University of Ostrava

Institute for Research and Applications of Fuzzy

Modeling

Algebraic Analysis of Fuzzy Systems

Antonio Di Nola, Ada Lettieri, Irina Perfilieva,Vilem Novak

Research report No. 72

Submitted/to appear:

Fuzzy Sets and Systems

Supported by:

Czech-Italian project 21/MC1, grant IAA1187301 of the GA AV CR, project MSM6198898701 of the MSMT CR

University of OstravaInstitute for Research and Applications of Fuzzy Modeling

30. dubna 22, 701 03 Ostrava 1, Czech Republic

tel.: +420-597 460 234 fax: +420-597 461 478e-mail: adinola@unisa.it,lettieri@unina.it,{Irina.Perfilieva,Vilem.Novak}@osu.cz

Algebraic Analysis of Fuzzy Systems 1

Antonio Di Nola, a Ada Lettieri, b Irina Perfilieva, c,∗

Vilem Novak c

aUniversita di Salerno,

Facolta di Scienze, Dipt. di Matematica e Informatica,

Via S. Allende, 84081 Baronissi,Salerno, Italy

bUniversita di Napoli, Dipt. di Costruzioni e Metodi Matematici in Architettura,

Via Monteoliveto 3, 80134 Napoli, Italy

cUniversity of Ostrava

Institute for Research and Applications of Fuzzy Modeling

30. dubna 22, 701 03 Ostrava 1, Czech Republic

Abstract

In this paper, we have developed an algebraic theory, suitable for the analysis offuzzy systems. We have used the notions of semiring and semimodule, introducedthe notion of semilinear space and gave numerous examples of them and definedalso the notions of linear dependence and independence. Then, we have shown thatthe composition operation, which plays an essential role in the analysis of fuzzysystems because of its role in the compositional rule of inference, can be interpretedas a homomorphism between special semimodules. Consequently, this operation is,in a certain sense, a linear operation. This property formally explains why fuzzysystems are attractive for the applications.

Key words: fuzzy logic, MV-algebra, semimodule, semilinear space, semiring,fuzzy systems

∗ Corresponding author: Irina Perfilieva, University of Ostrava,Institute for Research and Applications of Fuzzy Modeling,30. dubna 22, 701 03 Ostrava 1, Czech Republic

Email addresses: adinola@unisa.it, lettieri@unina.it,{Irina.Perfilieva,Vilem.Novak}@osu.cz (Vilem Novak).1 The paper has been supported by the bilateral Czech-Italian project 21/MC1 and partiallyalso by the grant IAA1187301 of the GA AV CR and the project MSM 6198898701 of the MSMTCR.

Preprint submitted to Elsevier Science 29 July 2005

1 Introduction

In performing complex tasks or in modeling complex systems or complex decision situa-tions, we very often recur to rules that have the form of a conditional statements

R = IF x is A THEN x is B. (1)

The part before THEN is called antecedent (a conditional part) and the part after itis called consequent (conclusion or action part). The antecedent usually describes someinput characteristics (independent parameters) while consequent describes the output(dependent) ones. Using rules, we can formalize rule based systems that can be takenas a special case of knowledge based systems. In practice, however, we very often face aproblem that the phenomena we want to describe are known only imprecisely, or are evenhereditarily vague. Then, another formalism is necessary to cope with such a situation.Namely, fuzzy logic is a reasonable tool which proved its usefulness in many real worldapplications. In this case, A and B in the rules of the form (1) are some vague expressions(e.g. fuzzy numbers) and therefore, (1) are called fuzzy IF-THEN rules.

A crucial question is how fuzzy IF-THEN rules can be mathematically modeled. In fuzzylogic theory, two essential approaches can be recognized. The first one, originally proposedby L. A. Zadeh [24, 25] and elaborated further by many authors (cf., e.g. [6, 12, 13]),is algebraic. In this case, the antecedent of a fuzzy rule is represented by a fuzzy setA ⊂

∼X and the consequent by a fuzzy set B ⊂

∼Y where X,Y are some sets. Each rule

R expresses a relation between elements from X and Y and so, it is modeled by a fuzzyrelation R ⊂

∼X×Y that is usually constructed using the fuzzy sets A and B. If more rules

Ri, i ∈ I, are considered then the corresponding fuzzy relations Ri are joined (usuallyusing the union operation) into one fuzzy relation R which represents a model of the givensystem and which is often called a fuzzy system.

The second model of fuzzy rules (1) is based on formal logic, possibly extended even bylinguistic aspect. This approach has been elaborated in some special formal fuzzy logicsystems, for example, by G. Gerla in [10] or in BL-fuzzy logic by P. Hajek [14] withoutlinguistics and with linguistic considerations in fuzzy logic with evaluated syntax by V.Novak et al. [18], or newly in fuzzy type theory by V. Novak and S. Lehmke [17].

In this paper, however, we will consider only the algebraic model of the fuzzy IF-THENrules. To apply the fuzzy system R, let us consider a situation that a new knowledge isgiven by a fuzzy set A′ ⊂

∼X. Then a conclusion based on A′ and R is a new fuzzy set

B′ ⊂∼

Y computed using the equality

B′(y) = (A′ ◦ R)(y) =∨

x∈X

(A′(x) ⊙ R(x, y)), y ∈ Y (2)

where ⊙ is a suitable operation in the chosen algebra of membership degrees (usually at-norm, or more generally, a product operation in a residuated lattice). Equation (2) has

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been called by L. A. Zadeh (see [24, 25]) a compositional rule of inference because of itsclose relation to the modus ponens inference rule in classical logic.

From the algebraic point of view, compositional rule of inference is a composition of a fuzzyset A′ and a fuzzy relation R. Note that for finite universes X,Y , equation (2) reducesto a matrix product analogous to the classical matrix product where the summation isreplaced by the supremum, and the product by the operation ⊙.

Our aim in this paper is to clarify the algebraic motivation supporting the choice of(2). Although fuzzy logic provides a basis for the approximate description of differentdependencies, including, of course, nonlinear ones, we will see how the above formula canbe understood as a model of a fuzzy function. Furthermore, a very important conceptin classical system theory is that of linearity since linear models demonstrate the best(most transparent) behavior. Unfortunately, fuzzy systems are from this point of viewnon-linear. Therefore, a surprising and important conclusion of this paper is that fuzzysystems behave in a way similar to that of classically linear systems. We are convincedthat this fact well justifies the use of fuzzy systems and helps in explanation why theybehave so well in applications.

We will use the machinery of semimodules theory in the MV-algebraic setting. In thiscontext, linearity means a natural morphism between algebraic structures of same type.In fact, we will show that using the semimodule structure living in the MV-algebraicsetting, some dependencies prima facie nonlinear, i.e., nonlinear in the sense of classicalalgebra, become linear in the MV-semimodule framework. Let us remark that an analogousphenomenon can be found also in Maslov’s idempotent analysis, see [15].

The paper is organized as follows. In Section 2, we overview the basic concepts of MV-algebras, fuzzy sets and fuzzy systems. Section 3 is devoted to the concept of semirings,semimodules and semilinear spaces. We will show that these concepts naturally generalizethe concepts of module and linear space where the latter is crucial for characterizationof classical linearity. These concepts are further used in Section 4 for elaboration of theconcept of MV-semimodule homomorphism. Finally, the latter is applied in Section 5to show that behavior of fuzzy systems can be described as a homomorphism betweensemimodules and so that it is semilinear (close to the classical linearity).

