Fuzzy Sets and Systems 30 (1989) 27-36 27 North-Holland

I S O M O R P H I S M S B E T W E E N D E M O R G A N T R I P L E T S *

Pete G A R C f A and L. V A L V E R D E Dept. de Matenu~tiques i Estadistica, E.T.& d'Arquitecmra de Barcelona, Av~ . D ~ 649, 08028 Barcelona, Spain

Received December 198.5

Abstract: This paper deals with the characterization of the isomorphims of the sm~cturcs related to De Morgan triplets. As is well known, the De Morgan triplets are used in order to define the basic connectives meet, join and negation in the setting of the Multiple-Valued Logic underlying Fuzzy Set Theory; therefore, such triplets will induce isomo~hisms on the structures defined in [0,1] x by these connectives. A classification of ~e De Morgan triplets is given for which the De Morgan triplets of the type (Min, Max, n)-and, therefore, the corresponding De Morgan algebras-constitute a class. The main resulting equivalence classes of De Morgan triplets are studied.

Keywords: t-norm; t-conorm; strong negation; De Morgan triplet.

1. Introduction

This article is concerned with the characterization of the isomorphisms of the structures related to De Morgan triplets, i.e. triplets (F, G, n) where F is a continuous t-norm, G a continous t-conorm and n a strong negation such that Fo (n × n) = n o G. As is well known [1, 6, 10], the De Morgan triplets are used in order to define the basic connectives meet , join and negation in the setting of the Multiple-Valued Logic underlying Fuzzy Set Theory; therefore, such iso- morphisms will induce isomorphisms on the structures defined in [0,1] x by these connectives.

As is also well known, when the De Morgan triplet is of the type (Min, Max, n), the resulting structure is a De Morgan algebra, for which isomorphisms have been largely studied [2, 3], but a similar study for general De l~organ triplet is, so far, missing.

It turns out that, in contrast with the De Morgan algebra case, there exists only one au tomorph i sm- the i d e n t i t y - f o r Archimedean De Morgan triplets. There- fore, any isomorphism will change the underlying De Morgan triplet. A classification of the De Morgan triplets is given for which those of the type (Min, Max, n ) - and, therefore, the corresponding De Morgan a lgebras- form a class. The main resulting equivalence classes of Archimedean De Morgan triplets are also studied.

* Research partially supported by the C.A.I.C.Y.T., project number 1503/82.

0165-0114/89/$3.50 ¢~) 1989, Elsevier Science Publishers B.V. (North-Holland)

28 P. Garcfa, L. Valverde

Coming, mainly, from their application in the field of Probabilistic Metric Spaces 18] there exists a useful classification of t-norms (and t-conorms), namely the classification given through the set

E(F) = (x E [0, 1]; F(x, x) = x )

of idempotent elements for the t-norm F, and

Nil(F) = {x ¢ (0, 1); F(x, y) - 0 for some y ¢ (0, 1))

of nilpotent elements for F. The former set allows one to split the set of continuous t-norms into two classes: Archimedean t-norms (i.e. t-norms F for which E(F)--{0, 1)) and non-Archimedean t-norms (i.e. E(F) is a proper super-set of {0,1}). In the case of E(F) --- 10, 1] then F - Min and, as was proven in I7], any other continuous t-norm is an ordinal sum of the t-norm Min and Archimedean t-norms. On its own, the set Nil(F) allows one to split the set of Archimedean t-norms into two subclasses: for any Archimedean t-norm, Nil(F) is either the empty set (strict t-norms), or the open unit interval (Nilpotent t-norms). Obviously, E(F) and Nil(F) further split the set of non-Archimedean t-norms into a large number of classes [5]. Anyway, after the above mentioned result by Ling I7], many properties of the ordinal sums may be derived from the properties of Archimedean t-norms and the t-norm Min. For that reason we have restricted our attention to the case of Archimedean t-norms.

It should be noticed that if only meets and joins are considered-i.e, pairs (F, G), F being a t-norm and G a t-conorm- then the above classification is also a classification of the structure related to these connectives. But, as shown in this paper, when a negation n is added such that (F, G, n) is a De Morgan triplet, there is a considerable enrichment of the structure, since a large number of classes appears even taking into account that if (F, G, n) is a De Morgan triplet then they are of the same type, i.e., F and G are Archimedean or non- Archimedean simultaneously a n d - i n tL~e former case- they are strict or nil- potent simultaneously as well, which justifies the use of terms like strict De Morgan triplet or nilpotent De Morgan triplet made throughout this paper. On the other hand, all results are given in terms of the t-norms: obviously similar results would be obtained considering the t-conorms.

