Algebraic properties of fuzzy sets

Algebraic Properties of Fuzzy Sets*

A. De Luca and S. Termini
Laboratorio di Cibernetica del Consiglio Nazionale Ricerche, Arco Felice, Napoli, Italy
Submitted by L. Zadeh

Abstract

Some new algebraic properties of the class $\mathscr{L}(I)$ of the “fuzzy sets” are stressed; in particular it is pointed out that the class of the generalized characteristic functions furnished with the lattice operations proposed by Zadeh is a Brouwerian lattice. The possibility of inducing other different lattice operations to the whole class $\mathscr{L}(I)$ or to a suitable subclass of it is considered. The problem of the relationship between “fuzzy sets” and classical set theory is finally remarked. A qualitative comparison with similar situations appearing in the axiomatic formulation of quantum mechanics and in the classical theory of probability is made.

1. Introduction

In this work some “algebraic aspects” of “fuzzy sets” theory, as developed by Zadeh [1] and afterwards generalized by Goguen [2], are considered. We believe that an algebraic analysis of such a theory may be relevant both for a deeper understanding of the connections between Zadeh’s theory and classical set theory, and for obtaining useful concrete realizations of the theory itself.

The latter is a crucial problem because the formalism of the generalized characteristic functions has not been developed in such a way as to allow the construction of a suitable mathematical calculus.

One of the greatest difficulties in such a direction essentially depends on the fact that in Zadeh’s theory the class of generalized characteristic functions is a distributive but noncomplemented lattice (we will show, in particular, that it is a Brouwerian lattice).

The nonexistence of a complementation, and so the impossibility of using the principle of “tertium non datur”, involves a substantial modification of Boolean logic greater than the one arising from the absence of distributivity,

^[1]

as in the case of the logical structure of Quantum Mechanics (Birkhoff and von Neumann [3]; Piron [4]).

At this point we recall Watanabe’s recent paper [5], in which an extension of the domain of application of the formalism and the logic of quantum theory in fields such as “information theory” and “pattern recognition,” is proposed. Such a formalism is very suitable for describing all those complex systems in which a strong (and uncontrollable) interaction between observed objects and measuring apparatus exists (De Luca and Termini [6]). The usefulness of the previous formalism is based on the existence of mathematical realizations (as the class of the subspaces of a linear space) of the corresponding algebraic structure allowing to obtain quantitative predictions.

Similar realizations have not yet been found in the case of Zadeh’s theory.
We stress that a noncomplemented logic, like the one underlying Zadeh’s theory, once a mathematical calculus has been constructed, may be of very great use in the analysis of many complex systems that are not easily described by means of the usual concepts and methods of the classical theory of probability. A useful application of this to the problem of “pattern recognition” is shown in De Luca and Termini [7].

An analysis of the algebraic structure of “fuzzy sets” makes evident some questionable points of the theory and also indicates some possible extensions. The lattice operations proposed by Zadeh are certainly not the only possible ones; it is, in fact, easy to show that one can provide in various ways convenient lattice structures for the class of generalized characteristic functions (or for a suitable subclass of it). Thus one can obtain lattice structures with different properties: Boolean, noncomplemented, nondistributive ones; the use of one structure or another being determined by the particular system under study.

The relationship between the previous algebraic structure and ordinary set theory may be of different kinds. It may, in fact, be an “isomorphism” or, more generally, simply a recovering of the original structure by means of a suitable composition of other “set-isomorphic” algebras.

From a theoretical point of view such a problem is not at all trivial; the classical theory of probability, for instance, in its axiomatic formulation, given by Kolmogorov, is reduced to the measure theory of ordinary sets. An analogous description of the probability appearing in quantum mechanics is by no means straightforward, but more complex. Essentially this depends on the nondistributivity of the lattice structure of quantum-mechanical propositions (Varadarajan [8]; Finch [9, 10]).

A similar problem is also present in the context of Zadeh’s theory, whose basic algebraic structure is noncomplemented, and in the case of more general theories making use of both nondistributive and noncomplemented lattice structures.

In the next section some remarks are made about the algebraic properties of the class of fuzzy sets (i.e., of all the maps from a universal class I of objects to a lattice $L$ ), confining ourselves to lattice operations induced to the whole class, point by point, from $L$.