2 Preliminaries

2.1 MV-algebras

A crucial algebraic structure which stands in the basis of the development of fuzzy logicas a formal system is that of residuated lattice (see, e.g. [10, 14, 18]). It is an algebraic

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structure of type (2, 2, 2, 2, 0, 0)

L = 〈L,∨,∧,⊙,→,0,1〉 (3)

where 〈L,∨,∧,0,1〉 is a lattice with the least element 0 and the greatest element 1. Theoperation ⊙ is a multiplication operation such that 〈L,⊙,1〉 is a commutative monoidand → is a residuation operation fulfilling the adjunction property

x ⊙ y ≤ z iff x ≤ y → z, x, y, z ∈ L. (4)

An MV-algebra (see [2, 3]) is an algebraic structure of type (2, 2, 1, 0, 0)

L = 〈L,⊕,⊙,¬,0,1〉 (5)

satisfying the following axioms:

(i) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z),(ii) x ⊕ y = y ⊕ x,

(iii) x ⊕ 0 = x,(iv) ¬¬x = x,(v) x ⊕ 1 = 1,

(vi) ¬0 = 1,(vii) x ⊙ y = ¬(¬x ⊕ ¬y),

(viii) ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x

for all x, y ∈ L.

Let L be an MV-algebra and put

x ∨ y = (x ⊙ ¬y) ⊕ y,

x ∧ y = (x ⊕ ¬y) ⊙ y, x, y ∈ L.

Then, as proved in [3], 〈L,∨,∧,0,1〉 is a bounded distributive lattice and we will refer toit as the reduct lattice of the MV-algebra L. As usual, we will define

x ≤ y iff x ∧ y = x.

We say that an MV-algebra is complete, if the reduct lattice 〈L,∨,∧,0,1〉 is complete.

Proposition 1Let L be an MV-algebra (5) and put

x → y = ¬x ⊕ y (6)

for all x, y ∈ L. ThenL = 〈L,∨,∧,⊙,→,0,1〉

is a residuated lattice.

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MV-algebras originated from an algebraic analysis of Lukasiewicz many-valued logic doneby C. C. Chang in [2]. They are non-idempotent generalizations of Boolean algebras.Indeed, Boolean algebras are just the MV-algebras obeying the additional equation x⊕x =x.

Let B(L) = {x ∈ L | x ⊕ x = x} be the set of all idempotent elements of L. Then,BL = 〈B(L),⊕,⊙,0,1〉 is a subalgebra of L, which is also a Boolean algebra. Moreover,it is the greatest Boolean subalgebra of L.

MV-algebras have been studied with respect to their relationship to other parts of math-ematics, as well as to various structures. D. Mundici in [16] proved that there exists anequivalence functor Γ between the category of MV-algebras and the category of lattice-ordered abelian groups (abelian ℓ -groups) with strong unit. For every abelian ℓ-groupwith strong unit 〈G, +, 0, u〉 the functor Γ equips the unit interval [0, u] with the followingoperations:

(i) 0 = 0,(ii) 1 = u,

(iii) x ⊕ y = u ∧ (x + y),(iv) x ⊙ y = 0 ∨ (x + y − u),(v) ¬x = u − x.

It is easy to see that the resulting structure L = 〈[0, u],⊕,⊙,¬, 0, u〉 is an MV-algebra.

Let L = [0, 1] (interval of real numbers) and define the MV-operations on [0, 1] as follows:

(i) 0 = 0,(ii) 1 = 1,

(iii) x ⊕ y = min{1, x + y},(iv) x ⊙ y = max{0, x + y − 1}(v) ¬x = 1 − x.

Then the resulting structure L L = 〈[0, 1],⊕,⊙,¬, 0, 1〉 is MV-algebra that is called LukasiewiczMV-algebra. It can be proved that the variety MV of MV-algebras coincides with the vari-ety HSP ([0, 1]), generated by the MV-algebra Γ(R, +, 0, 1), where (R, +, 0) is the totallyordered group of the real numbers (i.e. by Lukasiewicz MV-algebra). More details aboutMV-algebras can be found in the book [3].

2.2 Fuzzy sets and fuzzy systems

Let X be an arbitrary set called a universe and L be a residuated lattice (3). A fuzzy setA is identified with the function

A : X −→ L. (7)

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The function (7) is also called the membership function of the fuzzy set A (i.e. fuzzy setis identified with its membership function). We will occasionally use the symbol A ⊂

∼X

to express that A is a fuzzy set in the universe X. The set of all fuzzy sets is the set LX

of all functions (7). If L = [0, 1] or it is known from the context then we will simply writeF(X).

Operations on fuzzy sets are defined pointwise using the operations from the residuatedlattice L. This means that for arbitrary fuzzy sets A,B we put (A∪B)(x) = A(x)∨B(x)and similarly for the other operations. As a special case, we may take L to be an MV-algebra and define pointwise the operations ⊕, ⊙, ¬ on fuzzy sets. Then

〈LX ,⊕,⊙,¬,0X ,1X〉 (8)

is an MV-algebra of all fuzzy sets on X. The zero 0X and unit 1X elements of this algebraare functions identically equal to 0 and 1, respectively.

We may define also ordering of fuzzy sets by

A ≤ B iff A(x) ≤ B(x)

for all A,B ⊂∼

X and x ∈ X.

Let r ∈ L, A ∈ LX be a fuzzy set and let ◦ denote either ⊙ or ⊕. The external operationrA is defined by

(rA)(x) = r ◦ A(x)

for all x ∈ X.

If X × Y is a cartesian product of universes then the fuzzy set R ⊂∼

X × Y is a fuzzy

relation. We will write R(x, y) instead of R(〈x, y〉). A special fuzzy relation is the cartesian⊙-product of fuzzy sets A ⊂

∼X and B ⊂

∼Y defined by

(A ×⊙ B)(x, y) = A(x) ⊙ B(y).

The domain and codomain (range) of a fuzzy relation R are fuzzy sets given by themembership functions

dom(R)(x) =∨

y∈Y

R(x, y),

rng(R)(y) =∨

x∈X

R(x, y).

Let R ⊂∼

X × Y , S ⊂∼

Y × Z be two fuzzy relations. Then their composition is a fuzzy

relation R ◦ S ⊂∼

X × Z defined by

(R ◦ S)(x, z) =∨

y∈Y

(R(x, y) ⊙ S(y, z)). (9)

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Let A ⊂∼

X be a fuzzy set and R ⊂∼

X × Y a fuzzy relation. A relevant role in the theory

of fuzzy systems is played by an image of A under the fuzzy relation R, that is, a fuzzyset defined by

B(y) =∨

x∈X

(A(x) ⊙ R(x, y)). (10)

Note (10) can be also seen as a special case of a composition (9) of a fuzzy set and afuzzy relation. Clearly, (10) coincides with (2) and it is a principal formula characterizingbehavior of fuzzy systems constructed using fuzzy IF-THEN rules (cf. Examples 3 and 4in Section 5).

Another view on (10) is to understand it as a function g assigning every element A ∈ F(X)an element g(A) = B ∈ F(Y ) so that B is defined by the equation (10) (see Perfilieva[19]). Let us stress that this definition can be understood as a generalization of the classicalconcept of a function. Indeed, let g be a function from a finite set X = {x1, . . . , xn} toa finite set Y = {y1, . . . , ym} and let G = {(x, g(x)) | x ∈ X} ⊂ X × Y be the graph ofg. Obviously, (x, y) ∈ G iff y = g(x). Hence the graph of g is a (boolean) binary relationdefined by

I(x, y) = 1 iff y = g(x).