In fact, with the usual definition of a morphism between algebraic structures, any strict t-norm is isomorphic to the t-norm F(x, y ) = x . y and any nilpotent t-no~lrn is isomorphic to the t-norm F(x,y)-'Min(x+y-l,O). This is a straightforward corollary of the representation theorem for Archimedean t-norms 17]: For any Archimedean t-norm there exists a strictly decreasing and continuous function f, called additive generator of F, from [0,1] into/]~" with f(1) -- 0, such that

F(x, y) =fl-~l(,f(x) + f(y)), f t - l l being the pseudo-inverse o f f (i.e. fl-ll(X) =f-t(X) if X ¢ [0, f(0)]; fl-ll(X) -- 0, otherwise). Such additive generators are unique up to a positive multiplicative constant. Then F is strict if, and only if, f(0) -- +0o and F is nilpotent if, and only if, f(0) < +co. In the sequel of this paper only normalized additive generators (i.e. f(0) -- 1) of nilpotent t-norms will be considered.

Isomorpi~sms between De Morfan triplets 29

Strongly related to the above results is the representation theorem for the strong negation functions (i.e. continuous involutions in [0,1]) given in [9]: If h is a normalized additive generator of a nilpotent t-norm, then

n(x) = h-t(1 -h(x)) is a negation function and vice-versa. Such a fun~on is called additive generator of the negation. In [11] it is shown that if m is an additive generator of a strict t-norm, then

- i 1

is also a negation function and vice-versa, m is then called multiplicadve generator of the negation n.

After some basic.'~finitions and properties given in the next section, Section 3 is devoted to the study of the isomorphisms between nilpotent De Morgan triplets and the last section is concerned with the characterization of the isomorphisms between strict triplets. It would be noticed that in both characterizations the additive generators of the t-norms play a fundamental role. The same applies to the additive generators of the negation functions with respect to the former characterization, but the study of the isomorphisms between strict triplets requires the use of multiplicative generators of the negations.

2. Fumlmnentgls

Definition 2.1. An isomorphism between two De Morgan triplets (F, G, n), (F °, G', n') is a bijection ~p from [0,1] into itself satisfying

~ , o F ' = F o ( ~ , × ~) (2.1) and

~p o n' = n o lp (2.2)

Two De Morgan triplets are called isomorphic if there exists an isomorphism between them.

From the representation theorem for strong negation functions [9] it turns out that if n and n ' are two negations with additive generators h and h ' respectively, then Ip = h- toh ' satisfies (2.2) and, therefore, the following proposition holds.

Proposition 2.1. For any pair n, n' o f strong negation functions, the De Morgan triplets (Min, Max, n) and (Min, Max, n') are isomorphic.

The next proposition summarizes some other basic properties of isomoprhic De Morgan triplets.

Proposition 2.2. Let Ip be an isomorphism between (F, G, n) and (F', G', n'). Then:

(a) ~ ,o6' = 6o(~ , x ~,).

30 P. Garda, L. Valverde

(b) I f s and s' stand for the ftxed points of n and n' respectively, then ~p(s') = s. (c) I f F, F' are nilpotent and f, f ' are their respective normalized additive

generators, then ~ = f - l of,. Therefore, F is nilpotent if, and only if, F' is nilpotent. (d) I f F, F' are strict and generated by f, f ' respectively, then ~p =f- lo l~ of'

where ~ =f(s)]f ' (s ' ) and l=(x) = of. x. Z~erefore~ F is s~ct if, and only if, so is F'. (e) ~p is a continuous and increasing function.

P ~ f o Straightforward calculation allows one to check (a), (b) and (e). Now, in order to check (c) and (d) let it be noticed that the equality (2.1)

rewritten in terms of the additive generators f and f ' , yields

(f,)t-1]~,(x) + f ( y ) ) = ~p-1 of[-ll((f o ~,)(x) + ( f o ~p)(y)),

i.e. f ' and fo~p generate the same t-norm; therefore there exists ~ in R ÷ such that l~o f ' - - fo ~p. So if F and F ' are strict, from (b) it follows that ~ =f(s) / f ' (s ) , i.e. (d) holds.

Now, if F and F ' are nilpotent, f and f ' may be choosen in such way that f(O) =if(O) = 1. In that case 0~ = 1 and therefore (c) also holds.

Let it be noticed that from properties (b), (c) and (d) follows that the identity is the only pc/ssible automorphism from a De Morgan Archimedean triplet into itself (i.e. aatomorphism).

