsberg
algebras. All proper ideals P1={O,A,B}, P2={O,D} are prime.
We also have:
-W156/P1={O,E} and W156/P2={O,B,E}, where O={O,A,B},E={C,D,E},O={O,D},B={B,C},E={A,E};
-W156≃W156/P1×W156/P2, as in Remark 3.7, iv).
If we consider the map
[TABLE]
[TABLE]
we obtain the Wajsberg algebra W166=(W2×W1,∇166). In this algebra, we have C≤D,C≤A,B≤Aand the other elements can’t be compared. We denote this order
relation with ≤166. The algebras W116=(W2×W1,∇116) and W166=(W2×W1,∇166) are isomorphic only as ordered
sets. All proper ideals P1={O,C,D}, P2={O,B} are prime.
We also have:
-W166/P1={O,E} and W166/P2={O,A,E}, where O={O,C,D},E={A,B,E},O={O,B},A={A,C},E={D,E};
-W166≃W166/P1×W166/P2, as in Remark 3.7, iv).
If we take the map
[TABLE]
[TABLE]
we obtain the Wajsberg algebra W176=(W2×W1,∇176). In this algebra, we have C≤A,C≤B,D≤Band the other elements can’t be compared. The algebras W116=(W1×W2,∇116) and W176=(W2×W1,∇176) are isomorphic
as Wajsberg algebras.We denote this order relation with ≤176.
All proper ideals P1={O,A,C}, P2={O,D} are prime ideals.
We also have:
-W176/P1={O,E} and W176/P2={O,B,E}, where O={O,A,C},E={B,D,E},O={O,D},B={B,C},E={A,E};
-W176≃W176/P1×W176/P2, as in Remark 3.7, iv).
From the above, we have that there are only two types of nonisomorphic
Wajsberg algebras of order 6. Using connections between MV-algebras and
Wajsberg algebras, we obtain that there are two nonisomorphic MV-algebras of
order 6. The same number was found in [BV; 10], Table 8, by using other
method. Since we gave enough examples, now we wonder how we can find all
Wajsberg algebras of order 6? For this purpose, we must find all
isomorphisms of ordered sets f:W116→W116, such that f(O)=O,f(E)=E. Therefore,
there are 4!=24 isomorphisms and, in turn, 24 partially ordered Wajsberg
algebras: 8 are isomorphic with W116 as Wajsberg
algebras and ordered sets and the next 16 are isomorphic with W116 only as ordered sets. In total, there are 25 Wajsberg algebras of
order 6, as we can see in the below table in which are described all
isomorphisms f1j6:W116→W1j6, j∈{1,...,24}, such that f1j6(x∇116y)=f1j6(x)∇1j6f1j6(y) .
W1j6
The isomorphism and order relation
Isomorpism with W116
W116f116(A)=A,f116(B)=B,f116(C)=C,f126(D)=D
isomorphism of Wajsberg
algebras
W126f126(A)=A,f126(B)=C,f126(C)=B,f126(D)=D
isomorphism of Wajsberg
algebras
W136f136(A)=B,f136(B)=D,f136(C)=C,f136(D)=A
only isomorphism of ordered sets
W146f146(A)=B,f146(B)=D,f146(C)=A,f146(D)=C
isomorphism of Wajsberg algebras
W156f156(A)=B,f156(B)=A,f156(C)=D,f156(D)=C
isomorphism of Wajsberg algebras
W166f166(A)=C,f166(B)=D,f166(C)=B,f166(D)=A
only isomorphism of ordered sets
W176f176(A)=C,f176(B)=A,f176(C)=D,f176(D)=B
isomorphism of Wajsberg algebras
W186f186(A)=C,f186(B)=B,f186(C)=D,f186(D)=A
only isomorphism of ordered sets
W196f196(A)=C,f196(B)=B,f196(C)=A,f196(D)=D
only isomorphism of ordered sets
W1,106f1,106(A)=C,f1,106(B)=D,f1,106(C)=A,f1,106(D)=B
isomorphism of
Wajsberg algebras
W1,116f1,116(A)=C,f1,116(B)=A,f1,116(C)=B,f1,116(D)=D
only isomorphism of
ordered sets
W1,126f1,126(A)=B,f1,126(B)=C,f1,126(C)=D,f1,126(D)=A