Thus, by identifying every xi ∈ X and every g(xi) ∈ Y with their characteristic functionsχxi

and χg(xi), respectively, we obtain

χg(xi)(y) =∨

x∈X

(χxi(x) ◦ I(x, y))

which is a special case of (10).

3 Semirings, semimodules and semilinear spaces

In this section we will further develop special algebraic means useful for analysis of fuzzysystems. Namely we will concentrate on the concepts of a semimodule and a semilinearspace.

3.1 Semirings and semimodules

Definition 1A semiring R = 〈R, +, ·,0R,1R〉 is an algebraic structure (cf. [4, 11]) such that:

(i) 〈R, +,0R〉 is a commutative monoid.(ii) 〈R, ·,1R〉 is a monoid.

(iii) r · (s + t) = r · s + r · t holds for all r, s, t ∈ R.(iv) 0R · r = r · 0R = 0R holds for all r ∈ R.

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A semiring is called commutative if 〈R, ·,1R〉 is a commutative monoid.

Proposition 2 ([4])Let L be an MV-algebra. Then the reducts L∨ = 〈L,∨,⊙,0,1〉 and L∧ = 〈L,∧,⊕,1,0〉are commutative semirings.

proof: This follows from the definition. 2

The following definition of a semimodule is taken from J. S. Golan [11].

Definition 2Let R = 〈R, +, ·,0R,1R〉 be a semiring. A left R-semimodule is a commutative monoidA = 〈A, +A ,0A〉 for which an external multiplication R × A −→ A denoted by ra isdefined with the following properties: all r, r′ ∈ R and a, a′ ∈ A satisfiy the equalities

(i) (r · r′)a = r(r′a),(ii) r(a +A a′) = ra +A ra′,

(iii) (r + r′)a = ra +A r′a,(iv) 1R a = a,(v) 0R a = r 0A = 0A.

The definition of right R-semimodule is analogous, where the external multiplication isdefined as a function A × R −→ A. An R-bisemimodule is both right as well as left R-semimodule, i.e. it satisfies the equality (ra)r′ = r(ar′). A nonempty subset N of a leftR-semimodule M is called a subsemimodule if N is closed under addition and externalmultiplication.

Example 1(a) Let L = 〈L,⊕,⊙,¬,0,1〉 and B = 〈B,⊕,⊙,¬,0,1〉 be MV-algebras, L∨ = 〈L,∨,⊙,0,1〉

the semiring reduct of L and 〈B,∨,0〉 the sup monoid reduct of B. Moreover, leth : L −→ B be an MV-homomorphism and define the external multiplication by

pb = h(p) ⊙ b

for all p ∈ L and b ∈ B. Then 〈B,∨,0〉 is an L∨-semimodule.(b) Let L be an MV-algebra and L∨ its semiring reduct. Put A = Ln to be the set of all

n-dimensional vectors for some n ≥ 1 and define

(a1, . . . , an) +A (b1, . . . , bn) = (a1 ∨ b1, . . . an ∨ bn),

p(a1, . . . , an) = (p ⊙ a1, . . . , p ⊙ an)

where ai, bi ∈ L, i = 1, . . . , n and p ∈ L. Then A = 〈A, +A ,0A〉 is an L∨-semimodulewhere the zero (neutral) element is the vector 0A = (0, . . . , 0).

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(c) Let X 6= ∅, L be an MV-algebra and L∨ = 〈L,∨,⊙,0,1〉 its semiring reduct. PutA = LX = {f | f : X −→ L} and for all f, g ∈ A define

f(x) +A g(x) = f(x) ∨ g(x),

p f(x) = p ⊙ f(x), x ∈ X, p ∈ L.

The zero element 0A is the function 0A : x 7→ 0. Then AX = 〈A, +A ,0A〉 is anL∨-semimodule.

(d) Let L = 〈L,∨,∧,⊕,⊙,¬,0,1〉 be an MV-algebra, L∧ = 〈L,∧,⊕,1,0〉 its semiringreduct. Put A = Ln to be a set of all n-dimensional vectors for some n ≥ 1 and

(a1, . . . , an) +A (b1, . . . , bn) = (a1 ∧ b1, . . . an ∧ bn),

p(a1, . . . , an) = (p ⊕ a1, . . . , p ⊕ an),

where ai, bi ∈ L, i = 1, . . . , n and p ∈ L. Then A = 〈A, +A ,0A〉 is an L∧-semimodulewhere the zero element is the vector 0A = (1, . . . , 1).

(e) Let X 6= ∅, L be an MV-algebra and L∧ = 〈L,∧,⊕,1,0〉 its semiring reduct. PutA = LX = {f | f : X −→ L} and for all f, g ∈ A define

f(x) +A g(x) = f(x) ∧ g(x),

p f(x) = p ⊕ f(x), x ∈ X, p ∈ L.

The zero element 0A is the function 0A : x 7→ 1. Then AX = 〈A, +A ,0A〉 is anL∧-semimodule.

(f) Let L = 〈L,⊕,⊙,¬,0,1〉 be an MV-algebra and L∧ = 〈L,∧,⊕,1,0〉 its reduct semi-ring. Furthermore, let X be a non-empty set and put A = LX = {f | f : X −→ L}.Let us now define

f(x) +A g(x) = f(x) ∨ g(x),

p f(x) = ¬p ⊙ f(x), x ∈ X, p ∈ L.

where f, g ∈ A.The zero element 0A is the function 0A : x 7→ 0. Then the monoid AX = 〈A, +A ,0A〉

is an L∧-semimodule.We will prove the last claim by verification of the properties (i)–(v) of Definition 2.

Let p, q ∈ L and f, g ∈ LX . Then(i) (pq)f = ¬(p ⊕ q) ⊙ f = ¬p ⊙ (¬q ⊙ f) = p(¬q ⊙ f) = p(qf),

(ii) p(f +A g) = ¬p ⊙ (f ∨ g) = (¬p ⊙ f) ∨ (¬p ⊙ g) = pf +A pg,(iii) (p ∧ q)f = ¬(p ∧ q) ⊙ f = (¬p ∨ ¬q) ⊙ f = (¬p ⊙ f) ∨ (¬q ⊙ f) = pf +A qf ,(iv) 1 f = ¬0 ⊙ f = f ,(v) 0f = ¬1 ⊙ f = 0A = ¬p ⊙ 0A = p0A.

Examples (c), (e) and (f) demonstrate that it is possible to introduce a semimodule struc-ture on the set LX of all fuzzy sets on X. Therefore, any of the resulting semimodules willbe called an MV-semimodule associated with L. More precisely, the semimodules (c) and(f) will be called MV-sup-semimodules and denoted by A∨

X = 〈LX ,∨,0X〉. Similarly, the

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semimodule (e) will be called MV-inf-semimodule and denoted by A∧X = 〈LX ,∧,1X〉. Note

that due to the commutativity of the operations on MV-algebra, all these semimodulesare at the same time bisemimodules.