A careful examination is then made of the algebraic properties of Brouwerian lattices; it is, in fact, easy to see that the class of fuzzy sets furnished with Zadeh’s operations of composition is a Brouwerian lattice.

In Section 3 we take into account also other lattice structures that may be given to $\mathscr{L}(I)$, or to a subclass of it, by means of operations not induced, point by point, from $L$. In such a way one obtains classes of functions with different algebraic properties. It is then possible to look at the theory of probability or the formalism of quantum mechanics as some noteworthy particular cases of “classes of generalized characteristic functions” with defined operations of composition yielding specific lattice structures.

In Section 4 the relationship between fuzzy sets and classical set theory is considered, taking into account the different lattice structures of $\mathscr{L}(I)$ or of the considered subclass and an isomorphism’s theorem (McKinsey and Tarski [11]) between Brouwerian lattices and a subalgebra of the open sets of a topological space is reported.

Finally, the general problem of obtaining some ordered structures by suitable compositions of other algebraic structures with fewer elements is briefly sketched.

2. Fuzzy Sets and Brouwerian Lattices

In this section we will synthesize some relevant algebraic properties of fuzzy sets and point out new algebraic aspects of the theory, connecting it with Brouwerian lattices.

We recall some preliminary definitions and properties.
Let $I$ be a (nonempty) universal class whose general element is denoted by $x$. We assume the following definition of fuzzy set on $I[2]$.
(1) A fuzzy set (on $I$ ) is a map

$$f: I \rightarrow L$$

where $L$ is a partially ordered set (poset).
We will denote by $\mathscr{L}(I)$ the class of all maps from $I$ to $L$. If $L$ consists of two elements only $L \equiv\{0,1\}$, then each fuzzy set is an ordinary characteristic function defining an ordinary subset of $I$. If $L$ is the closed interval $[0,1]$ of the real field we have the generalized characteristic functions or fuzzy sets as introduced by Zadeh [1].

The most interesting posets that may be considered are the lattices. In such a case the following proposition holds:
(2) If $L$ is a lattice, then so is $\mathscr{L}(I)$ with respect to the operations $\vee($ join $)$ and $\wedge$ (meet) defined as follows:

$$\begin{array}{ll} (f \vee g)(x)=\operatorname{lub}\{f(x), g(x)\}, & \text { for all } x \in I \\ (f \wedge g)(x)=\operatorname{glb}\{f(x), g(x)\}, & \text { for all } x \in I \end{array}$$

where lub and glb respectively, denote the least upper bound and the greatest lower bound of $f(x)$ and $g(x)$ in the lattice $L$. In $\mathscr{L}(I)$ the partial order relation $\leqslant$ is then defined:

$$f \leqslant g \Leftrightarrow f(x) \leqslant g(x), \quad \text { for all } x \in I$$

(where $\Leftrightarrow$ means “if and only if”) having

$$f=g \Leftrightarrow f(x)=g(x), \quad \text { for all } x \in I$$

(3) More generally any binary operation $(\cdot)$ defined on $L$ can be induced point by point on $\mathscr{L}(I)$ by

$$(f \cdot g)(x)=f(x) \cdot g(x), \quad \text { for all } x \in I$$

The same is, of course, also true for unary or generally for $n$-ary operations.
Confining ourselves to the case of posets no more general than lattices, we emphasize that the most relevant algebraic properties of the lattice $L$ are induced on $\mathscr{L}(I)$ by the lattice operations (2.1). In particular one may easily see that
(4) The modularity or the distributivity of $L$ is transmitted on $\mathscr{L}(I)$; moreover, if $L$ is a complete, complemented or Boolean lattice, then so is $\mathscr{L}(I)$.

We explicitly note that not every property of $L$ can be induced in $\mathscr{L}(I)$ by (2.1); if $L$ is a chain, for instance, $\mathscr{L}(I)$ is not so (except the pathological case in which $I$ consists of only one element).

Let us now assume $L$ to be a Brouwerian lattice. The following noteworthy property holds:
(5) If $L$ is a Brouwerian lattice then so is $\mathscr{L}(I)$.