only isomorphism of ordered sets
W1,136f1,136(A)=B,f1,136(B)=C,f1,136(C)=A,f1,136(D)=D
only isomorphism of ordered sets
W1,146f1,146(A)=B,f1,146(B)=A,f1,146(C)=C,f1,146(D)=D
only isomorphism of ordered sets
W1,156f1,156(A)=A,f1,156(B)=C,f1,156(C)=D,f1,156(D)=B
only isomorphism of
ordered sets
W1,166f1,166(A)=A,f1,166(B)=D,f1,166(C)=C,f1,166(D)=B
only isomorphism of
ordered sets
W1,176f1,176(A)=A,f1,176(B)=D,f1,176(C)=B,f1,176(D)=C
only isomorphism of
ordered sets
W1,186f1,186(A)=A,f1,186(B)=B,f1,186(C)=D,f1,186(D)=C
only isomorphism of
ordered sets
W1,196f1,196(A)=D,f1,196(B)=B,f1,196(C)=C,f1,196(D)=A
isomorphism of Wajsberg algebras
W1,206f1,206(A)=D,f1,206(B)=C,f1,206(C)=B,f1,206(D)=A
only isomorphism of ordered sets
W1,216f1,216(A)=D,f1,216(B)=A,f1,216(C)=C,f1,216(D)=B
only isomorphism of ordered sets
W1,226f1,226(A)=D,f1,226(B)=C,f1,226(C)=A,f1,226(D)=B
only isomorphism of ordered sets
W1.236f1,236(A)=D,f1,236(B)=A,f1,236(C)=B,f1,236(D)=C
isomorphism of Wajsberg algebras
W1,246f1,246(A)=D,f1,246(B)=B,f1,246(C)=A,f1,246(D)=C
only isomorphism of ordered sets
4.3.Wajsberg algebras of order 8
**Totally ordered case. **Let W={O≤X≤Y≤Z≤T≤U≤V≤E} be a totally ordered set. On W we define a multiplication
as in relation (3.1). We have X=V, Y=U, Z=T. Therefore the algebra W has the
following multiplication table:
[TABLE]
2) **Partially ordered case. **There is only two types of
partially ordered Wajsberg algebra with 8 elements, up to an isomorphism
of ordered sets. Indeed, π8=2. Let
(W1={0,a,b,e},∘,,e) and (W2={0,1},⋅,′,1) be two finite totally ordered
Wajsberg algebras. Using relation (3.1), on W1 we have
that b=a and a=b. We consider W1×W2={(0,0),(0,1),(a,0),(a,1),(b,0),(b,1),(e,0),(e,1)}=
={O,X,Y,Z,T,U,V,E}. On W1×W2 we obtain a Wajsberg algebra
structure by defining the multiplication as in relation (3.2), namely W118=(W1×W2,∇118). The multiplication ∇118 is given in the
following table:
[TABLE]
(see [FV; 19], Example 4.13, relation (4.8))
In W118 we have that O≤X≤Z≤U≤E, O≤Y≤T≤V≤E, O≤Y≤Z≤U≤E,O≤Y≤T≤U≤E
and the other elements can’t be compared in this algebra. We denote this
order relation with ≤118. All proper ideals P1={O,Y,T,V},
P2={O,X} are prime ideals.
We also have:
-W118/P1={O,E} and W118/P2={O,Y,U,E}, where O={O,Y,T,V},E={X,Z,U,E},O={O,X},Y={Y,Z},
U={U,T},E={V,E};
-W118≃W118/P1×W118/P2, as in Remark 3.7, iv).
Now, we consider W2×W1={(0,0),(0,a),(0,b),(0,e),(1,0),(1,a),(1,b),(1,e)}=
={O,X,Y,Z,T,U,V,E}. On W2×W1 we obtain a Wajsberg algebra
structure by defining the multiplication as in relation (3.2), namely W128=(W2×W1,∇128). The multiplication ∇128 is given in
relation (4.8) from [FV;19], Example 4.13. We have that O≤X≤Y≤Z≤E, O≤X≤Y≤V≤E,
O≤X≤U≤V≤E, O≤T≤U≤V≤E, and the other
elements can’t be compared in this algebra. We denote this order relation
with ≤128. These two structures, W118 and W128, are isomorphic as Wajsberg algebras. The morphism is
[TABLE]
All proper ideals P1={O,X,Y,Z}, P2={O,T} are prime ideals.
We also have:
-W128/P1={O,E} and W128/P2={O,Y,U,E}, where O={O,X,Y,Z},E={T,U,V,E},O={O,T},U={X,U},
Y={V,Y},E={Z,E};
-W128≃W128/P1×W128/P2, as in Remark 3.7, iv).