3.2 Semilinear spaces

A central role in mathematics and its applications is played by the concept of a linearspace. This is a special case of a module which has been introduced to characterize thelinearity property in an abstract way. Since the concept of semimodule is weakening ofthe concept of module, we may introduce semilinear space as a semimodule over an MV-algebra L = 〈L,⊕,⊙,¬,0,1〉.

From the point of view applications, these spaces may be suitable for solving semilinearequations and systems of semilinear equations with fuzzy coefficients. Moreover, as shownin this paper, they play also important role in the demonstration that the fuzzy systemscan be taken as semilinear and so, they serve as a powerful tool useful in applications.The concepts semilinear space, linear dependence and basis, as well as some results of thissection are taken from [20].

To define a linear space, it is necessary to have at disposal a commutative group accom-plished with the inverse operation with respect to the main operation of addition. In thecase of a semiring (and not a group), we may consider the operation of residuum → (in-troduced by (6)) in order have the semi-inverse operation with respect to ⊙. Thereforewe may state the following definition.

Definition 3Let semiring R be a reduct of an MV-algebra L = 〈L,⊕,⊙,→,¬,0,1〉 extended by theresiduation operation → (6). Then a semimodule over L is called a semilinear space.

Elements of a semilinear space will be called vectors and elements of an MV-algebrascalars. The former will be denoted by bold letters to distinguish them from scalars.Examples of semilinear space are in Example 1.

3.3 Linear dependence and independence

The notions of linear dependence and independence can be extended also to semilinearspaces. However, some properties and criteria become easier if we confine to semilinearspaces with linearly ordered semirings. Therefore, we will formulate all basic results forboth cases in parallel.

Let L be an MV-algebra and A = 〈A, +A ,0A〉 be some of its semilinear spaces. By a linear

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combination of vectors a1, . . . , an ∈ A we mean the expression

α1 · a1 + · · · + αn · an

where α1, . . . , αn ∈ R are scalars (called also coefficients). Each linear combination uniquelydetermines a certain vector from A.

Definition 4A single vector a is linearly independent. Vectors a1, . . . , an, n ≥ 2, are linearly indepen-dent if none of them can be represented by a linear combination of the others. Otherwise,we say that vectors a1, . . . , an are linearly dependent. An infinite set of vectors is linearlyindependent if any finite subset of it is linearly independent.

Example 21. Let L be an MV-algebra and A the L∨-semilinear space of n-dimensional vectors

A = Ln. Then the following vectors are linearly independent:

e1 = (1, 0, 0, . . . , 0)

e2 = (0, 1, 0, . . . , 0) (11)

. . . . . . . . . . . . . . .

en = (0, 0, 0, . . . , 1)

2. Let L be an MV-algebra and A the L∧-semilinear space of n-dimensional vectors A =Ln. Then the following vectors are linearly independent:

f1 = (0, 1, 1, . . . , 1)

f2 = (1, 0, 1, . . . , 1)

. . . . . . . . . . . . . . .

fn = (1, 1, 1, . . . , 0)

In the rest of this section, we will confine ourselves to the semilinear space Ln over L∨

where L is an MV-algebra.

The following theorem characterizes coefficients of a linear combination of vectors a1, . . . , am ∈Ln representing a vector b (provided that the latter is expressible by at least one of suchcombinations).

Theorem 1Let A = Ln be the L∨-semilinear space of n-dimensional vectors where L is an MV-algebra.Let the vector b ∈ Ln be represented by a linear combination of vectors a1, . . . , am ∈ Ln.Then b can be represented by the linear combination of a1, . . . , am with coefficients

xi =n∧

j=1

(aij → bj), i = 1, . . . ,m. (12)

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proof: Let us form the (m × n) matrix A from components of vectors a1, . . . , am:

A =

a11 a12 · · · a1n

......

......

am1 am2 · · · amn

. (13)

By the assumption, the vector b ∈ Ln can be represented by

b = x1 ⊙ a1 ∨ · · · ∨ xm ⊙ am

with coefficients constituting the vector x = (x1, . . . , xm). Then we can rewrite the lastequality in the matrix form

b = x ◦ A

where ◦ stands for the sup−⊙ composition.

From the theory of fuzzy relation equations and their solvability (see e.g. [5, 23]) it followsthat the vector x with components

xi =n∧

j=1

(aij → bj),

i = 1, . . . ,m, is the greatest solution of the matrix equality. 2

It is worth noticing that if a vector b ∈ Ln can be represented by a linear combination ofvectors a1, . . . , am ∈ Ln then the representation is not necessarily unique.

Corollary 1Let the conditions of Theorem 1 be fulfilled and b = (b1, . . . , bn) ∈ Ln be a vectorrepresented by a linear combination of a1, . . . , am. Then for each j = 1, . . . , n

bj ≤ a1j ∨ · · · ∨ amj. (14)

proof: By the assumption, the vector b can be represented by a linear combinationof vectors a1, . . . , am ∈ Ln :

bj = x1 ⊙ a1j ∨ · · · ∨ xm ⊙ amj.

where j = 1, . . . n. On the basis of the inequality x⊙a ≤ a we easily come to the conclusionthat

bj ≤m∨

i=1

aij, j = 1, . . . n.

2

13

Corollary 2Let A = Ln be the L∨-semilinear space of n-dimensional vectors where L is an MV-algebra.Then the zero vector 0 = (0, . . . , 0) ∈ Ln is representable by the linear combination ofarbitrary vectors a1, . . . , am ∈ Ln with the respective coefficients

xi =n∧

j=1

¬aij, i = 1, . . . ,m. (15)

proof: The statement easily follows from Theorem 1, formula (12) and is substantiatedby the fact that the zero vector is always representable by the linear combination withzero coefficients and the fact that ¬a = a → 0. 2

Remark 1We refer again to the theory of systems of fuzzy relation equations (cf. [5, 23]) and canstate even more: coefficients (15) are the largest among all those which give zero linearcombination of vectors a1, . . . , am.

By the criterion suggested below, it is possible to investigate whether the given system ofvectors is linearly independent.

Theorem 2Let A = Ln be the L∨-semilinear space of n-dimensional vectors where L is an MV-algebra. Vectors a1, . . . , am ∈ Ln, m ≥ 2, are linearly independent if and only if

(∀l ∈ {1, . . . ,m})(∃i ∈ {1, . . . , n})

ali �m∨

j=1,j 6=l

aji ⊙

(n∧

k=1

ajk → alk

)

. (16)

proof: Suppose that the vectors a1, . . . , am ∈ Ln are linearly dependent. Then at leastone of them is representable by a linear combination of the others. Therefore, there existsa vector of coefficients x = (x1, . . . , xm) such that

(∃l ∈ {1, . . . ,m})(∀i ∈ {1, . . . , n})

ali =m∨

j=1,j 6=l

xj ⊙ aji

. (17)

Let us fix the value l for which (17) holds. Analogously to the proof of Theorem 1, thevector x with components

xj =n∧

i=1

(aji → ali), j = 1, . . . ,m, j 6= l,

fulfils the equality (17), i.e.

(∃l ∈ {1, . . . ,m})(∀i ∈ {1, . . . , n})

ali =m∨

j=1,j 6=l

xj ⊙ aji

. (18)

14

It is easy to verify that the following inequality holds in general:

m∨

j=1,j 6=l

xj ⊙ aji =m∨

j=1,j 6=l

((n∧

k=1

(ajk → alk)

)

⊙ aji

)

≤ ali.