To prove this we have to show that for any pair of elements $f$ and $g$ of $\mathscr{L}(I)$ the set of all functions of $\mathscr{L}(I)$, such that

$$f \wedge \psi \leqslant g$$

contains a greatest element $f^{g}$ called relative pseudocomplement of $f$ in $g$. We can construct $f^{g}$ defining, for any $x,\left(f^{g}\right)(x)$ as the greatest element

(that by hypothesis exists) of the set of the elements $z$ of $L$ satisfying the relation

$$f(x) \wedge z \leqslant g(x)$$

If we introduce on $\mathscr{L}(I)$ the binary operation $\div$

$$f \div g=f^{g}, \quad \text { for all } f, g \in \mathscr{L}(I)$$

it is easily seen, by the definition of Brouwerian lattice, that

$$\begin{gathered} f \geqslant g \wedge(f \div g), \quad(f \wedge g) \div g \geqslant f \\ f \div h \geqslant(f \wedge g) \div h \end{gathered}$$

Conversely if $\mathscr{L}(I)$ is closed with respect to a binary operation $\div$ such that the previous relations are satisfied then $\mathscr{L}(I)$ is a Brouwerian lattice having

$$f^{g}=f \div g$$

If $L$ is a chain with 0 and 1 , then $\mathscr{L}(I)$ will be a noncomplemented lattice, whose only complemented elements are the classical characteristic functions forming a narrow sublattice of $\mathscr{L}(I)$. However, since a chain with 1 is a Brouwerian lattice, such will be, by (5), $\mathscr{L}(I)$. In this case the relative pseudocomplement of $f$ on $g$ is the function defined, for any $x$, as

$$f^{g}(x)= \begin{cases}1, & \text { if } \quad f(x) \leqslant g(x) \\ g(x), & \text { if } \quad f(x)>g(x)\end{cases}$$

If $g=0$ the pseudocomplement (or Brouwerian complement) of $f$, e.g., $f^{0}$, usually denoted by $f^{*}$, is given by

$$f^{*}(x)= \begin{cases}1, & \text { if } \quad f(x)=0 \\ 0, & \text { if } \quad f(x)>0\end{cases}$$

Let us now remember some useful theorems of lattice theory (Birkhoff [12]) allowing us to derive some remarkable propositions about fuzzy sets.
(6) A complete lattice is Brouwerian if and only if the meet operation is completely distributive on joins, that is,

$$a \wedge\left(\bigvee x_{a}\right)=\bigvee\left(a \wedge x_{a}\right), \quad \text { for any set }\left\{x_{a}\right\}$$

and for any a.
(7) From (6) and (5) it follows that if $L$ is a complete Brouwerian lattice (as a chain with 0 and 1 ), $\mathscr{L}(I)$ will be a complete Brouwerian lattice having the complete distributivity of meet on joins, that is,

$$f \wedge\left(\bigvee g_{a}\right)=\bigvee\left(f \wedge g_{a}\right), \quad \text { for any set }\left\{g_{a}\right\}$$

and any $f$.

We can also consider a complete distributive law of join on meets, that is,

$$a \vee\left(\bigwedge x_{\alpha}\right)=\bigwedge\left(a \vee x_{\alpha}\right), \quad \text { for any set }\left\{x_{\alpha}\right\}$$

and for any $a$.
While the formulas (2.5) and (2.7) hold for arbitrary sets in any complete Boolean lattice, they do not hold in every complete distributive lattice. There exist easy examples of lattices for which (2.4) holds and (2.6) does not. Then we cannot generally invoke the principle of duality of the lattices to say that (2.5) is equivalent to its dual (2.7) (as done by Goguen [2]). In fact, for Brouwerian lattice there does not hold even a weak duality principle: i.e., if we change $v$ and $\wedge$ we do not still obtain a Brouwerian lattice [11].