In [10] ~t is shown that the automorphisms of the De Morgan triplets (Max, Min, n) are given by functions ~p from [0,1] into itself of the form

= ~3,(x) ° i f x ~ s , ~p(x) [(n y. n)(x) otherwise,

where s is the fixed point of n and Y is any increasing bijection from [0, s] into itself.

Taking into account these remarks it is easy to characterize the automorphisms of non-Archhnedean triplets. Such an automorphism would be given by a function ~p satisfying

(a) ~'Ito,~1-~(x)=x, (b) ~l~(x)=~3'(~ ) o ifx~<s,

[(n 3' n)(x) otherwise,

where D is the set of elements x in [0,1] such that both x and n(x) are nilpotent for F (i.e. F(x, x) =x and F(n(x), n(x)) -- n(x)).

3. homo~l~isms between nilpotent Mp|ets

Throughout this section the isomorphisms between nHpotent De Morgan triplets are characterized. This characterization is given by means of the additive

lsomovphlsms bem~en De Morg4n ~iptets 31

generators of the corresponding t-norms as well as the additive generators of the negation functions.

Given a nilpotent triplet (F, G, n), it is shown that, for ~ny nilpotent t-norm F' there exists a De Morgan triplet (F', G', n ' ) isomorphic to the given one. It is also shown that, given a nilpotent triplet (F, G, n), for any negation n ' there exists a family (Fp, Gp, n ' ) of niipotent De Morgan triplets such that any of such triplets is isomorphic to the given one.

In what follows, H is the set of continuous decreasii~g bijections from the unit interval into itself, i.e., the set of normalized additive generators of nilpotent t-norms; C stands for the group of the continuuus bijections p from the unit interval into itself such that

N o p = p o N ,

where N(x)= 1 - x . C* is the set of functions of C which are increasing. First of all, we give alemma which shows the relationship between the set W of

negation functions, H and C. Let ~ c be the equivalence relation defined in H by hi ~chz if hloh£ 1 e C; then:

l a ~ 3.1. The map Z from H l ~ c into W defined by

Y,(h) = h -~ oNoh

is a bi]ection.

The main results of this section are straightforward corollaries of the following:

Theorem 3.1. Let (F, G, n), (F', G', n') be two nilpotent De Morgan triplets and let f, f ' , h, h' be normalized additive generators of F, F', n, n' respectively. Then (F, G, n) and (F', G', n') are isomorphic if, and only if,

h o f - ' o f , o(h,) -1 e ¢. (3.1)

PmOfo Let ~p = f - i of,; considering Proposition 2.1(c) it suffices to show that the above condition (3.1) holds if, and only if,

~pon' =no~p.

That can be immediately checked writing this equality using the additive generators of n and n' , and taking into account that f -1 of, o (f , ) l -q =f~-~l.

Corelhn~, 3.1. Let (F, G, n) be a nilpotent De Morgan triplet and let F' be a nilpotent t-norm. Then there exists a negation n' such that (F', G', n') is isomorphic to (F, G, n). This negation n' is unique and h' = h o f - t o f ° is one of its additive generators.

Proof. Straightforward by taking into account the above Theorem (3.1) and the Lemma (3.1).

32 P. Garda, L. Valverde

Enml~e 1o Let F(x, y) = Max(x + y - 1, 0); n(x) = 1 - x and

F'~(x, y ) = 1 - ~ n ( 1 , [ ( 1 - x)~ + ( 1 - yp]"~).

Then n'~(x)= 1 - ( 1 - ( 1 - x ) ° ) lm. In this example we have f ( x ) = 1 - x = h(x) and f ' ( x ) = ( 1 - x)~ = h'(x).

l g ~ p I ¢ 2° Let F(x, y) = Max(x + y - 1, 0); n(x) = 1 - x and

F'~(x, y) ffi (Max(O, x ~ + .V ~ - 1)p '~.

Then n~(x)= (1 -x~) 1/~. In this case, we have f(x)ffi 1 - x =h(x) and f~(x) = 1 - x ~.

Summing up, the above Corollary 3.1 can be represented by the following schema:

~v A H ~:Y~ H n.~ H / ~ c A W

F'--~f ' ~ h of -1 of'--~ [h of-1 of']---~ n' ,

where A is the bijection which assignes to any niipotent t-norm its normalized additive generator and Hi s the canonical projection. It is worth noting that in the case that (F, G, n) is a Lukasiewicz triplet, i.e. F and n additively generated by the same function, then OF, n iS the identity map. Therefore n ' is also generated by f ' , the generator of F ' , so all Lukasiewicz triplets are isomorphic.

