769–785 DOI: 10.18514/MMN.2018.2637 DERIVATIONS IN ŁUKASIEWICZ SEMIRINGS IVAN CHAJDA AND HELMUT L ¨ANGER Received 29 May, 2018 Abstract

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Vol. 19 (2018), No. 2, pp. 769–785 DOI: 10.18514/MMN.2018.2637

DERIVATIONS IN ŁUKASIEWICZ SEMIRINGS

IVAN CHAJDA AND HELMUT L ¨ANGER Received 29 May, 2018

Abstract. An axiomatization of classical propositional logic is provided by means of Boolean algebras which are term equivalent to Boolean rings. This is important because rings form a classical part of algebra whose tools can be used for the investigations. The Łukasiewicz many- valued logic was axiomatized via so-called MV-algebras by C. C. Chang in 1950’s. MV-algebras are successfully applied in the logic of quantum mechanics and hence they are considered as quantum structures. It is a natural question if also MV-algebras have their alter ego among classical structures. For this reason the so-called Łukasiewicz semirings were introduced by the first author and his collaborators in [3] – [4]. As shown, Łukasiewicz semirings are term equivalent to MV-algebras and we can use with advantage several developed tools for their study.

In particular, we investigate derivations in semirings which were introduced formerly but here these semirings are enriched by an involution.

2010Mathematics Subject Classification: 13N15; 16Y60; 06D35 Keywords: Łukasiewicz semiring, MV-algebra, derivation

1. INTRODUCTION

The concept of derivation in rings was introduced already in the 1960’s. For semirings, it was defined in J. S. Golan’s book ([9]). However, derivations were studied also in lattices (see e.g. [7] and [11]). Let us note that bounded distributive lattices are special semirings. Starting with 2010, several authors extended the study of derivations to certain algebras forming an algebraic semantic for non-classical logics including the logic of quantum mechanics. In this context let us mention the papers [1] and [10]. It was shown in [5] that every MV-algebra can be converted into a so- called Łukasiewicz semiring. The advantage of this approach is that we can apply also methods from the theory of semirings. This was done by the authors in [6].

The main difference between our approach to derivations in Łukasiewicz semirings and the approach by J. S. Golan in [9] is that in Łukasiewicz semirings we have a unary operation such that addition can be expressed by both multiplication and this

Support of the research of both authors by ¨OAD, project CZ 04/2017, of the first author by IGA, project PˇrF 2018 012, and of the second author by the Austrian Science Fund (FWF), project I 1923- N25, is gratefully acknowledged.

c 2018 Miskolc University Press

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unary operation. Moreover, Łukasiewicz semirings are additively idempotent and ordered by the induced semilattice ordering, see [1] – [4]. On the other hand, these semirings are not lattices and hence we cannot use results and methods from [7] or [11].

2. ŁUKASIEWICZ SEMIRINGS

We start with the following definition:

Definition 1. A Łukasiewicz semiringis an algebraRD.R;C;;⁰; 0; 1/of type .2; 2; 1; 0; 0/such that

.R;C; 0/is a commutative monoid, .R;; 1/is a commutative monoid, .xCy/´x´Cy´,

x00,

xCxxandxC11, .x⁰y/⁰y.y⁰x/⁰x, xyimpliesy⁰x⁰, .x⁰/⁰x.

An algebra.R;C;; 0; 1/of type.2; 2; 0; 0/satisfying the first four conditions is called a commutative semiring. So .R;C/ can be considered a join-semilattice and we denote byits corresponding partial order relation. We call it theinduced orderof R. Put

x^yWD.x⁰Cy⁰/⁰

for allx; y2R. Since.R;C; 0; 1/is a bounded join-semilattice and⁰is an antitone involution on.R;/,.R;C;^; 0; 1/is a bounded lattice.

Example1. The algebraRD.Œ0; 1;_;ˇ;⁰; 0; 1/with x_yWDmax.x; y/;

xˇyWD.xCy 1/_0;

x⁰WD1 x

for allx; y2Œ0; 1is a Łukasiewicz semiring. HereŒ0; 1denotes the unit interval of the real numbers.

Example2. For every positive integernthe algebraRnD.f0; 1; 2; : : : ; ng;_;ˇ;

0; 0; n/with

x_yWDmax.x; y/;

xˇyWD.xCy n/_0;

x⁰WDn x

for allx; y D0; : : : ; nis a Łukasiewicz semiring. Note that in this case the greatest element isn.

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Example3. Every Boolean algebraBD.B;_;^;⁰; 0; 1/is a Łukasiewicz semiring.

Lemma 1. Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring and a; b; c2R.