In order that the completely distributive laws (2.5) and (2.7) be both true in a complete lattice $L$ we have to suppose $L$ to be not only Brouwerian but also dually Brouwerian. This means that for any pair $a$ and $b$ of elements of $L$, the set of elements $x$ of $L$, such that

$$a \vee x \geqslant b$$

must admit a least element denoted by $a^{-b}$.
Let us remark, at this point, that, as can easily be verified,
(8) A lattice at the same time Brouwerian and dually Brouwerian is a Boolean algebra if and only if it is

$$a^{*}=a^{-1}, \quad \text { for all } a \in L$$

If $L$ is a chain (with 0 and 1 ) then $\mathscr{L}(I)$ is also dually Brouwerian having

$$f^{-g}=\begin{aligned} & g(x), \\ & 0, \end{aligned} \quad \begin{aligned} & \text { if } \\ & \text { if } \end{aligned} \quad f(x)<g(x)$$

it is therefore

$$f^{-1}=\begin{aligned} & 1, \\ & 0, \end{aligned} \quad \begin{aligned} & \text { if } \\ & \text { if } \end{aligned} \quad f(x)<1$$

The condition (2.8) then becomes, in $\mathscr{L}(I)$,

$$f^{*}=f^{-1}=\begin{aligned} & 1, \\ & 0, \end{aligned} \quad \begin{aligned} & \text { if } \\ & \text { if } \end{aligned} \quad f(x)=0$$

Relation (2.9) can be satisfied, as we said before, only by the elements of the subset of $\mathscr{L}(I)$ formed by the classical characteristic functions defined on $I$. Let us observe that in any case we have

$$f \wedge f^{*}=0, \quad f \vee f^{-1}=1$$

We finally report the important theorem [11]
(9) “The set of all open elements of a closure algebra is a Brouwerian lattice and vice versa any Brouwerian lattice can be embedded in this algebra that is considered as the algebra of open elements of a closure algebra.”

A dual theorem in which closed elements are substituted for open ones and Brouwerian lattices for dual Brouwerian ones, of course holds. Concluding the section we note that in the particular case of Zadeh’s theory, in which the range of $f$ 's is the interval $[0,1]$ of the real field, the algebraic structure of $\mathscr{L}(I)$ may be further enriched, inducing by (2.3) other operations of $[0,1]$ also. This is the case, for instance, of the usual multiplication which is induced by (2.3). This operation [in $\mathscr{L}(I)$ ] satisfies the associative property and the distributivity on joins (and on meets). This is expressed in algebraic terms by saying that $\mathscr{L}(I)$ is an $L$ semigroup with respect to the operations $v, \wedge, \cdot$.

3. Further Algebraic Structures of Fuzzy Sets

The algebraic properties of fuzzy sets that have been discussed in the previous section refer to the natural lattice structure that can be induced in $\mathscr{L}(I)$ by the point-by-point definition (2.1). If $L$ is a lattice with respect to more than one pair of operations then more lattice structures can simultaneously be induced on $\mathscr{L}(I)$.
If the class $I$ has some algebraic structures, as poset or lattice, it is possible to determine, as we will see in the following, suitable subclasses of $\mathscr{L}(I)$ to which we may give lattice structures.
Let $I$ and $L$ be posets, a map $f: I \rightarrow L$ is called isotone if

$$x \leqslant y \Rightarrow f(x) \leqslant f(y)$$

The set $L^{I}$ of all isotone functions from $I$ to $L$ (usually called cardinal power with base $L$ and exponent $I$ ) is partly ordered by setting

$$f \leqslant g \Leftrightarrow f(x) \leqslant g(x), \quad \text { for all } x \in I$$

The following noteworthy theorem holds (Birkhoff [12]):
(10) If $L$ is a lattice and $I$ a poset, then $L^{I}$ is a lattice with respect to the operations induced from $L$ by (2.1). If $L$ is modular or distributive then so is $L^{I}$.

We explicitly observe that it is not, in general, possible if $L$ is complemented to induce point by point a complementation in $L^{I}$, since if $f$ is isotone, the complement $f^{\perp}$ of $f$ is anti-isotone [and so belongs to $\mathscr{L}(I)$ but not to $L^{I}$ ].

If $I$ is a lattice we induce a lattice structure into $L$ (or a subset of $L$ ) in the following way; for any map $w$ from $I$ into $L$ we define

$$\begin{aligned} & w(x) \vee w(y)=w(x \vee y) \\ & w(x) \wedge w(y)=w(x \wedge y) \end{aligned}$$

If $I$ and $L$ are both lattices then any one-to-one ${ }^{1}$ map $w$ for which (3.1) holds is called morphism.

