θ−valued Lukasiewicz–Moisil algebras with and without negation
A. V. Figallo, I. Pascual and A. Ziliani
Departamento de Matem´atica, Universidad Nacional del Sur, Instituto de Ciencias B´asicas, Universidad Nacional de San Juan,
Argentina.
Abstract
Philosophical problems arising from the idea that there are state- ments which are neither true nor false, led to the formulation of many–
valued logics by Lukasiewicz. Since then, plenty of research has been developed in this area. In 1968, G. Moisil found an example which gave him the motivation he had been looking for in order to legitimate the introduction and study of infinitely–valued Lukasiewicz algebras, so he definedθ−valued Lukasiewicz algebras, whereθis the order type of a chain.
In this article, we determine a topological duality for θ−valued Lukasiewicz algebras (or Lkθ−algebras) equivalent to the one given by Filipoiu in 1980. Not only does the duality enable us to obtain a description of the Lkθ−congruences on an Lkθ−algebra, but also to characterize the subdirectly irreducible Lkθ−algebras. Furthermore, we extend the above study to the case of Lkθ−algebras with nega- tion arriving through a different method at the results indicated by V.
Boisescu et al ( Lukasiewicz–Moisil Algebras, Annals of Discrete Math- ematics 49, North–Holland, 1991).
1 Introduction and preliminaries
In 1940, G. Moisil introduced 3−valued and 4−valued Lukasiewicz alge- bras with the purpose of obtaining the algebraic counterpart of the corre- sponding Lukasiewicz logics. A year later, he generalized these algebras by definingn−valued Lukasiewicz algebras ([14]) and he studied them from the algebraic point of view. It is well-known that these algebras are not the al- gebraic counterpart ofn−valued Lukasiewicz propositional calculi forn≥5 (see [3, 4]). R. Cignoli ([5, 6]) found algebraic counterparts forn≥5 and he called them proper n−valued Lukasiewicz algebras. It is worth mentioning that in 1997, M. Fidel ([10]) described a propositional calculus which has n−valued Lukasiewicz algebras as the algebraic counterpart.
On the other hand, in 1968 G. Moisil ([15]) definedθ−valued Lukasiewicz algebras, whereθis the order type of a chain. The membership to a fuzzy set
in the sense of Zadeh is the example where this author found the motivation in order to legitimate the introduction and study of these algebras.
Letθ≥2 be the order type of a totally ordered set J with least element 0 beingJ ={0}+I (ordinal sum). Following Boisescu et al ([3]) recall that:
A θ−valued Lukasiewicz–Moisil algebra (or Lkθ−algebra) is an algebra hA,∨,∧,0,1,{φi}i∈I,{φi}i∈Iiof type (2,2,0,0,{1}i∈I,{1}i∈I) wherehA,∨,∧, 0,1iis a bounded distributive lattice and for alli∈I,φi and φi satisfy the following conditions:
(L1) φi is an endomorphism of bounded distributive lattices, (L2) φix∨φix= 1, φix∧φix= 0,
(L3) φiφjx=φjx,
(L4) i6j implies φix6φjx.
(L5) φix=φiy for alli∈I implyx=y.
Aθ-valued Lukasiewicz–Moisil pre-algebra (or Lkθ-pre-algebra) is an al- gebra hA,∨,∧,0,1,{φi}i∈I,{φi}i∈Ii of type (2,2,0,0,{1}i∈I,{1}i∈I) where hA,∨,∧,0,1iis a bounded distributive lattice and for all i ∈ I, φi and φi fullfil (L1)–(L4).
The idea of considering these notions separately is due to Beznea ([1]) in the caseθ=n.
We shall denote by PLkθ the category of Lkθ−pre-algebras and their corresponding homomorphisms and by Lkθ the category of Lkθ−algebras which is a full subcategory of the first one.
Suppose now that the set J defined above has both least element 0 and greatest element 1.
A θ−valued Lukasiewicz–Moisil algebra (pre-algebra) with negation (or nLkθ−algebra (pre-algebra)) is an algebrahA,∨,∧,∼,0,1,{φi}i∈I,{φi}i∈Ii of type (2,2,1,0,0,{1}i∈I,{1}i∈I) where
(NL1) hA,∨,∧,∼,0,1iis a De Morgan algebra,
(NL2) hA,∨,∧,0,1,{φi}i∈I,{φi}i∈Ii is an Lkθ−algebra (pre-algebra), (NL3) φix=∼φix,
(NL4) there is a decreasing involutiond:I →I such thatφi∼x=∼φdix.
The category of the Lkθ−algebras with negation and their corresponding homomorphisms will be denoted by nLkθ.