Therefore, (18) is equivalent to the inequality

ali ≤m∨

j=1,j 6=l

((n∧

k=1

(ajk → alk)

)

⊙ aji

)

. (19)

Summarizing the above reasoning, we come to the conclusion that the vectors a1, . . . , am ∈Ln are linearly dependent if and only if (19) holds true. This implies that the vectorsa1, . . . , am ∈ Ln are linearly independent if and only if the negation of (19) holds true. Itremains to realize that the negation of (19) is precisely the condition (16). 2

Corollary 3Let A = [0, 1]n be the L∨-semilinear space of n-dimensional vectors and L be an MV-algebra on [0, 1]. Vectors a1, . . . , am ∈ Ln are linearly independent if and only if

(∀l ∈ {1, . . . ,m})(∃i ∈ {1, . . . , n})

ali >m∨

j=1,j 6=l

aji ⊙

(n∧

k=1

ajk → alk

)

. (20)

Remark 2(1) Theorems 1, 2 and their Corollaries 1 and 3 are valid also in the case of a L∨-

semilinear space of L-valued functions A = LX where L is an MV-algebra.(2) Let us remind that in the case of a linear space, we distinguish linearly dependent and

linearly independent vectors by analyzing coefficients of their linear combinationsleading to zero vectors. As we will see below, this characterization is unhelpful inthe case of semilinear spaces where we care about the expressibility property. Toexemplify this claim, let us take the reduct L∨ of Lukasiewicz algebra on [0, 1] andfor a ∈ (0, 1) consider the following set of linearly independent vectors from [0, 1]n :

a1 = (a, 0, 0, . . . , 0)

a2 = (0, a, 0, . . . , 0) (21)

. . . . . . . . . . . .

an = (0, 0, 0, . . . , a).

It is easy to see that the linear combination

¬a ⊙ a1 ∨ · · · ∨ ¬a ⊙ an = 0

with non-zero coefficients ¬a gives the zero vector.On the other hand, the vectors a1, . . . , an, a1 + a2 are linearly dependent, and

again, their linear combination with all coefficients equal to ¬a gives the zero vector.Therefore, independently on the fact whether the vectors are linearly dependent ornot (in the sense of our definition), their linear combination with non-zero coefficients

15

may be equal to the zero vector. Note that this may happen if at least one of thecoefficients given by (15) is non-zero.

3.4 Basis in a semilinear space

In this subsection, we will show that the concept of basis known from the theory of linearspaces can be introduced in semilinear ones. However, we must be careful since not allproperties are transferred in a straightforward way.

Let us fix an MV-algebra L and consider the L∨-semilinear space of n-dimensional vectorsA = Ln.

Definition 5A linear independent set of generators of a semi-linear space A is called a basis of A.

An example of a basis in Ln is given by vectors e1, . . . , en in Example 2 (cf. (11)). Itimmediately follows from the last definition that an arbitrary element from a semi-linearspace Ln can be represented by a linear combination of elements of its basis.

We investigate conditions assuming or guaranteeing that vectors a1, . . . , am ∈ Ln form abasis. Let us denote

ai = (ai1, . . . , ain), i = 1, . . . ,m.

Suppose that a vector b = (b1, . . . , bn) ∈ Ln is represented by a linear combination ofa1, . . . , am. Then there exist coefficients x1, . . . , xm ∈ L such that

b = x1 ⊙ a1 ∨ · · · ∨ xm ⊙ am; (22)

coordinatewise, (22) is equivalent to

bj = x1 ⊙ a1j ∨ · · · ∨ xm ⊙ amj, j = 1, . . . n.

If we take b = (1, . . . , 1) then from the last equality, we easily obtain the followingnecessary condition the the basic vectors :

a1j ∨ · · · ∨ amj = 1, j = 1, . . . n. (23)

It turns out that when L is linearly ordered, a criterion that vectors a1, . . . , am constitutea basis of Ln has a very simple form.

Theorem 3Let a support set L of an MV-algebra be the interval [0, 1]. Then the unique basis in thesemilinear space of n-dimensional vectors [0, 1]n over the semiring L∨ is given by (11).

16

proof: Let a1, . . . , am ∈ Ln be basic vectors in [0, 1]n and i, 1 ≤ i ≤ n, be a fixednatural number. The vector ei = (0, . . . , 1

︸︷︷︸

i

, . . . , 0) can be represented by a linear com-

bination of basic vectors. Moreover, the components

xk =n∧

j=1

(akj → eij), k = 1, . . . ,m, (24)

constitute the vector x which is a solution of the equation

ei = x ◦ A (25)

where x is an unknown vector and the matrix A is given by (13).

We will prove that for the chosen i there exists l, 1 ≤ l ≤ m, such that ei = al. Substitutingthe coefficients given by (24) into (25), we obtain for the i-th component:

1 = x1 ⊙ a1i ∨ · · · ∨ xm ⊙ ami =

= (¬a11 ∧ · · · ∧ 1︸︷︷︸

i

∧ · · · ∧ ¬a1n) ⊙ a1i ∨ · · ·

· · · ∨ (¬am1 ∧ · · · ∧ 1︸︷︷︸

i

∧ · · · ∧ ¬amn) ⊙ ami.

Therefore, there exists l, 1 ≤ l ≤ m, such that

1 = (¬al1 ∧ · · · ∧ 1︸︷︷︸

i

∧ · · · ∧ ¬aln) ⊙ ali

which implies

al1 = 0, . . . ali = 1, . . . , aln = 0.

Thus,

ei = al

which proves the theorem. 2

Let us investigate the problem whether a system of linearly independent vectors can beextended to a basis. The following simple example shows that this is not always the case:Let us take the reduct L∨ of Lukasiewicz algebra on [0, 1] and for a ∈ (0, 1) consider theset of linearly independent vectors (21) from Ln. By Theorem 3, the only basis for thisspace is given by (11). However, we cannot add any of the vectors from (11) to the set(21) without destroying their linear independency. On the basis of this example, we mayprove the following lemma.

Lemma 1Let A = Ln be the L∨-semilinear space of n-dimensional vectors where L is an MV-algebra. Then the system of linearly independent vectors (21) cannot be extended to abasis of Ln.

17

proof: Suppose that contrary to the conclusion of the lemma, there are vectorsb1, . . . ,bk, k ≥ 1, which together with vectors a1, . . . , an from (21) form a basis of Ln.Then either the vector e1 = (1, 0, . . . , 0) is among the vectors b1, . . . ,bk, or it can berepresented by a linear combination of vectors a1, . . . , an,b1, . . . ,bk. In the first case, thevector a1 is represented by a1 = a ⊙ e1. In the second case, e1 can be represented by thelinear combination of vectors a1, . . . , an,b1, . . . ,bk which does not include the vector a1

so that a1 can be represented by the same linear combination multiplied by a. In bothcases, the obtained extension a1, . . . , an,b1, . . . ,bk cannot be a basis. This contradictionproves the lemma. 2

4 MV-semimodule homomorphisms

In this section, we will study the concept of semimodule homomorphism that plays acrucial role when characterizing semilinearity, analogously as is the role of the conceptof module homomorphism when characterizing classical linearity. Besides the general de-finition of semimodule homomorphism, we will define also the concepts of sup- and inf-semicontinuous homomorphisms that occur when dealing with fuzzy systems. We will alsoshow that homomorphisms give rise to a new semimodule.