If $w$ is onto, $L$ itself is a lattice. It is very easy to verify that if $I$ is modular, distributive, complemented or Boolean, then so will be L. By any bijection $w$ from $I$ to $L$ we can then induce a lattice structure in $L$ and by the point-bypoint definitions (2.1) a lattice structure on $\mathscr{L}(I)$, or $L^{I}$.

One may, however, define other order relations on $\mathscr{L}(I)$ (or on a convenient subset of it) taking into account also lattice operations not defined point by point.

Every bijection $W$ from $\mathscr{L}(I)$ to a lattice $M$, for instance, induces on $\mathscr{L}(I)$ the algebraic structure of $M$; if $W$ is a surjective mapping it is possible to define in $\mathscr{L}(I)$ the equivalence relation $\sim$ :

$$f \sim g \Leftrightarrow W(f)=W(g)$$

and order the equivalence classes of $\mathscr{L}(I)$ defined by $\sim$ or the subset of $\mathscr{L}(I)$ formed by a sample from each class.

A concrete example of an ordering of equivalence classes of $\mathscr{L}(I)$ by means of a functional defined on $\mathscr{L}(I)$ can be found in De Luca-Termini [7].

Moreover we can induce a lattice structure in a subset of $\mathscr{L}(I)$, starting from a lattice $M$ by any one-to-one map $w$

$$w: M \rightarrow \mathscr{L}(I)$$

in the following way:

$$\begin{aligned} & w(A) \wedge w(B)=w(A \cap B) \\ & w(A) \vee w(B)=w(A \cup B), \quad \text { for all } A, B \in M \end{aligned}$$

where $\cap$ and $\cup$ are the lattice operations in $M$. Denoting by $f_{A}(x)$ the value of $w(A)$ for a certain $x$, we can write (3.2) also in the form

$$\begin{aligned} & \left(f_{A} \vee f_{B}\right)(x)=f_{A \cup B}(x) \\ & \left(f_{A} \wedge f_{B}\right)(x)=f_{A \cap B}(x) \end{aligned}$$

^[1]

Every lattice property of $M$ is transmitted to a convenient subset of $\mathscr{L}(I)$ [or to $\mathscr{L}(I)$ if $w$ is onto]; if $M$ is complemented, this complementation is induced by

$$\overline{w(A)}=w(\bar{A})$$

Remarkable cases are obtained when $M$ is a ring $R$ (or a field $F$ ) of subsets of $I$ or when it is the class of the (closed) subspaces of a linear space (in particular $\mathbf{R}^{(n)}$ ). In the case in which $M \equiv F$, let us suppose that $L \equiv[0,1]$ and, for some $x \in I$,

$$\begin{aligned} f_{A \cup B}(x) & =f_{A}(x)+f_{B}(x) \\ f_{I}(x) & =1 \end{aligned}$$

for all $A, B \in F$ such that $A \subseteq B$, then the triplet $\{I, F, f(x)\}$ is a field of probability.

If $M$ is not a distributive lattice, as in the case of the class of elementary propositions of quantum mechanics, then if (3.5) and other specific assumptions are satisfied, the couple $\{M, f(x)\}$ is the kind of generalized probability appearing in the axiomatic formulation of quantum mechanics, as proposed by Varadarajan [8]. As regards the implications of a nondistributive structure see Watanabe [5].

Let us now consider, for instance, in the case $L \equiv[0,1]$, those functions $f_{A}$ that may be expressed in the form ${ }^{2}$

$$f_{A}(x)=\int_{A} \rho_{z}(y) d \mu(y), \quad A \in F$$

where $\mu$ is a measure on a field $F$ of subset of $I$ and $\rho$ a nonnegative function (probability density) such that

$$\int_{I} \rho_{z}(y) d \mu(y)=1$$

(3.5) is satisfied for each $x$.

Let $M$ now be the nondistributive lattice of the subspaces of a linear space in which the lattice operations $\cup$ and $\cap$ between two subspaces $A$ and $B$ consist of taking the smallest subspace containing both $A$ and $B$ and the greatest one contained in them, respectively. By using (3.3) and (3.6) a nondistributive lattice of functions of $\mathscr{L}(I)$ is, in such a way, easily obtained.