The notions and results announced here will be used throughout the paper.
It is well known that there are congruences in the classes of Lkθ−alge- bras and nLkθ−algebras such that the quotient algebra doesn’t satisfy the determination principle. That is the reason why a new notion is defined as follows: a θLkθ−congruence on an Lkθ−algebra is a bounded distributive lattice congruence ϑ such that (x, y) ∈ ϑ if and only if (φix, φiy) ∈ ϑ for alli∈I ([2, 13]).
On the other hand, the interest in the following example arises from the statement proved in [3], which we will show below in Theorem 2.29 and Corollary 2.17 by a different reasoning:
Let B2[I] be the set of all increasing functions from I to the Boolean algebra B2 with two elements and let θ be the order type of the totally ordered set {0}+I with least element 0. Then, B2[I] can be made into an Lkθ−algebra hB2[I],∧,∨,0,1,{φi}i∈I,{φi}i∈Ii where the operations of the latticehB2[I],∧,∨,0,1iare defined pointwise and for alli, j∈I, (φif)(j) = f(i) and (φif)(j) = (f(i))0, being x0 the Boolean complement of x. If the set I has least and greatest element and there is a decreasing involution d:I −→I, then the Lkθ−algebra B2[I] can be made into an nLkθ−algebra hB2[I],∧,∨,∼,0,1,{φi}i∈I,{φi}i∈Ii, where (∼f)(i) = (f(di))0 for alli∈I.
A. Filipoiu ([3, 11, 12]) presented a duality to the case of Lkθ−alge- bras. In order to do this, he considered the category Prθ whose objects areθ−valued Priestley spaces (or Pθ−spaces) and whose morphisms are in- creasing continuous functions with certain additional conditions. Besides, this author proved that this category is dually equivalent toLkθ. Further- more, he extended this duality and the one obtained by Cornish and Fowler ([9]), to the case of nLkθ−algebras.
For a general account of the origins and the theory of Lukasiewicz many valued logics and Lukasiewicz algebras the reader is referred to [3, 7, 8, 13].
This paper is organized as follows. In Section 1, we summarize the de- finitions and results necessary for further development. In Section 2, we determine a topological duality for θ−valued Lukasiewicz algebras equiva- lent to the one given by Filipoiu in [11]. This duality allows us to obtain the Lkθ−congruences on these algebras. As a consequence, we characterize the subdirectly irreducible Lkθ-algebras. Furthermore, we prove that the asso- ciated space of subdirectly irreducible Lkθ-algebras is a totally ordered set, and therefore, they are the subalgebras ofB2[I]described above. In Section 3, we extend the results obtained in Section 2, to the case of Lkθ−algebras with negation arriving through a different way at the results indicated in [3].
2 θ − valued Lukasiewicz–Moisil algebras without negation
Here we will obtain a characterization of the dual of Lkθ in terms of ordered topological spaces.
Definition 2.1 A θ−valued Lukasiewicz–Moisil pre-space(orlθP−pre-space) is a pair (X,{fi}i∈I) provided the following conditions are satisfied:
(lP1) X is a Priestley space,
(lP2) fi:X →X is a continuous function, (lP3) x6y implies fi(x) =fi(y),
(lP4) i6j implies fi(x)6fj(x), (lP5) fi◦fj =fi.
A θ−valued Lukasiewicz–Moisil space(or lθP−space) is an lθP−pre-space satisfying the following condition:
(lP6) for all U, V ∈ D(X), fi−1(U) = fi−1(V) for all i∈ I imply U = V, where D(X)is the lattice of all increasing clopen subsets of X.
Definition 2.2 Let(X,{fi}i∈I)and(X0,{fi0}i∈I)belθP−spaces. AnlθP− function is an isotone continuous function f :X → X0 satisfying fi0◦f = f◦fi for all i∈I.
The categorylθP haslθP−spaces as objects andlθP−functions as mor- phisms and it is a full subcategory of the categoryPlθP of lθP–pre–spaces.
Proposition 2.3 enables us to prove that the notion of Pθ−space, given by Filipoiu, is equivalent to that oflθP−space.
Proposition 2.3 Let (X,{fi}i∈I) be an lθP−space. Then for each U ∈ D(X) the following conditons are fulfilled:
(lP9) X\fi−1(U)∈D(X),
(lP10) i6j implies fi−1(U)⊆fj−1(U).
Theorem 2.4 LetX be a Priestley space and letfi :X→X be a function for each i∈I. Then the following conditions are equivalent:
(i) (X,{fi}i∈I) is an lθP−space, (ii) (X,{fi}i∈I) is a Pθ−space.