C o x ~ j 3.2. Let (F, G, n) be a nilpotent De Morgan triplet and let n' be a negation function additively generated by h'. Then, for any p ¢ C*, fp = f o b -lo p oh' generates a t-norm Fp such that the De Morgan triplet is isomorphic to (F, C;, n).

Like Corollary 3.1, Corollary 3.2 can be represented by the following schema: E-I

W ~ H l ~ c ~ Pc.(H)'->Pc..(H)-'~P(~;N)

n' ~-~[h'].~> {poh'}~,~c.--~ { f oh-Zopoh'}p~C.--> {Fp}p~c. where

Pc*(H) = {{p °h'}pec*; h ' ¢H},

ec. .(H) = { { f * h - l ° p oh'}p~c.; h ' e H}.

It is interesting to point out that, in the case of (F, G, n) being a Lukasiewicz triplet, the family (Fp, Gp, n ') is precisely the family of Lukasiewicz triplets associated to n' , i.e. any Fp is additively generated by a generator of n'.

4. bomorphlsms between strict De M o r g n Mplets

This section is concerned with the characterization of the isomorphisms between strict De Morgan triplets. As in the previous section, a necessary and sumcient condition for two of these De Morgan triplets to be isomorphic, is given

Isomovphisms between De Morgan ~ t s 33

by means of the additive generators of the corresponding t-norm and the multiplicafive generators of the negations. Then methods to build isomorphic De Morgan triplets to a fixed one are given.

In what follows Cz stands for the set of continuous bijections from ~+ into itself satisfying qoK = Koq, where K is the function from R÷ into itself defined by K(x) = l/x, C~ = {q ~ C~; q is increasing}./-/1 is the set of additive generators of strict t-norms, i.e. continuous and strictly decreasing bijections from the unit interval into/]+.

Like in the previous case the sets W,/-/1 and C~ are related in the following way. Let me1 the equivalence relation defined on//1 by m me, m' if m' om -~ ¢ C~. Then we have:

l~mma 4.1. The map A from Hz/mcl into W defined by

A(m) = m -1 o If o m

is a bijection.

Similar arguments to these used in Theorem 3.1 allow one to proof the following result.

Theorem 4.1. Let F, F' be two strict t~norms additively generated by )', f ' ¢ l'Iz and let n, n' be two negations multiplicatively generated by m, m' ¢ Hi. Then the strict De Morgan triplets (F, G, n) and (F', G', n') are isomorphic if, and only if

mof- tol~ o(m') -1 ¢ C1,

where l~ = or. x and ot =J'(s)/f'(s) with s and s' the fixed points by n and ~', respectively.

Corollm~j 4.1. Let (F, G, n) be a strict De Morgan triplet and let F' be a strict t-norm. I f f and f ' stand for the additive generators of F and F' respectively, then for any a e R + - {0}, the function

mo =mof - lo lao f '

generates multiplicatively a negation na, such that the De Morgan triplet (F', Go, n~) is isomorphic to (F, G, n).

Example. Let F(x, y ) = x .y and n(x )=e T M . Then the family of negations associated with F'(x, y) =x . y l (x + y - x y ) is na(x) = (1 -x ) / (1 + (lla 2 - 1)x), i .e . the family of negations of Sugeno. In this example, we have f ( x ) = - L n x = m(x) and f ' (x) = (1 - x)/x.

The content of the above lemma may be rephrased through the following schema: rF,.

~ n~/~ -----,r~,,,,(H~l=c,)"-, P(W)

F'--~ [f°]--~ {[m o/-~ ol, o/']; ~ ¢ R + - {0})--~ {n~},~eR+-mp

where ~s stands for the set of strict t-norms.

34 P. Garcfa, L. Valverde

C O ' m y 4.2. Let F be a strict t-norm additively generated by f and let n, n' be strong negations multiplieaavely generated by m, m', respectively. For any q ¢ C~ the.function

fq =fore -1 o q ore'

generates additively a strict t-norm Fq such that the De Morgan triplet (Fq, Gq, n') is isomorphic to (F, G, n).

In other words, given a De Morgan triplet (F, G, n) and a strong negation function n' , multiplicatively generated by m', a family of strict De Morgan triplets isomorphic to the given one may be built as follows:

."- '> [m'].-> { I f om- ' oq ore'I; q ¢ C~'} ,~ {Fe}q,ct.