^[1]

Finally we want to observe further that, inversely, if $L$ is a poset it is possible to induce a poset structure into $I$. To this end if $\Omega$ denotes a subset of $\mathscr{L}(I)$ we introduce the order relation

$$x \leqslant y \Leftrightarrow f(x) \leqslant f(y), \quad \text { for all } f \in \Omega$$

$I$ is a poset with respect to this partial order relation. If $\Omega \equiv \mathscr{L}(I)$ every element will be in relation $\leqslant$ with itself whereas two different elements $x$ and $y$ will never be comparable.

If $L$ has a greatest element 1 , and for every $x \in I$ there exists at least an $f \in \Omega$ such that $f(x)=1$, we may introduce into $I$ the order relation $\leqslant$ by decreeing

$$x \leqslant y \Leftrightarrow[f(x)=1] \Rightarrow[f(y)=1], \quad \text { for all } f \in \Omega$$

We have that

$$x \leqslant y \Rightarrow x \leqslant y$$

but the converse is not true.
Let us note that these ways of ordering the elements of a class $I$ are both used, even if in a more specific context, in the axiomatic formulations of quantum mechanics (see, for instance, Finch [9] and Jauch and Piron [13]). In these cases $I$ is the set of the propositions of a physical system and the elements of $\Omega$ are the quantum-mechanical states.

The previous discussion clearly shows that, apart from any interpretation, the class of generalized characteristic functions (or a suitable subclass of it) may be furnished with lattice structures in various ways. These may be Boolean, distributive but noncomplemented, as in the case of Zadeh’s theory, nondistributive but complemented, as occurs in the case of quantummechanical probability measures, and so on.

The choice of one structure or of another affects, and is affected by, of course, the interpretation of the functions themselves. The previous choice, therefore, strongly depends on the phenomena one wants to describe, since the use of a certain structure can certainly be more useful and effective than another.

4. Remarks on the Relationships Between Fuzzy Sets and Classical Set Theory

In this last Section we will briefly discuss some simple properties of fuzzy sets that we think useful in relating them with classical set theory. No claim to completeness is made at all.

The finding of such connections is very important from a theoretical point of view; in fact, it helps in clarifying if and how fuzzy set theory can be considered reducible to classical set theory.

To this purpose we firstly emphasize that any distributive lattice $L$ is isomorphic with a ring of sets (Birkhoff’s theorem). In particular, if $L$ is a Boolean lattice this isomorphism is with a field of sets (Stone’s theorem). This result means that we can always look at the partial order relation $\leqslant$ of a distributive lattice as the set-theoretic order relation $\subseteq$ (inclusion) and $v$ and $\wedge$ as the set-theoretic union $\cup$ and intersection $\cap$. From this theorem it follows that
(11) If $\mathscr{L}(I)$ is a distributive lattice, as in the case of Zadeh’s theory, we can always look at fuzzy sets in terms of suitable classical sets and the “join” and “meet” operations in terms of the set-theoretic “union” and “intersection.”

In the case of a Brouwerian lattice by Proposition (10) and other classical results one has that (see McKinsey and Tarski [11, Theorem 1.19, p. 134]) “The algebra of open sets of a topological space, and every subalgebra of this algebra, is a Brouwerian algebra and, conversely, every Brouwerian algebra is isomorphic to a subalgebra of the algebra of the open sets of a topological space.”

We observe that the previous isomorphisms are very important in principle but do not invalidate Zadeh’s approach since the translation of this last language in set-theoretical terms is not easily feasible and, therefore, does not provide useful suggestions for developing a calculus containing the meaningful features of the theory. Moreover if $\mathscr{L}(I)$ (or a suitable subset of it) is a nondistributive lattice an analog of the previous isomorphism theorem is not available.

In the case of nondistributive structures it is not generally possible to use isomorphism theorems; to establish a connection with set theory one can, anyway, make use of indirect mechanisms, which may be more complex. Such a problem has been studied by Finch [10] in the case of the (nondistributive) logic of quantum mechanics.

We now sketch this method of composition.
Let us consider a set $\mathscr{P} \equiv\left\{P_{\gamma}: \gamma \in \Gamma\right\}$ of posets partly ordered with respect to an order relation that we denote by $\omega_{\gamma}$.

