Semilinear Spaces - Basic Structures for Fuzzy Systems∗ Irina Perfilieva University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic e-mail: Irina.Perfi[email protected] Abstract A semiring (see [4]) has two associative op- erations that obey the law of distributivity. The notion of a linear space is gener- Semirings have been used to model formal alized to the case where the underly- languages and finite automata (see [3]). The ing algebra is a commutative monoid special semiring over real numbers is an un- and the set of scalars is a semiring derlying structure of the idempotent analysis reduct of a BL-algebra or an MV- [6]. In [1], semirings are introduced as restric- algebra. The notions of linear de- tions ofLukasiewicz algebra and more gener- pendence and independence are also ally, as restrictions of MV-algebra. Actually, introduced and necessary and suffi- two dual semiring restrictions have been in- cient conditions when vectors form a troduced in both algebras. basis of a semi-linear space are given. This contribution introduces a semi-linear A linear mapping between two semi- space over a semiring. A number of exam- linear spaces is defined and consid- ples of semirings which are restrictions of BL- ered in the case when it is repre- algebras, or dual BL-algebras are considered sented by a fuzzy relation. An ex- and the corresponding semi-linear spaces are ample of a fuzzy system which can introduced over them. A generalized notion of be characterized by fuzzy IF-THEN linear independence is introduced and some rules is modeled using the respective necessary and sufficient conditions of linear linear mapping. independency are found. A linear mapping Keywords: semi-linear space, between two semi-linear spaces is defined and semiring, BL-algebra, MV-algebra considered in the case when it is represented by a fuzzy relation. It is shown that each model of a fuzzy system which uses the com- Introduction positional rule of inference realizes a linear mapping between two semi-linear spaces. The notion of a linear space is one of the cen- tral notions in mathematics and its applica- Preliminaries tions. Therefore, generalization of the linear space to a weaker structure, such as a commu- In the sequel, we will use the BL, dual BL and tative monoid over a semiring or over a BL- MV algebras as basic structures for “arith- algebra is of interest. From the application metic” operations over elements of lattice point of view, these spaces may be suitable for ordered sets. The definitions given below solving semi-linear equations and systems of summarize definitions originally introduced in semi-linear equations with fuzzy coefficients. [5, 7]. The∗ research has been supported by project MSM 6198898701 of the MSMTˇ CR.ˇ Definition 1 (ii) L, ⊕, 0, ∧, ∨ is a commutative lattice A BL-algebra is an algebra ordered monoid, L = L, ∧, ∨, ∗, →, 0, 1 (iii) for all x, y, z ∈ L with four binary operations and two constants z ≥ (y x) iff x ⊕ z ≥ y, such that x ⊕ (y x)=x ∨ y, (i) L, ∧, ∨, 0, 1 is a lattice with 0 and 1 (x y) ∧ (y x)=0. as the least and greatest elements w.r.t. the lattice ordering, Let us give an example of the dual to G¨odel (ii) L, ∗, 1, ∨, ∧ is a commutative lattice algebra on [0, 1]: ordered monoid, ∗ → Example 1 (iii) and form an adjoint pair, i.e. d LG = [0, 1], ∨, ∧, ⊕G, G, 1, 0 and z ≤ (x → y) iff x ∗ z ≤ y y, x < y, ∈ x ⊕G y = x ∨ y, x G y = for all x, y, z L, 0,x≥ y. (iv) x ∗ (x → y)=x ∧ y, Finally, we will introduce MV-algebras as spe- → ∨ → (x y) (y x)=1 cial BL-algebras (see e.g. [5]). ∈ for all x, y L. Definition 3 A BL-algebra L = L, ∨, ∧, ∗, →, 0, 1 where The following operations of negation and the double negation law biresiduation can be additionally defined: x = ¬(¬x), ¬x = x → 0 x ↔ y =(x → y) ∧ (y → x). is valid for all x ∈ L, is called an MV-algebra. The well known examples of BL-algebra are A typical example of an MV-algebra is G¨odel,Lukasiewicz and product algebras on Lukasiewicz algebra on [0, 1]. [0, 1]. Moreover, it is known that in the case L =[0, 1], the operation ∗ is a continuous t- Main Definitions norm. The notion of a dual BL-algebra has been in- In this section we will give definitions of semi- troduced in [7] as a special case of a dually rings and