Example 1. As shown in [4], any f in/ /1 generates additively a strict t-norm F a~M multiplicatively a strong negation n, such that (F, G, n,) is a De Morgan triplet for any t ¢ R + - {0}, where

n,(x) =f-t( t / f(x)). Then we have:

M p o i t i o n 4.1. For any t ¢ R ÷ - {0}, (F, G, n) and (F, G, n,) are isomorphic De Morgan triplets.

Proof. Straightforward by taking into account that the multiplicative generator of n, is l,~ofwith a~ = 1 ! ~ .

Example 2. Le~ y be a continuous and decreasing bijection from R÷ into itself satisfying the property that there exists to ¢ (0, 1) such that

~,(tx) = ~,(x)/t if, and only if, t ¢ {t~;p E Z}; t h e n - as is also shown in [4]- for any f in/-/1, the family of negations

n~ =f-'°l~o°yOf is such that (F, G, np) is a De Morgan triplet for any p ~ Z, F being the strict t-norm additively generated by f. For these triplets we have:

~ s i ~ o n 4.2. (F, G, rip) and (F, G, np,) are isomorphic De Morgan triplets if, and only if, p - p ' (rood 2).

Pn~of. It is easy to check that if p - p ' (rood 2) then

~p=f-'olo, of with ~=6p'-prZ.

is an isomorphism between the given triplets.

lsomo~hisms between De Morgan triplets 35

Now suppose that ~p is an isomorphism be tween (F, G, rip) and (F, G, rip,). From Proposit ion 2.2(d) it fo | |ows that there exists ~ ¢ R ÷ - {0) such that

~, __f-1 o t,~of.

After rap|acing in (2.2) ~, by this expression we get

~,(~x) = ~ - to p-p ' . y(x) ,

which imp|ies

y(o fx ) - (o~-P')"y(x) for any n e Z. (4.1)

~f p ~ p ' ( m o d 2 ) then the set e = { ~ . t g ; m , n e Z } is dense in R÷ and therefor~ there exist sequences of integers (Pn).~z, (P~).~z such that

|ira a~P't p~ = c~. (4.2)

From (4.1) and the continuity of y it turns out that

o~. t ~p'-p) • y(1) = y(o~) -- lira y (~P" t p~) n - . . ¢ ~

= y(1) |ira (~" t~P'-P~F" .-.® ~ (4.3)

From the equa| i t ies (4.2) and (4.3) it follows that iimn_,o.p, = i ; since Pn ~ Z, there exists no such that for any n ~> no ho|ds p . - 1. Therefore |im~_.o.p~ = 0, which implies that o~ is an isolated point in the dense set P. This contradictio~ entai |s that when p ~ p ' ( rood2) , there is no isomorphism be tween (F, G, np) and (F, G, n~.).

Re feFen¢~

[1] C. Alsina, E. Trillas and L. Valverde, On some logica| connectives for fuzzy set theory, J. Math. Anal. Appl. 93 (1983) 15-26.

[2] F. Esteva, On some isomorphisms of the De Morgan algebras of fuzzy sets, Stochastica 7 (1983) 217-227.

[3] F. Esteva and N. Piera, Classification of the regular De Morgan algebras of fuzzy sets, Stochastica $ (1984) 249-265.

[4] P. Garcia and L. Valverde, Sobre una classe de tames de De Morgan, Acres de! IV Congr~s Catald de LOgica, 1984, Barcelona (1985) 73-76.

[5] J. Kamp6 de F6fiet, Une 6quation functionelle de la th6orie de Finfonnation g6n6ra|i~nt F6quafion de Caochy, Demonstratio Mathematica 6 (1973) 665-685,

[6] E.P. K|ement, Operations on fuzzy sets and fuzzy numbers related to triangular norms, Proc. of Xlth LS.M.V.L., Oklahoma (1981) 218-225.

[7] C.H. Ling, Representation of Associative Functions. Publ. Math. Decebren 12 (1965) 182-212. [8] B. Schweizer and A. Sklar, Probabilisti¢ Metric Spaces (North-Hotland, Amsterdam, 1983). [9] E. Trillas, Sobre funciones de negaci6n en la teofla de conjuntos difusos, Stochasgca 3 (1979)

47-60.

36 P. Garc~, L. Valverd¢

[I0] E. TriHas, C. A|sina and L. Valverde, Do we need Max, Min and I -j in Fuzzy Set Theory?, in: R.R. Yaser, Ed., Fuzzy Set and Possibil~y Theory: Recent Developments (Pergamon Press, New York, 1982) 275-287.

[11] L. Valverde, Contribuci6 a l'estudi deis models maternities per a 16giques multivalents, Ph.D. Thesis, Urfiversitat Polit~cnica de Catalunya Barcelona (1982).