If we take the map
[TABLE]
we obtain the Wajsberg algebra W138=(W2×W1,∇138). These two structures, W118
and W138, are isomorphic as ordered sets. In this algebra,
we have O≤Y≤X≤Z≤E,O≤U≤V≤T≤E,
O≤U≤X≤Z≤E,O≤U≤V≤Z≤E, and the other
elements can’t be compared in this algebra. We denote this order relation
with ≤138. The multiplication are given in the following table:
[TABLE]
[TABLE]
All proper ideals P1={O,U,V,T}, P2={O,Y} are prime ideals.
We also have:
-W138/P1={O,E} and W138/P2={O,U,Z,E}, where O={O,U,T,V},E={X,Z,Y,E},O={O,Y},Y={U,X},
Z={Z,V},E={V,E};
-W138≃W138/P1×W138/P2, as in Remark 3.7, iv).
If we take W2×W2×W2={(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),
(1,0,1),(1,1,0),(1,1,1)}={O,X,Y,Z,T,U,V,E}, on W2×W2×W2 we obtain a
Wajsberg algebra structure by defining the multiplication as in relation (3.2), namely W218=(W2×W2×W2,∇218). The multiplication ∇218 is given in the following table:
[TABLE]
(see [FV; 19], Example 4.13, relation (4.9)).
We have that X≤Z,X≤U,Y≤Z,T≤V,Y≤V,T≤U and the other
elements can’t be compared in this algebra. We denote this order relation
with ≤218. These two structures, W118 and W218, are not isomorphic as ordered sets, neither as
Wajsberg algebras. Here, we must remark that W2×W2×W2≃W218 and W2×(W2×W2)≃W118. All proper ideals P1={O,X}, P2={O,Y}, P3={O,T}, P4={O,X,Y,Z}, P5={O,X,T,U}, P6={O,Y,T,V} are also prime ideals.
We have that:
W218/P1={O,Y,T,E}, where O={O,X}, Y={Y,Z}, T={T,U},E={V,E};
-W218/P2={O,X,V,E}, where O={O,Y}, X={X,Z}, V={V,U}, E={T,E}.
-W218/P3={O,X,Y,E}, where O={O,T}, X={X,U}, Y={Y,V}, E={Z,E};
W218/P4={O,E}, where O={O,X,Y,Z}, E={U,V,T,E}; -W218/P5={O,E}, where O={O,X,T,U}, E={Y,Z,V,E};
-W218/P6={O,E},
where O={O,Y,T,V}, E={X,Z,U,E}.
Therefore
[TABLE]
and
[TABLE]
as in Remark 3.7, iv).
If we consider the map
[TABLE]
we obtain the Wajsberg algebra W228=(W2×W2×W2,∇228) given in the below table:
[TABLE]
These two structures, W218 and W228, are
isomorphic as Wajsberg algebras. We have that U≤V,U≤Z,T≤V,X≤Y,T≤Y,X≤Z and the other elements can’t be compared in this algebra.
We denote this order relation with ≤228.
All proper ideals P1={O,U}, P2={O,T}, P3={O,X}, P4={O,U,T,V}, P5={O,U,X,Z}, P6={O,T,X,Y} are also prime
ideals.
We obtain that:
W228/P1={O,T,X,E}, where O={O,U}, T={T,V}, X={X,Z},E={Y,E};
-W228/P2={O,U,Z,E}, where O={O,T}, U={U,V}, Z={Z,Y}, E={X,E}.
-W228/P3={O,Z,Y,E}, where O={O,X}, Z={Z,U}, Y={Y,T}, E={V,E};
W228/P4={O,E}, where O={O,U,T,V}, E={X,Y,Z,E}; -W228/P5={O,E}, where O={O,U,X,Z}, E={Y,V,T,E};
-W228/P6={O,E},
where O={O,T,X,Y}, E={V,Z,U,E}.
Therefore
[TABLE]
and
[TABLE]
as in Remark 3.7, iv).
If we take the map
[TABLE]
we obtain the Wajsberg algebra W238=(W2×W2×W2,∇238) given in the below table:
[TABLE]
These two structures, W218 and W238, are
isomorphic as ordered sets.We have that Z≤V,Z≤T,X≤V,U≤Y,X≤Y,U≤T and the other elements can’t be compared in this algebra.
We denote this order relation with ≤238.