Corollary 2.5 LethA,∨,∧,0,1,{φi}i∈I,{φi}i∈Ii be anLkθ−algebra and let X(A)be the Priestley space associated withA. Then Lθ(A) = (X(A),{fiA}i∈I) is an lθP−space, where for every i∈I, the function fiA :X(A)−→ X(A) is defined by fiA(P) =φi−1(P) for allP ∈X(A).
Next, we are going to denote by Lθ(A) or byX(A) the lθP-space asso- ciated withA.
Theorem 2.6 The category lθP is naturally equivalent to the dual of the categoryLkθ.
As a direct consequence of Theorem 2.6 we conclude that the categories PLkθand PlθP are dually equivalent.
Next we will determine some properties of lθP−spaces which will be quite useful in order to characterize the lattice of all congruences and the lattice of allθ−congruences of Lkθ−algebras. These results will be taking into account to obtain the subdirectly irreducible Lkθ−algebras and the subdirectly irreducibleθLkθ−algebras.
Proposition 2.7 Let(X,{fi}i∈I) be an lθP−space. Then for all x∈X it holds
(lP11) x6fi(x) or fi(x)6x for all i∈I.
Corollary 2.8 If(X,{fi}i∈I) is anlθP−space, then X is the cardinal sum of the sets[{fi(x)}i∈I)∪({fi(x)}i∈I]for x∈X.
Proposition 2.9 Let (X,{fi}i∈I) be an lθP−space where I has least ele- ment0and greatest element 1. Then the following conditions hold, for each x∈X:
(lP13) f0(x)6x 6f1(x),
(lP14) f0(x) is the unique minimal element inX that precedesx, (lP15) f1(x) is the unique maximal element inX that followsx.
Corollary 2.10 Let (X,{fi}i∈I) be an lθP−space where I has least ele- ment 0 and greatest element 1. Then, X is the cardinal sum of the sets [f0(x), f1(x)]for x∈X.
Definition 2.11 Let (X,{fi}i∈I) be an lθP−space. A subset Y of X is semimodal if fi(Y) ⊆ Y for all i ∈ I or equivalently Y ⊆ fi−1(Y) for all i∈I.
The closed and semimodal subsets of the lθP−space associated with an Lkθ−algebra play a fundamental role in the characterization of Lkθ−con- gruences on these algebras, as we shall show next.
Theorem 2.12 Let A ∈ Lkθ and let Lθ(A) be the lθP−space associated with A. Then, the lattice CS( Lθ(A)) of all closed and semimodal subsets of Lθ(A) is isomorphic to the dual lattice ConLkθ(A) of all Lkθ−congruences on A, and the isomorphism is the function ΘS defined by the prescription ΘS(Y) ={(a, b)∈A×A: σA(a)∩Y =σA(b)∩Y} (see[16, 17, 18]).
Definition 2.13 Let (X,{fi}i∈I) be an lθP−space. A θ−subset is a semi- modal subset Y of X which verifies that Y ⊆ S
i∈I
fi(Y).
Proposition 2.14 Let(X,{fi}i∈I) be an lθP−space and let Y be a subset of X. Then the following conditions are equivalent:
(i) Y is a closedθ−subset,
(ii) there is a subset Z of X such that Y = S
i∈I
fi(Z).
Corollary 2.15 Let (X,{fi}i∈I) be an lθP−space and let Cθ0(X) be the set of all non-empty, closed and θ−subsets of X. Then, a subset Y of X is a minimal element of Cθ0(X) if and only if there is x ∈ X such that Y ={fi(x)}i∈I.
Proposition 2.16 If(X,{fi}i∈I) is an lθP−space, then S
i∈I
fi(X) is dense in X.
The closedθ−subsets of thelθP−space associated with an Lkθ−algebra enable us to characterize theθLkθ−congruences on these algebras as Theo- rem 2.17 shows.
Theorem 2.17 Let A ∈ Lkθ and let Lθ(A) be the lθP−space associated with A. Then, the lattice Cθ( Lθ(A)) of all closed θ−subsets of Lθ(A) is isomorphic to the dual lattice ConθLkθ(A) of all θLkθ−congruences on A, and the isomorphism is the functionΘθ defined as in Theorem 2.12.
Corollary 2.18 Every maximal Lkθ−congruence on an Lkθ−algebra A is a maximal θLkθ−congruence onA.
Corollary 2.19 Every Lkθ−algebra is semisimple.