Definition 6Let R = 〈R, +, ·,0R,1R〉 be a semiring and 〈M, +M ,0M〉 and 〈N, +N ,0N〉 R-semimodules.A mapping H : M −→ N is an R-homomorphism if the followings holds:

(i) H(m +M m′) = H(m) +N H(m′), for all m,m′ ∈ M .(ii) H(rm) = rH(m), for all m ∈ M and r ∈ R.

The kernel of H is the set

Ker(H) = {m ∈ M | H(m) = 0N}.

Note that Ker(H) 6= ∅. Indeed, choose r = 0R. Then by (ii) of Definition 6 and (v) ofDefinition 2, we obtain H(0M) = 0N .

The set of all R-homomorphisms M −→ N is denoted by Hom(M,N). Note that Hom(M,N) 6=∅ because the function H0 : m 7→ 0N , m ∈ M , is an element of Hom(M,N).

Let us now define the addition +H on Hom(M,N) by

(H +H K)(m) = H(m) +N K(m) (26)

for every H,K ∈ Hom(M,N) and m ∈ M . Furthermore, let us define the external multi-plication by

(r ◦H H)(m) = rH(m) (27)

for every H ∈ Hom(M,N), m ∈ M and r ∈ R.

18

Lemma 2If R is a commutative semiring and r ∈ R then H +H K and r ◦H H are elements ofHom(M,N).

proof: It is immediate that

(H +H K)(m + m′) = (H +H K)(m) + (H +H K)(m′).

Moreover,

(H +H K)(rm) = H(rm) + K(rm) = rH(m) + rK(m) = r((H +H K)(m)).

Thus H +H K ∈ Hom(M,N).

Consider now r ◦H H. Then

(r ◦H H)(m + m′) = rH(m + m′) = rH(m) + rH(m′) = (r ◦H H)(m) + (r ◦H H)(m′).

Moreover,

(r ◦H H)(sm) = rH(sm) = r(sH(m)) = (s · r)H(m) = s(rH(m)) =

= s((r ◦H H)(m))

and so, r ◦H H ∈ Hom(M,N). 2

Theorem 4Let R be a commutative semiring and the addition “+H” and the external multiplication“◦H” be defined by (26) and (27), respectively. Then

〈Hom(M,N), +H , H0〉

is an R-semimodule.

proof: Clearly, 〈Hom(M,N), +H , H0〉 is a commutative monoid. Let us now verify theproperties (i)–(v) of Definition 2.

Let r, s ∈ R and H,K ∈ Hom(M,N). Then for every m ∈ M , we have the following:

(i)

((r · s) ◦H H)(m) = (s · r)H(m) = H(s(rm)) = sH(rm)

= (s ◦H H)(rm) = r ◦H (s ◦H H)(m).

Hence, ((r · s) ◦H H) = r ◦H (s ◦H H).

19

(ii)

(r ◦H (H +H K))(m) = r(H +H K)(m) = r(H(m) + K(m))

= rH(m) + rK(m) = (r ◦H H)(m) + (r ◦H K)(m).

Hence, r ◦H (H +H K) = (r ◦H H) +H (r ◦H K).(iii)

((r + s) ◦H H)(m) = (r + s)H(m) = rH(m) + sH(m)

= (r ◦H H)(m) + (s ◦H H)(m)

Hence, (r + s) ◦H H) = r ◦H H +H s ◦H H.(iv) (1R ◦H H)(m) = H(1R m) = H(m). Hence, 1R ◦H H = H.(v) (r◦H H0)(m) = rH0(m) = r 0N = 0N . Hence, r◦H H0 = H0. Finally, (0R◦H H)(m) =

0R H(m) = 0N . Hence, 0R ◦H H = H0.

2

Corollary 4Let L be an MV-algebra and 〈LX , +,0〉, 〈LY , +,0〉 be MV-semimodules associated withL. Then 〈Hom(LX , LY ), +H , H0〉 is a semimodule.

proof: This follows from Lemma 2, Theorem 4 and Examples 1 (c), (e), (f). 2

We will now introduce a special kind of semimodule homomorphism that preserves eithersupremum or infimum.

Definition 7Let H ∈ Hom(LX , LY ). If

H(∨

i∈I

fi) =∨

i∈I

H(fi)

holds for every family {fi | i ∈ I} of elements in LX such that∨

i∈I fi exists then H iscalled sup-semicontinuous. If

H(∧

i∈I

fi) =∧

i∈I

H(fi)

holds for every family {fi | i ∈ I} such that∧

i∈I fi exists then H is called inf-semicontinuous.

If a homomorphism is both sup- as well as inf-semicontinuous then we will call it contin-uous.

The set of all the sup-semicontinuous homomorphism from LX to LY is denoted byHomsc(L

X , LY ) and inf-semicontinuous ones by Homic(LX , LY ). It holds that both Homsc(L

X , LY )as well as Homic(L

X , LY ) are nonempty because they both contain the homomorphismH0 (note that in the second case, the H0 denotes a homomorphism assigning to each fuzzyset f ∈ LX the function 0LY identically equal to 1).

20

Proposition 3Let L be an MV-algebra, A∨

X = 〈LX ,∨,0X〉 and A∨Y = 〈LY ,∨,0Y 〉 the associated MV-

sup-semimodules. Then 〈Homsc(LX , LY ),∨H , H0〉 is a subsemimodule of

〈Hom(LX , LY ),∨H , H0〉.

proof: Let H,K ∈ Homsc(LX , LY ) and {fi | i ∈ I} be a set of elements in LX such

that∨

i∈I fi exists and r ∈ L. Then, using sup-continuity, we can prove:

(H ∨HK)(∨

i∈I

fi) = H(∨

i∈I

fi) ∨ K(∨

i∈I

fi) =∨

i∈I

H(fi) ∨∨

i∈I

K(fi) =

=∨

i∈I

(H(fi) ∨ K(fi)) =∨

i∈I

(H ∨HK)(fi).

This implies that H ∨HK ∈ Homsc(LX , LY ).

Let r ◦H H ∈ Homsc(LX , LY ). Then

(r ◦H H)(∨

i∈I

fi) = q ⊙ H(∨

i∈I

fi) = q ⊙∨

i∈I

H(fi) =

=∨

i∈I

(q ⊙ H(fi)) =∨

i∈I

(r ◦H H)(fi)

where q = p or q = ¬p dependently on the semimodule of Example 1(c) or (f). Thisimplies that r ◦H H ∈ Homsc(L

X , LY ). 2

Proposition 4Let L be an MV-algebra, A∧

X = 〈LX ,∧,1X〉 and A∧Y = 〈LY ,∧,1Y 〉 the associated MV-

inf-semimodules. Then 〈Homic(LX , LY ),∧H , H0〉 is a subsemimodule of

〈Hom(LX , LY ),∧H , H0〉.

proof: We will proceed analogously to the proof of Proposition 3. Let H,K ∈ Homic(LX , LY )

and {fi | i ∈ I} be a set of elements in LX such that∧

i∈I fi exists and r ∈ L. Then, usinginf-continuity, we can prove:

(H ∧HK)(∧

i∈I

fi) = H(∧

i∈I

fi) ∧ K(∧

i∈I

fi) =∧

i∈I

H(fi) ∧∧

i∈I

K(fi) =

=∧

i∈I

(H(fi) ∧ K(fi)) =∧

i∈I

(H ∧H K)(fi).

This implies that H ∧HK ∈ Homsc(LX , LY ).