Then

(i) 0⁰D1and1⁰D0, (ii) aa⁰D0,

(iii) ab impliesacbc, and henceaba^b, especially,aaa, (iv) a.aCb/⁰D0,

(v) aCbD..ab⁰/⁰b⁰/⁰, (vi) ab if and only ifab⁰D0.

Proof.

(i) This follows immediately from the fact that0is the smallest and1the greatest element of.R;/and that⁰is an antitone involution of.R;/.

(ii) According to (i) we haveaa⁰D.1a⁰/⁰a⁰D.a0/⁰0D0.

(iii) abimpliesaCbDbwhenceacCbcD.aCb/cDbc, i.e.acbc.

Sinceaba1Daandab1bDb, we haveaba^b.

(iv) According to (iii) and (ii) we have a.aCb/⁰.aCb/.aCb/⁰D0and hencea.aCb/⁰D0.

(v) According to (iv), (i) and (ii) we have

aCbD..a.aCb/⁰/⁰.aCb/⁰/⁰D...aCb/a⁰/⁰a⁰/⁰D..ba⁰/⁰a⁰/⁰D..ab⁰/⁰b⁰/⁰: (vi) Ifab then according to (iii) and (ii) we haveab⁰bb⁰D0 showing

ab⁰D0. If, conversely ab⁰D0 then according to (v) and (i) we have aaCbD..ab⁰/⁰b⁰/⁰Db.

An elementaofRis calledBooleanifaaDa. This is equivalent toaCa⁰D1 and to a^a⁰D0. Hence, a is Boolean if and only if a⁰ has this property. The mentioned equivalence can be seen as follows: IfaaDathenaCa⁰D..a.a⁰/⁰/⁰ .a⁰/⁰/⁰D..aa/⁰a/⁰D.a⁰a/⁰D0⁰D1. If, conversely, aCa⁰D1 then aaD aaC0DaaCaa⁰Da.aCa⁰/Da1Da. Let BoolRdenote the set of all Boolean elements ofR. It is easy to see that BoolRis a subuniverse of.R;;⁰; 0; 1/.

AnMV-algebrais an algebra.A;˚;:; 0/of type.2; 1; 0/satisfying the following identities:

.x˚y/˚´x˚.y˚´/;

x˚yy˚x;

x˚0x;

x˚11;

:.:x/x;

:.:x˚y/˚y :.:y˚x/˚x:

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Here and in the following1WD :0. We define x˝yWD :.:x˚ :y/;

x_yWD :.:x˚y/˚y;

x^yWD :.:x_ :y/

for allx; y 2A. Then˝is distributive with respect to _. Moreover, for x; y2A we definexyif:x˚yD1. Then.A;_;^; 0; 1/is a bounded distributive lattice whose induced order coincides with. Moreoverxyimplies:y :x(x; y2A).

It is known that Łukasiewicz semirings are in a natural one-to-one correspondence with MV-algebras (cf. e.g. [2], [3] and [8]). For the convenience of the reader we provide a short direct proof.

Theorem 1. IfRD.R;C;;⁰; 0; 1/is a Łukasiewicz semiring and x˚yWD.x⁰y⁰/⁰

for all x; y 2R thenM.R/WD.R;˚;⁰; 0/ is anMV-algebra. If, conversely, AD .A;˚;:; 0/is anMV-algebra thenR.A/WD.A;_;˝;:; 0; 1/is a Łukasiewicz semiring. This correspondence is one-to-one.

Proof. First assumeRD.R;C;;⁰; 0; 1/to be a Łukasiewicz semiring and define x˚yWD.x⁰y⁰/⁰for allx; y2R. Then we have

.x˚y/˚´...x⁰y⁰/⁰/⁰´⁰/⁰..x⁰y⁰/´⁰/⁰.x⁰.y⁰´⁰//⁰ .x⁰..y⁰´⁰/⁰/⁰/⁰x˚.y˚´/;

x˚y.x⁰y⁰/⁰.y⁰x⁰/⁰y˚x;

x˚0.x⁰0⁰/⁰.x⁰1/⁰.x⁰/⁰x;

x˚1.x⁰1⁰/⁰.x⁰0/⁰0⁰1;

.x⁰/⁰x;

.x⁰˚y/⁰˚y...x⁰/⁰y⁰/⁰/⁰/⁰y⁰/⁰...x⁰/⁰y⁰/⁰y⁰/⁰...y⁰/⁰x⁰/⁰x⁰/⁰ ...y⁰/⁰x⁰/⁰/⁰/⁰x⁰/⁰.y⁰˚x/⁰˚x:

HenceM.R/D.R;˚;⁰; 0/is an MV-algebra. Conversely, letAD.A;˚;:; 0/be an MV-algebra. Then.A;_; 0/is a commutative monoid. Moreover,

.x˝y/˝´ :.:.:.:x˚ :y//˚ :´/ :..:x˚ :y/˚ :´/

:.:x˚.:y˚ :´// :.:x˚ :.:.:y˚ :´///x˝.y˝´/;

x˝y :.:x˚ :y/ :.:y˚ :x/y˝x;

x˝1 :.:x˚ :1/ :.:x˚0/ :.:x/x:

Hence,.A;˝; 1/is a commutative monoid, too. It is clear that the partial order rela- tions inAandR.A/coincide. Now fora; b2A,abimplies:b :a. Moreover,

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we have

.x_y/˝´.x˝´/_.y˝´/;

x˝0 :.:x˚ :0/ :.:x˚1/ :10;

x_xx;

x_11;

:.:x˝y/˝y :.:.:.:.:.:x/˚ :y///˚ :y/ :.:.:.:x/˚ :y/˚ :y/

:.:.:.:y/˚ :x/˚ :x/ :.:.:.:.:.:y/˚ :x///˚ :x/

:.:y˝x/˝x:

This shows that R.A/ D .A;_;˝;:; 0; 1/ is a Łukasiewicz semiring. If RD.R;C;;⁰; 0; 1/is a Łukasiewicz semiring,M.R/WD.R;˚;⁰; 0/andR.M.R//D .R;_;˝;⁰; 0; 0⁰/then

x_y.x⁰˚y/⁰˚y...x⁰/⁰y⁰/⁰/⁰/⁰y⁰/⁰..xy⁰/⁰y⁰/⁰xCy;

x˝y.x⁰˚y⁰/⁰...x⁰/⁰.y⁰/⁰/⁰/⁰xy;

0⁰1;

i.e. R.M.R// D R. If, finally, A D .A;˚;:; 0/ is an MV-algebra, R.A/WD.A;_;˝;:; 0; 1/andM.R.A//D..A;˚¹;:; 0/then

x˚¹y :.:x˝ :y/ :.:.:.:x/˚ :.:y///x˚y

and henceM.R.A//DA.

Due to this correspondence, for every Łukasiewicz semiringRD.R;C;;⁰; 0; 1/, .R;C;^; 0; 1/is a bounded distributive lattice.

We are going to show when an interval in a Łukasiewicz semiring can be converted into a bounded lattice with an antitone involution.

Lemma 2. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring andb2BoolRand put

xWDx⁰^b;

x\yWD.xCy/

for all x; y2Œ0; b. Then.Œ0; b;C;\;; 0; b/ is a bounded lattice with an antitone involution.

Proof. Sinceb2BoolR, i.e.b^b⁰D0, we have

.x/D.x⁰^b/⁰^bD.xCb⁰/^bD.x^b/C.b⁰^b/DxC0Dx for allx2Œ0; b. Obviously,is antitone. The rest of the proof is clear.

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Lemma 3. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring anda2BoolRand put

xWDx⁰Ca;

x\yWD.xCy/

for all x; y2Œa; 1. Then.Œa; 1;C;\;; a; 1/is a bounded lattice with an antitone involution.

Proof. Sincea2BoolR, i.e.aCa⁰D1, we have

.x/D.x⁰Ca/⁰CaD.x^a⁰/CaD.xCa/^.a⁰Ca/Dx^1Dx for allx2Œa; 1. Obviously,is antitone. The rest of the proof is clear.

Theorem 2. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring anda; b2BoolR, assumeaband put

xWD.x⁰Ca/^b;

x\yWD.xCy/

for allx; y 2Œa; b. Then.Œa; b;C;\;; a; b/is a bounded lattice with an antitone involution.

Proof. This follows from the fact that.R;C;^/is a distributive lattice.

Remark1. Theorem2remains valid if one definesxWD.x⁰^b/Cafor allx2 Œa; b. Note that.x⁰^b/Ca.x⁰Ca/^bfor allx2R.

3. DERIVATIONS

Although Łukasiewicz semirings are in a one-to-one correspondence with MV- algebras, we do not define derivations like in [1] or [10] since there the MV-operation

˚is used instead ofC. Hence also our results differ essentially from those obtained in the mentioned papers.

Now we can adopt the definition of a derivation from [9].

Definition 2. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring. Aderivationon Ris a mappingd fromRtoRsatisfying the following identities:

d.xCy/dxCdy;

d.xy/.dx/yCx.dy/:

Let DerR denote the set of all derivations onR, for everya2R letdNa denote the mapping fromR toR defined bydNa.x/WDax for all x2Rand let id denote the identical mapping fromRtoR. We call aderivationd onRprincipalif it is of the formdNa for somea2R. Hence0D Nd0and idD Nd1are principal derivations. For any mappingd fromRtoRdefine FixdWD fx2RjdxDxg.