All proper ideals P1={O,Z}, P2={O,X}, P3={O,U}, P4={O,Z,X,V}, P5={O,Z,U,T}, P6={O,X,U,Y} are also prime
ideals.
We obtain that:
W238/P1={O,X,T,E}, where O={O,Z}, X={X,V}, T={T,U},E={Y,E};
-W238/P2={O,V,Y,E}, where O={O,X}, V={V,Z}, Y={Y,T}, E={U,E}.
-W238/P3={O,Z,Y,E}, where O={O,U}, ={Z,T}, Y={Y,X}, E={V,E};
W238/P4={O,E}, where O={O,Z,X,V}, E={U,Y,T,E}; -W238/P5={O,E}, where O={O,Z,U,T}, E={Y,X,V,E};
-W238/P6={O,E},
where O={O,X,U,Y}, E={V,Z,T,E}.
Therefore
[TABLE]
and
[TABLE]
as in Remark 3.7, iv).
From the above, we have that there are only three types of nonisomorphic
Wajsberg algebras of order 8. Using connections between MV-algebras and
Wajsberg algebras, we obtain that there are two nonisomorphic MV-algebras of
order 8. The same number was found in [BV; 10], Table 8, by using other
method. Now, we count the number of all Wajsberg algebras of order 8. For
this purpose, we must find all isomofirsms of ordered sets f:W118→W118, such that f(O)=O,f(E)=E and all isomofirsms of ordered sets f:W218→W218, such that f(O)=O,f(E)=E. Therefore, there are 2×6!=1440 isomorphisms
and, in turn, 1440 partially ordered Wajsberg algebras. In total, there
are 1441 Wajsberg algebras of order 8.
4.4.Wajsberg algebras of order 9\vskip12.0ptplus4.0ptminus4.0pt
**Totally ordered case. **Let W={O≤X≤Y≤Z≤T≤U≤S≤V≤E} be a totally ordered set. On W we define a
multiplication as in relation (3.1). We have X=V, Y=S, Z=U,T=T.
Therefore the algebra W has the following multiplication table:
[TABLE]
2) **Partially ordered case. **There is only one type of
partially ordered Wajsberg algebra with 9 elements, up to an isomorphism
of ordered sets. Indeed, π9=1(W1={0,a,e},∘,,e) and (W2={0,b,1},⋅,′,1).We have
a=a and b′=b. We have
We obtain a Wajsberg algebra structure, namely W119=(W1×W2,∇119). We remark that O≤X≤Y≤U,O≤X≤T≤U,O≤X≤T≤V,
O≤Z≤T≤U,O≤Z≤T≤V,O≤Z≤S≤V and the other
elements can’t be compared. We denote this order relation with ≤119. All proper ideals P1={O,X,Y}, P2={O,Z,S}, are also
prime ideals.
We also have:
-W119/P1={O,U,E} and W119/P2={O,T,E}, where O={O,X,Y},U={U,T,Z},E={S,V,E},
O={O,Z,S},T={T,V,X},E={Y,U,E};
-W119≃W119/P1×W119/P2, as in Remark 3.7, iv).
From the above, we have that there are only two types of nonisomorphic
Wajsberg algebras of order 9. Using connections between MV-algebras and
Wajsberg algebras, we obtain that there are two nonisomorphic MV-algebras of
order 9. The same number was found in [BV; 10], Table 8, by using other
method. Now, we count the number of Wajsberg algebras of order 9. For this
purpose, we must find all isomofirsms of ordered sets f:W119→W119, such that f(O)=O,f(E)=E, such that f(O)=O,f(E)=E. Therefore, there are 7!=5040 isomorphisms and, in turn, 5040 partially
ordered Wajsberg algebras. In total, there are 5041 Wajsberg algebras of
order 9.
**Remark 4.1. **For n∈{2,3,5,7}, the Wajsberg algebras of order n are totally ordered, since n is a prime number. The multiplication
tables are easy to obtain.
From the above, we obtain the following proposition.
Proposition 4.2.With the above notations we have:
The number of nonisomorphic Wajsberg algebras types of ordernisπn.
The total number of Wajsberg algebras of ordernisπn(n−2)!+1, forn≥4,nnot a prime number.□
[TABLE]
Conclusions
Starting from some results obtained in [FV;19], in this paper we provided a
Representation Theorem for finite Wajsberg algebras. Using this theorem, we
describe all finite Wajsberg algebras of order n≤9.
Wajsberg algebras have many applications in various domains and their study
can provide us new and interesting properties of them.
[TABLE]
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[TABLE]
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