21

Let r ◦H H ∈ Homic(LX , LY ). Then

(r ◦H H)(∧

i∈I

fi) = r ⊕ H(∧

i∈I

fi) = r ⊕∧

i∈I

H(fi) =

=∧

i∈I

(r ⊕ H(fi)) =∧

i∈I

(r ◦H H)(fi).

This implies that r ◦H H ∈ Homic(LX , LY ). 2

5 Compositional rule of inference as an MV-semimodule homomorphism

5.1 General theory

In this section we will show that the composition of fuzzy relations can be seen as asemimodule homomorphism. Since this operation is the core of semantics of the compo-sitional rule of inference that is the main tool for manipulation with fuzzy systems, wemay conclude from our results that the latter is a semilinear operator whose behavior, ascan be seen from our results in previous sections, is close to linear. Recall that by a fuzzysystem, we understand a special fuzzy relation. We have proved that fuzzy systems form asemimodule that is isomorphic to the semimodule of semicontinuous semilinear operators.Consequently, fuzzy systems can be treated as (almost) linear that is a good news for theapplications (and their justification). This opens a space for application, at least some ofimportant results from the functional analysis in the theory of fuzzy systems.

Let L = 〈L,⊕,⊙,¬,0,1〉 be a complete MV-algebra and X,Y two nonempty sets. Notethat in the case, the MV-algebras (8) with the respective supports LX and LY are completeas well.

Let us now consider the MV-sup-semimodules A∨X = 〈LX ,∨,0X〉 and A∨

Y = 〈LY ,∨,0Y 〉.We will now show how the theory presented above allows us to interpret the compositionoperation as a homomorphism between A∨

X and A∨Y .

Let A ⊂∼

X and R ⊂∼

X × Y (i.e. A ∈ LX and R ∈ LX×Y ). Set

HR(A)(y) =∨

x∈X

(A(x) ⊙ R(x, y)) = B(y), y ∈ Y, (28)

i.e. B ∈ LY . It is easy to verify that HR ∈ Hom(LX , LY ).

Lemma 3Let H ∈ Hom(LX , LY ). Then r ⊙ H(A) = H(r ⊙ A) holds for every r ∈ L and A ∈ LX .

proof: By (ii) of Definition 6, r ⊙ H(A) = qH(A) = H(qA) = H(r ⊙ A) where q = r

22

or q = ¬r dependently on the semimodule of Example 1(c) or (f). 2

Theorem 5The mapping HR given in (28) is a homomorphism from the semimodule A∨

X = 〈LX ,∨,0X〉to the semimodule A∨

Y = 〈LY ,∨,0Y 〉.

proof: Let A,B ∈ LX and r ∈ L. We will verify the properties (i) and (ii) of Defini-tion 6. For all y ∈ Y we prove:

(i)

HR(A ∨ B)(y) =∨

x∈X

((A ∨ B)(x) ⊙ R(x, y)) =

=∨

x∈X

((A(x) ⊙ R(x, y)) ∨ (B(x) ⊙ R(x, y))) =

=∨

x∈X

(A(x) ⊙ R(x, y)) ∨∨

x∈X

(B(x) ⊙ R(x, y)) = H(A)(y) ∨ HR(B)(y).

Thus, HR(A ∨ B) = HR(A) ∨ HR(B).

(ii)

HR(rA)(y) = HR(q ⊙ A)(y) =∨

x∈X

((q ⊙ A(x)) ⊙ R(x, y)) =

= q ⊙∨

x∈X

(A(x) ⊙ R(x, y)) = q ⊙ HR(A)(y) = rHR(A)(y).

where q = r or q = ¬r dependently on the semimodule of Example 1(c) or (f). ThusHR(rA) = rHR(A).

2

We have shown that a fuzzy composition bears a homomorphism between semimodules sothat indeed, the compositional rule of inference can be taken as a (semi)linear mapping.

Now we will study the kernel Ker(HR). Recall that HR(0X) = 0Y . Thus 0X ∈ Ker(HR),so that Ker(HR) 6= ∅. The following proposition tells us explicitly what is the content ofthis kernel.

Proposition 5Put B(x) =

∧

y∈Y ¬R(x, y) for all x ∈ X. Then

Ker(HR) = {A | A ∈ LX ,0X ≤ A ≤ B}.

proof: Observe that the fuzzy relation equation ξ ◦ R = 0X (with respect to theunknown ξ) is solvable. Therefore, B defined in the assumption is the greatest solution.

23

Let A ≤ B. Then we get HR(A) ≤ HR(B) = 0X , which implies HR(A) = 0X . 2

An interesting question is the following: given a homomorphism F : LX −→ LY , whatconditions on F assure us that there exists a fuzzy relation R(x, y) such that F = HR.If such a relation exists, we say that F is representable by a fuzzy relation R or, shortly,that F is representable. We will answer this question in the following theorem.

Let a ∈ X. By fa we denote the singleton fuzzy set on a, i.e. it is an element of LX definedby

fa(x) =

1 if x = a,

0 if x 6= a,

for all x ∈ X. Let F ∈ Hom(LX , LY ). We put RF (x, y) = F (fx)(y) for all (x, y) ∈ X ×Y .

Theorem 6If F ∈ Hom(LX , LY ) is sup-semicontinuous then F is representable and F = HRF

.

proof: The function (HRF)(A) is defined by

(HRF)(A)(y) =

∨

x∈X

(A(x) ⊙ F (Ax)(y)), y ∈ Y.

Fix x0 ∈ X. Since F is a homomorphism, by Lemma 3, A(x0)⊙F (fx0) = F (A(x0)⊙ fx0

).From this, we conclude that

∨

x∈X

(A(x) ⊙ F (fx)) =∨

x∈X

(F (A(x) ⊙ fx)) = F

(∨

x∈X

(A(x) ⊙ fx)

)

= F (A)

by the semicontinuity assumption. Consequently, HRF(A)(y) = F (A)(y) holds for all

y ∈ Y and so, F is representable. 2

Lemma 4The function H : R 7→ HR where R ∈ LX×Y and HR ∈ Hom(LX , LY ) is an injec-tive homomorphism from the R-semimodule A∨

X×Y = 〈LX×Y ,∨,0〉 to the R-semimodule〈Hom(LX , LY ),∨H , H0〉.

proof: First, we will verify the properties (i) and (ii) of Definition 6. Let R,S ∈ LX×Y

and A ∈ LX .

(i) Since H(R ∨ S) = HR∨S, we get for y ∈ Y

H(R ∨ S)(A)(y) =∨

x∈X

(A(x) ⊙ (R(x, y) ∨ S(x, y))) =

=∨

x∈X

((A(x) ⊙ R(x, y)) ∨ (A(x) ⊙ S(x, y)) =

=∨

x∈X

(A(x) ⊙ R(x, y)) ∨∨

x∈X

(A(x) ⊙ S(x, y)) = HR(A)(y) ∨ HS(A)(y).

24

(ii) Let r ∈ L. Since H(rR) = HrR, we get for y ∈ Y

H(rR)(A)(y) =∨

x∈X

(A(x) ⊙ (q ⊙ R(x, y))) =

q ⊙∨

x∈X

(A(x) ⊙ R(x, y)) = q ⊙ HR(A)(y) = r HR(A)(y)

where q = r or q = ¬r dependently on the semimodule of Example 1(c) or (f).