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Here and in the following we writedxinstead ofd.x/if no confusion can arise.

Lemma 4. Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring, a; b 2R and d; d1; d22DerR. Then

(i) dNa2DerR, (ii) d 0D0, (iii) daa,

(iv) ab impliesdadb, (v) a.d1/da,

(vi) d1D1if and only ifd Did,

(vii) .d1a/.d2b/C.d1b/.d2a/.d1ıd2/.ab/whereıdenotes the composi- tion of mappings,

(viii) d1 ı d2 2 DerR if and only if for all x; y 2 R we have .d1x/.d2y/C.d1y/.d2x/..d1ıd2/x/yCx..d1ıd2/y/.

Proof.

(i) This is clear.

(ii) We haved 0Dd.00/D.d 0/0C0.d 0/D0.

(iii) According to (ii) and Lemma1(ii) we have .da/a⁰d.aa⁰/Dd 0D0

and hence.da/a⁰D0whencedaaaccording to Lemma1(vi).

(iv) ab impliesdadaCdbDd.aCb/Ddb.

(v) We havea.d1/d.a1/Dda.

(vi) Ifd1D1then aDa.d1/d.a1/Ddaa according to (iii) showing dDid. The converse is trivial.

(vii) We have

.d1a/.d2b/C.d1b/.d2a/

.d1a/.d2b/C.d1b/.d2a/C..d1ıd2/a/bCa..d1ıd2/b/D D.d1.d2a//bC.d2a/.d1b/C.d1a/.d2b/Ca.d1.d2b//D

Dd1..d2a/b/Cd1.a.d2b//Dd1..d2a/bCa.d2b//Dd1.d2.ab//D D.d1ıd2/.ab/:

(viii) According to the proof of (vii) we have

.d1ıd2/.ab/D.d1a/.d2b/C.d1b/.d2a/C..d1ıd2/a/bCa..d1ıd2/b/:

Lemma 5. Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring and d 2 DerR.

Then the following are equivalent:

(i) d is principal,

(ii) dxDx.d1/for allx2R, i.e.d D Ndd1.

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Moreover, ifx^.d1/x.d1/for allx2Rthend is principal.

Proof. The equivalence of (i) and (ii) is clear. If, finally,x^.d1/x.d1/for all x2Rthen

dxx^.d1/x.d1/d.x1/Ddx

for allx2Rand hencedD Nd_d1.

Lemma5shows that the Łukasiewicz semiring from Example3has only principal derivations since in this semiring multiplication coincides with lattice meet.

It was shown in [10], Lemma 5, that basic algebras have only principal derivations.

We are going to show that this is not the case for Łukasiewicz semirings. In these semirings there exist non-principal derivations, see the following example.

Example4. Consider the Łukasiewicz semiringRD.f0; a; b; c; 1g;C;;⁰; 0; 1/with the following operations:

C 0 a b c 1

0 0 a b c 1

a a a b c 1

b b b b c 1

c c c c c 1

1 1 1 1 1 1

0 a b c 1

0 0 0 0 0 0

a 0 0 0 0 a

b 0 0 0 a b

c 0 0 a b c

1 0 a b c 1

x 0 a b c 1

x⁰ 1 c b a 1

The principal derivations look as follows:

x 0 a b c 1

dN₀x 0 0 0 0 0 dNax 0 0 0 0 a dN_bx 0 0 0 a b dNcx 0 0 a b c dN1x 0 a b c 1

However, there exists a derivationd ofRwhich is not principal, namely the following:

x 0 a b c 1

dx 0 0 0 a a

The Łukasiewicz semiring considered in Example4 is isomorphic to the Łuka- siewicz semiring R4 from Example 2. The following proposition shows that the situation described in Example4can be generalized toRnfor arbitraryn > 1.

Proposition 1. For eachn > 1the Łukasiewicz semiringRnhas the non-principal derivationd defined by

d.x/WD

0 ifx < n 1 1 otherwise:

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Proof. Leta; b2 f0; : : : ; ng. Thend.a_b/D.da/_.db/since.f0; 1; : : : ; ng;/ is a chain. Moreover,

d.aˇb/D

0 ifaCb < 2n 1 1 otherwise;

.da/ˇbD

0 ifa < n 1orb < n 1 ifan 1andbDn;

aˇ.db/D

0 ifa < norb < n 1 1 ifaDnandbn 1:

Henced 2DerRn. Since dNa.b/D

0 ifbn a

b .n a/ otherwise;

d is not principal.