semi-linear spaces and consider nu- residuated lattice ordered monoids. merous examples of both structures. Definition 2 Definition 4 A dual BL-algebra is an algebra A semiring R = R, +, ·, 0, 1 is an algebra where Ld = L, ∨, ∧, ⊕, , 1, 0 (SR1) R, +, 0 is a commutative monoid, with four binary operations and two constants such that (SR2) R, ·, 1 is a monoid, (i) L, ∨, ∧, 1, 0 is a lattice with 1 and 0 (SR3) for all a, b, c ∈ R as the greatest and least elements w.r.t. the lattice ordering, a·(b+c)=a·b+a·c, (b+c)·a = b·a+c·a. A semiring is commutative if (R, ·, 1) is a com- Let us remark that the semirings in Example 2 mutative monoid. are the idempotent semirings ([6]), because A semiring is a semiring with annihilator if their operation of “addition” is idempotent. the neutral element of R, +, 0 is an annihi- lator, i.e. Definition 5 0 · a = a · 0=0 Let A = ∅ be a set of elements and R = R, +, ·, 0, 1 a semiring. We say that A is ∈ for all a R. a (left) semimodule over R if two operations are defined: A typical example of a commutative semi- ring is a set N of non-negative integers with (a) addition + such that for each two ele- addition and multiplication. Below, we will ments a, b ∈ A there is a uniquely deter- give examples of other semirings which can mined element a+b ∈ A called their sum, be taken as reducts of BL-algebras, dual BL- algebras and MV-algebras. (b) multiplication · by an element from R ∈ ∈ Example 2 such that for any a A and p R there is a uniquely determined element p · a called 1. Let L = L, ∨, ∧, ∗, →, 0, 1 be a BL- their product. algebra. Then its ∨-reduct L∨ = L, ∨, ∗, 0, 1 These operations fulfil the following proper- ties for all a, b, c ∈ A and p, q ∈ R: is a commutative semiring. (SL1) a + b = b + a, 2. Let again L be a BL-algebra. The following ∧ L algebra ( -reduct of ) is a commutative (SL2) a +(b + c)=(a + b)+c, semiring: (SL3) there exists the (neutral) element 0 ∈ L∧ = L, ∧, ∗, 1, 1. A such that a + 0 = a, 3. Let Ld = L, ∨, ∧, ⊕, , 1, 0 be a dual BL- (SL4) p · (a + b)=p · a + p · b, algebra. Then its ∧-reduct (SL5) (p + q) · a = p · a + q · a, Ld = L, ∧, ⊕, 1, 0 ∧ (SL6) p · (q · a)=(p · q) · a, is a commutative semiring. (SL7) 1 · a = a. 4. Let again Ld be a dual BL-algebra. The following algebra (∨-reduct of Ld)isacom- A right semimodule over R maybedefined mutative semiring: analogously. d A nonempty subset B of a left (right) semi- L∧ = L, ∧, ∗, 1, 1. module A over R is called a subsemimodule if B is closed under addition and multiplication. 5. Let L = L, ∨, ∧, ⊗, ⊕, ¬, 0, 1 be an MV- algebra. Then its ∨-reduct Definition 6 Let semiring R be a reduct of a BL- L ∨ ⊗ ∨ = L, , , 0, 1 algebra (dual BL-algebra, MV-algebra) L = L, ∨, ∧, ∗, →, 0, 1. Then a semimodule over and its ∧-reduct L is called a semilinear space. L∧ = L, ∧, ⊕, 1, 0. The elements of a semilinear space are called are commutative semirings. vectors and elements of L scalars. Example 3 (a1 ∧ b1,...an ∧ bn), 1. Let L = L, ∨, ∧, ∗, →, 0, 1 be a BL- p · (a1,...,an)=(p ∗ a1,...,p∗ an) algebra (MV-algebra) on L, L∨ = L, ∨, ∗, 0, 1 its semiring reduct. Let us where p ∈ L. The neutral element in A is consider the set of all n-dimensional vec- n the vector (1,...,1). It is easy to see that tors A = L , n ≥ 1, and define n L is a semi-linear space over L∧. (a1,...,an)+(b1,...,bn)= Linear Dependence and (a1 ∨ b1,...an ∨ bn), Independence p · (a1,...,an)=(p ∗ a1,...,p∗ an) Let A be some left semi-linear space over a where p ∈ L. The neutral element in A is semiring R. We will denote vectors from A the vector (0,...,0). It is easy to see that n by bold characters to distinguish them from L is a semilinear space over L∨. scalars. ∅ L 2. Let X = , be a BL-algebra (MV- By a linear combination of vectors L ∨ ∗ algebra) on L and ∨ = L, , , 0, 1 its a1,...,an ∈ A we mean the following semiring reduct. Let us consider the set of expression all L-valued functions A = LX and put α1 · a1 + ···+ αn · an A(x)+B(x)=A(x) ∨ B(x), · ∗ p A(x)=p A(x) where α1,...,αn ∈ R are scalars called also coefficients. This linear combination uniquely ∈ where p L. The neutral element in A is determines a certain vector from A. the function which is identically equal to 0. It is easy to see that LX is a semi-linear Definition 7 space over L∨.