If

$$P=\bigcup_{\gamma \in \Gamma} P_{\gamma}$$

is the set-theoretic union of the $P_{\gamma}$ 's we introduce suitable hypotheses on the $P_{\gamma}$ 's in order to give a convenient structure to $P$.

Let us begin by making the following assumptions:

(i) If $x$ and $y$ belong to $P_{\alpha} \cap P_{\beta}$ then $x \omega_{\alpha} y$ if and only if $x \omega_{\beta} y$.
(ii) If $x \omega_{\alpha} y$ and $y \omega_{\beta} z$ there exists an index $\gamma$ such that $x \omega_{\gamma} z$.

From these two assumptions it follows that we can introduce an order relation $\omega$ into $P$ defining $x \omega y$ if and only if there exists an index $\gamma$, at least, such that $x \omega_{\gamma} y$.

It is obvious that $P$ is a poset with respect to the relation $\omega$.
By (i) and (ii) one is able to provide a poset structure to the union (4.1) of a class of posets. The following problems then naturally arise:
(a) What hypotheses have to be added to (i) and (ii) in order to provide a required structure to $P$, as defined by (4.1)?
(b) Under what conditions may a given $P$ be obtained by the previous compositions starting from a suitable class of $P_{\gamma}$ 's?

These questions have been answered by Finch [9] in the case in which the $P_{\gamma}$ 's are Boolean algebras and $P$ is an orthomodular poset.

Concluding this section we note that, confining ourselves to the specific case of Zadeh’s theory, any generalized characteristic function $f(x)$ can be expressed by means of classical characteristic functions. In general, the required number of these functions becomes infinite if we wish to compute $f(x)$ exactly. However, for any fixed precision, we are able to compute $f(x)$ by means of a finite number of characteristic classical functions (independent of the particular $x$ ).

In fact, for any $f \in \mathscr{L}(I)$ we can write the binary expansion of $f(x)$ as

$$f(x)=\sum_{i=1}^{\infty} \psi_{i}(x) 2^{-i}$$

where $\psi_{i}$ are functions of $x$ assuming only the values 0 and 1 , that is, classical characteristic functions.

Let us note that the expansion (4.2) only makes use of the constant functions $2^{-i}(i=1,2, \ldots)$ and the ordinary operations + (sum) and $\cdot$ (product) that are induced on $\mathscr{L}(I)$ as

$$(f \cdot g)(x)=f(x) \cdot g(x)$$

and

$$(f+g)(x)=f(x)+g(x)$$

if $f(x)+g(x) \leqslant 1$ for all $x$.
Moreover, if we cut the expansion (4.2) to the $n$-th term we make an error on evaluation of $f(x)$ less than or equal to $2^{-n}$.

5. Concluding Remarks

In the second Section we stressed that $\mathscr{L}(I)$ is a Brouwerian lattice, provided that $L$ is. Such Brouwerian lattices express in algebraic terms those substantial modifications of two-valued logic proposed by the intuitionistic school mainly represented by Brouwer and Heyting (see, for instance, Heyting [14]).

A Brouwerian logic is a noncomplemented propositional calculus which does not admit the validity of the “by contradiction” proofs. As can easily be seen, a Brouwerian logic is a Brouwerian lattice and vice versa every Brouwerian lattice may be regarded as a Brouwerian logic. We believe that the previous connection between $\mathscr{L}(I)$ and Brouwerian logics is an important feature of Zadeh’s theory.

In the subsequent sections, however, we made a sharp distinction between Zadeh’s theory and those other ones in which the operations of composition among the generalized characteristic functions are not necessarily related to those of the range of the functions themselves.

This second way gives us the possibility of selecting some noteworthy classes of functions with operations of composition naturally arising from the specific considered situations without being conditioned by pre-assigned operations that in some cases may not be most useful. An example of this is given by Watanabe’s approach, to which we referred in the Introduction.

We finally remark that it is not possible to give an answer once and for all to every question previously introduced (as the analysis of the relationships with the classical theory of sets or the finding of concrete mathematical realizations or even the ties with the theory of probability) since they are strictly related to the particular algebraic structure of the specific considered class of generalized characteristic functions.

Acknowledgment

Many interesting suggestions by Professor E. R. Caianiello are gratefully acknowledged.

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