Theorem 2.20 Let(X,{fi}i∈I)be anlθP−space andILθ(X)theLkθ−algebra associated with X. Then the following conditions are equivalent:
(i) X is a totally ordered set, (ii) ILθ(X)is a simple Lkθ−algebra, (iii) ILθ(X)is a simple θLkθ−algebra,
(iv) ILθ(X)is a subdirectly irreducible θLkθ−algebra, (v) ILθ(X)is a subdirectly irreducible Lkθ−algebra.
Corollary 2.21 The algebra B2[I] is a subdirectly irreducibleLkθ−algebra.
In order to prove that every subdirectly irreducible Lkθ−algebra is iso- morphic to an Lkθ−subalgebra of B2[I], we will study some properties of totally ordered lθP−spaces and in particular, of the lθP−space associated withB2[I].
Let (D,≺) be a directed set andY a topological space. In what follows, for a net ϕ:D→Y, we shall write ϕ→y0 whenever ϕ converges toy0. Proposition 2.22 Let (X,{fi}i∈I) be a totally ordered lθP−space and for each x ∈ X, x 6= fi(x) for all i ∈ I, let Kx = {k ∈ I : fk(x) < x} and Jx = {i ∈ I : x < fj(x)}. If the order on Kx is the one induced by the order defined on I, and the order onJx is the one induced by the dual order defined onI, then fj(x)→x or fk(x)→x.
Proposition 2.23 Let P ∈ X(B2[I]) and let P 6=φi−1(P) for all i∈I. If KP = {k ∈ I : φk−1(P) ⊂ P}, JP = {j ∈ I : P ⊂ φj−1(P)}, where the order onKP is the one induced by the order defined on I, and the order onJP
is the order induced by the dual order defined onI, then eitherφj−1(P)→P or φk−1(P)→P, but not both.
Corollary 2.24 Let P ∈ X(B2[I]) and let P 6= φi−1(P) for all i ∈ I. If {φid
−1(P)}d∈D is a net andφid
−1(P)→P, then there isdo∈Dsuch that ei- ther{φid
−1(P)}d∈Td0 ⊆ {φj−1(P)}j∈JP or{φid
−1(P)}d∈Td0 ⊆ {φk−1(P)}k∈KP, but not both, where Td={c∈D: d≺c} for all d∈D.
Lemma 2.25 If P ∈ X(B2[I]), then φi−1(P) 6= φj−1(P) for all i, j ∈ I, i6=j.
Proposition 2.26 LetP ∈X(B2[I]). Then, P =φi0
−1(P) for some i0 ∈I if and only if P = [f), where f :I −→ B2 is defined by f(i) = 1 if i0 6i andf(i) = 0 otherwise.
Proposition 2.27 LetP ∈X(B2[I])and letP =φi0
−1(P)for some i0 ∈I. If {φid
−1(P)}d∈D is a net and φid
−1(P) → P, then there is d0 ∈ D such that φid
−1(P) =φi0
−1(P) for all d∈Td0.
Proposition 2.28 If(X,{fi}i∈I)is a totally orderedlθP−space, then there is an lθP−function fromX(B2[I]) onto X.
Theorem 2.29 LetAbe an Lkθ−algebra. ThenA is subdirectly irreducible if and only if A is isomorphic to an Lkθ−subalgebra of B2[I].
3 θ − valued Lukasiewicz–Moisil algebras with ne- gation
In the sequel, taking into account the results established in [9] we will extend the duality obtained in the Section 2 for Lkθ−algebras to the case of nLkθ−algebras. This duality leads us determine the congruences and the θ−conguences on nLkθ−algebras. Finally, we will describe the subdirectly irreducible nLkθ−algebras and the subdirectly irreducibleθnLkθ−algebras.
In order to this, we introduce the following notion:
Definition 3.1 A θ−valued Lukasiewicz–Moisil space(pre-space)with nega- tion or N lθP−space (pre-space) is a triple (X, g,{fi)}i∈I) satisfying the following conditions:
(nlP1) (X, g)is a De Morgan space([9]),
(nlP2) (X,{fi)}i∈I) is an lθP−space (pre-space), (nlP3) fi◦g=fi,
(nlP4) g◦fi =fdi, where d:I →I is a decreasing involution.
Definition 3.2 Let(X, g,{fi}i∈I)and(X0, g0,{fi0
}i∈I)be NlθP–spaces(pre- spaces). An NlθP–function from(X, g,{fi}i∈I)to(X0, g0,{fi0}i∈I)is an lθP–
functionf :X→X0 satisfying that f◦g=g0◦f.
The category NlθP of NlθP−spaces is a full subcategory of P NlθP which has NlθP−pre-spaces as objects and N lθP−functions as morphisms.