Now will will show that H is an injective map. Assume R 6= S. Hence, there is a couple(x0, y0) ∈ X × Y such that R(x0, y0) 6= S(x0, y0). Then we have

HR(fx0)(y0) = R(x0, y0) 6= S(x0, y0) = HS(fx0

)(y0).

Thus, HS(fx0) 6= HR(fx0

), which implies HR 6= HS. 2

Theorem 7The semimodules A∨

X×Y = 〈LX×Y ,∨,0X×Y 〉 and 〈Homsc(LX , LY ),∨H , H0〉 are isomor-

phic.

proof: In Lemma 4, we proved that 〈H(LX×Y ),∨,0X×Y 〉 is a subsemimodule of〈Hom(LX , LY ),∨, H0〉 (up to isomorphism). Moreover, by Theorem 6, it is the subsemi-module of all semicontinuous homomorphisms from LX to LY and each semicontinuoushomomorphism is representable by some fuzzy relation. The theorem then follows fromLemma 4. 2

Note that by Theorem 6, H(LX×Y ) contains the subsemimodule of all semicontinuoushomomorphisms from LX to LY . Then we obtain the following theorem.

Theorem 8Let X,Y be finite sets, R ∈ LX×Y and HR ∈ Hom(LX , LY ). Then the function H :R 7→ HR is an isomorphism from the R-semimodule A∨

X×Y = 〈LX×Y ,∨,0X×Y 〉 to theR-semimodule 〈Hom(LX , LY ),∨, H0〉.

proof: This follows from Theorem 6 and Lemma 4. 2

5.2 Examples

Example 3Let X,Y ⊂ R be compact sets, h : X −→ Y be a continuous function and ε > 0 bea real number. Using the methods described in [18], Chapter 5, we may find finite sets{Aj | Aj ⊂∼

X, j ∈ J} and {Bj | Bj ⊂∼X, j ∈ J} such that each fuzzy relation

Rj = Aj ×⊙ Bj,

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j ∈ J , represents a fuzzy rule and the following properties are fulfilled:

(a)⋃

j∈J Supp(Aj) = X,⋃

j∈J Supp(Bj) = rng h.(b) If Rj(x, y) > 0 then |y − h(x)| < ε for all x ∈ X and y ∈ Y .

Since Rj ∈ LX×Y is an element of the semimodule A∨X×Y the fuzzy relation R =

⋃

j∈J Rj

is its element as well. Then by Theorems 7 and 5, R determines a (semicontinuous)semilinear operator HR.

Let us consider a fuzzy function F : LX −→ LY such that F (A) = HR(A) for all A ⊂∼

X.

Since HR is a semilinear operator, the fuzzy function F is semilinear as well.

Take x0 ∈ X and let A′ ⊂∼

X be a fuzzy set such that A′(x0) = 1 (i.e. A′ is normal) and

there is j ∈ J such that Supp(A′) ⊆ Supp(Aj) (we can take, for example, the singletonA′ = fx0

). Then the construction of R described in [18] assures that F (A′)(h(x)) > 0and so, |y − h(x)| < ε for every y ∈ Supp(F (A′)). Hence, each y ∈ Supp(F (A′)) is areasonable approximation of h(x). We may conclude that a generally nonlinear functionh is approximated by some semilinear fuzzy function with arbitrary precision ε > 0. Notethat for A′ = fx0

, we have, in fact, obtained the classical approximation of the function h

by means of the technique of fuzzy approximation theory (cf. also Perfilieva [21]).

Example 4Let us consider the inverted pendulum control problem. This system can be approximatelymodeled by a non-linear differential equation

y − 10 sin y = 0

where y depends on time t. The system can be controlled using a non-linear functionu(t) = h(e(t), e(t)) where e(t) = −y(t) is a pendulum deviation from the vertical position.We will replace it by means of a fuzzy PD-controller.

In our terms this means that we must construct a fuzzy system R ⊂∼

X×Y ×Z where X is

the range of deviation e(t), Y is the range of its velocity e(t) and Z is the range of controlaction u(t) (this is a momentum sent to the inverted pendulum). For the control, we haveused the software system LFLC (see [7]). The universes are set as follows: X = [−1.5, 1.5],Y = [−10, 10], and Z = [−90, 90]. The fuzzy system (i.e. fuzzy PD-controller) consists ofthe following fuzzy IF-THEN rules:

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number e(t) e(t) u(t)

1 Ze Ze Ze

2 RoZe - Sm +VeSm

3 RoZe + Sm -VeSm

4 -NoZe -NoZe +Me

5 +NoZe +NoZe -Me

6 -NoSm +NoZe +Ro17

7 +NoSm -NoZe -Ro17

Each rule is a fuzzy relation of the form Rj = Aj ×⊙ Bj ×⊙ Cj where ⊙ is in this caseeither the operation ∧ or the MV-product, and Aj, Bj, Cj, j = 1, . . . , 7 are fuzzy setsoccurring in the corresponding rule. Their shapes are depicted on Figure 1. They are, infact, meanings of the corresponding linguistic expressions zero (Ze), roughly zero (RoZe),not zero (NoZe), small (Sm), very small (VeSm), not small (NoSm), roughly 20% (Ro20)(the latter is a fuzzy number).

Ze+NoZe-NoZe

+Ro20-Ro20

RoZe+VeSm-VeSm

+Sm-Sm

0

+NoSm-NoSm

Fig. 1. Shapes of fuzzy sets for the fuzzy control of inverted pendulum. They are symmetricallyplaced around 0.

The fuzzy control is given by a semilinear fuzzy function F defined by

F (fe(t), fe(t)) = HR(fe(t), fe(t))

where fe(t) ∈ X, fe(t) ∈ Y are singletons and for each t, F (fe(t), fe(t)) ⊂∼

Z is a fuzzy

set that approximates value of the control function u(t) = h(e(t), e(t)), i.e. each elementz ∈ Z belonging to it in non-zero membership degree approximates u(t). Concretely, wetake z = z0 where z0 is obtained by a defuzzification of F (fe(t), fe(t)) (usually Center OfGravity).

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Samples

0 10 20 30 40 50 60 70 80 90

Devia

tion

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

Fig. 2. Fuzzy control of inverted pendulum.

Figure 2 demonstrates that this simple fuzzy system indeed controls the inverted pen-dulum. Of course, the control can be much improved when adding more rules. Our goal,however, was to demonstrate the method and not the control itself. Again, we have ap-proximated nonlinear function by a semilinear fuzzy function.

6 Conclusion

In this contribution, we have developed an algebraic theory suitable for the analysis offuzzy systems. Namely, we have used the notions of semiring and semimodule and intro-duced the notion of semilinear space. We gave numerous examples of these notions, definedthe notions of linear dependence and independence and suggested conditions (necessaryor sufficient) when vectors form a basis of a semilinear space. Then, we have consideredthe composition operation of fuzzy relations which plays an essential role in the analysisof fuzzy systems because using it, the compositional rule of inference is defined. We haveshown that it can be interpreted as a homomorphism between special semimodules. Be-cause the classical concepts of module and linear space are abstract characterization ofthe linearity property, it follows from our results that the compositional rule of inferencecan be taken as a linear operation as well. This fact makes it even more attractive forthe applications because linearity is a desirable property that makes models simpler andmore transparent.

The basic algebraic structure considered in this paper is MV-algebra. However, it is clearfrom all the constructions that we can take even a more general algebra such as BL- orIMTL-algebra.

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