Theorem 3.3 The categoryNlθP is dually equivalent to the categorynLkθ. Theorem 3.3 allows us to assert that the categoriesnPLkθandP NlθP are dually equivalent.
The following properties of N lθP−spaces enable us to determine the subdirectly irreducible nLkθ−algebras.
First, recall that a subsetY of a De Morgan space (X, g) is involutive if g(Y) =Y.
Proposition 3.4 Every N lθP−space X is the cardinal sum of the closed, semimodal and involutive sets[f0(x), f1(x)], forx∈X.
Lemma 3.5 IfX∈NlθP, then every closedθ−subset of X is involutive.
Theorem 3.6 LetA∈nLkθand letN Lθ(A) be theN lθP-space associated withA. Then, it holds:
(i) the lattice CIS(N Lθ(A)) of all involutive, semimodal and closed sub- sets of N Lθ(A) is isomorphic to the dual lattice ConnLkθ(A) of all nLkθ−congruences on A,
(ii) the latticeCθ(N L(A)) of all closedθ−subsets ofN L(A) is isomorphic to the dual latticeConθnLkθ(A) of all θnLkθ−congruences onA, where in both cases, the isomorphism is defined as in Theorem 2.12.
Corollary 3.7 LetA∈nLkθand let ϑbe a binary relation on A. Thenϑ is aθLkθ−congruence on A if and only if ϑ is aθnLkθ−congruence on A.
Theorem 3.6 and Corrollary 3.7 combine, as we have already indicated, to yield our promised characterization of the subdirectly irreducible objects innLkθ.
Theorem 3.8 Let A ∈ nLkθ. Then the following conditions are equiva- lent:
(i) A is a simple nLkθ−algebra, (ii) A is a simple θnLkθ−algebra,
(iii) A is a subdirectly irreducibleθnLkθ−algebra, (iv) A is a subdirectly irreduciblenLkθ−algebra,
(v) X(A) is totally ordered,
(vi) A is isomorphic to an nLkθ−subalgebra of B2[I].
References
[1] L. Beznea θ−valued Moisil algebras and dual categories, (Romanian), Master Thesis, University of Bucharest, 1981.
[2] C. Boicescu, Contributions to the study of Lukasiewicz algebras, (Ro- manian). Ph.D. Thesis. Univ. of Bucharest, 1984.
[3] C. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu, Lukasiewicz–
Moisil Algebras, North–Holland, 1991.
[4] R. Cignoli,Moisil Algebras, Notas de L´ogica Matem´atica 27, Instituto de Matem´atica, Universidad Nacional del Sur, 1970.
[5] R. Cignoli,Some algebraic aspects of many-valued logics, Brazilian Con- ference on Mathematical Logic 3(1980), 49–69.
[6] R. Cignoli, Proper n−valued Lukasiewicz algebras as S−algebras of Lukasiewicz valued propositional calculi, Studia Logica 41(1982), 3-16.
[7] R. Cignoli, I. D’Ottaviano and D. Mundici, Algebras das Logicas de Lukasiewicz, Campinas UNICAMP - CLE, 1995.
[8] R. Cignoli, I. D’Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer, 2000.
[9] W. Cornish and P. Fowler, Coproduts of De Morgan Algebras, Bull.
Austral. Math. Soc., 16(1977), 1–13.
[10] M. Fidel,Semantic models of modal logic inn–valued logics, Actas del Cuarto Congreso Dr. Antonio A.R. Monteiro (1997), 167–177.
[11] A. Filipoiu,Representation theorems for Lukasiewicz algebras. Discrete Math., 27(1980), 107–110.
[12] A. Filipoiu, θ valued Lukasiewicz-Moisil algebras and logics. (Ro- manian). Ph.D. Thesis, Univ. of Bucharest, 1981.
[13] A. Iorgulescu, (1+θ)−valued Lukasiewicz–Moisil algebras with negation (Romanian). Ph.D. Thesis. Univ. of Bucharest, 1984.
[14] G. Moisil, Notes sur les logiques non-chrysippiennes, Ann. Sci. Univ.
Jassy, 27(1941), 86–98.
[15] Gr. Moisil, Lukasiewiczian algebras. Computing Center, Univ.
Bucharest (preprint)(1972), 311–324.
[16] H. Priestley,Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc., 2(1970), 186–190.
[17] H. Priestley, Ordered topological spaces and the representation of dis- tributive lattices, Proc. London Math. Soc., 3(1972), 507–530.
[18] H. Priestley,Ordered sets duality for distributive lattices, Ann. Discrete Math., 23(1984), 39–60.