For the unary operation⁰, we can prove the following theorem:

Theorem 3. Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring andd 2DerR.

Then

(i) dx⁰D.dx/⁰for allx2Rif and only ifdDid,

(ii) .dx/.dx⁰/x.dx⁰/x⁰.dx/0and hencedx⁰.dx/⁰for allx2R, (iii) ifa2BoolRthendaCda⁰Dd1,

(iv) d..d1/⁰/.d1/^.d1/⁰,

(v) ifd12BoolRthend..d1/⁰x/D0for eachx2R.

Proof.

(i) Ifdx⁰D.dx/⁰for allx2Rthen for allx2Rwe havexD.x⁰/⁰.dx⁰/⁰D ..dx/⁰/⁰Ddxxaccording to Lemma4(iii) and hencedDid.

(ii) Due to Lemma4 (iii) and Lemma 1 (iii) and (ii) we have.dx/.dx⁰/x .dx⁰/xx⁰D0andx⁰.dx/x⁰xD0for allx2R. By Lemma1(vi), dx⁰.dx/⁰for allx2R.

(iii) Ifa2BoolRthendaCda⁰Dd.aCa⁰/Dd1.

(iv) Since .d1/⁰1, we have d..d1/⁰/d1 by Lemma 4 (iv). According to Lemma4(iii) we haved..d1/⁰/.d1/⁰. This showsd..d1/⁰/.d1/^.d1/⁰. (v) Ifd12BoolRthen by (iv), Lemma4(iv) and Lemma1(ii) we haved..d1/⁰ x/D.d..d1/⁰//xC.d1/⁰.dx/ D0xC.d1/⁰.dx/D .dx/.d1/⁰ .d1/.d1/⁰D0for allx2R.

Theorem 4. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,a2R andd 2 DerR. Then

(i) .DerR;C/ is a join-semilattice with the least element dN0 and the greatest elementid,

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(ii) Fixd is a subuniverse of.R;C;; 0/, (iii) .Fixd /\d ¹.f0g/D f0g,

(iv) ifa2BoolRthendNa2End.R;C;; 0/.

Proof.

(i) It is easy to see that DerRis closed with respect toC. Hence.DerR;C/is a semilattice which is considered as a join-semilattice. We then haved1d2if and only ifd1xd2xfor allx2R. Obviously,dN0;id2DerRanddN0is the smallest element of.DerR;/. Because of Lemma4(iii), id is the greatest element of.DerR;/.

(ii) Ifa; b2Fixd thend.aCb/DdaCdbDaCband d.ab/D.da/bCa.db/DabCabDab:

Moreover,d 0D0according to Lemma4(ii).

(iii) a2.Fixd /\d ¹.f0g/impliesaDdaD0.

(iv) This can be proved by a straightforward computation.

For Łukasiewicz semirings we adopt the concept of an ideal as introduced for semirings in [9]. Hence, we define

Definition 3. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring. AnidealofR is a subsetI ofRsatisfying

02I;

ifx; y2I thenxCy2I;

ifx2I andy2Rthenxy2I:

Let IdRdenote the set of all ideals ofR.

It is well known and easy to prove that.IdR;/is a complete lattice the with the smallest elementf0gand the greatest elementR.

Proposition 2. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,d 2DerRand I 2IdR. Then

(i) d ¹.f0g/2IdR,

(ii) d ¹.I /WD fx2Rjdx2Igis a subuniverse of.R;C;; 0/.

Proof.

(i) We have 02d ¹.f0g/. Moreover, d ¹.f0g/ is closed with respect to C. Ifa2d ¹.f0g/and b2Rthen d.ab/d.a1/DdaD0according to Lemma1(iii) and Lemma4(iv) and henced.ab/D0, i.e.ab2d ¹.f0g/.

(ii) The proof is straightforward.

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Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring and 2ConR. Denote by Œ0 the class of 0with respect to , the so-calledkernelof. It is well-known that in any semiring, the kernel of any congruence is an ideal, but not every ideal is a kernel of some congruence, see e.g. [9]. Now letd 2DerR. We are interested in the question ifd ¹.Œ0/2IdR.

Theorem 5. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,2ConRand d 2DerR. Then

(i) d ¹.Œ0/WD fx2Rjdx2Œ0g 2IdR,

(ii) d.Œ0/Œ0andd.Œ0/is a subuniverse of.R;C; 0/.

Proof.

(i) IfIWDd ¹.Œ0/,a; b2I andc2Rthenda; db2Œ0 and hence d.aCb/DdaCdb2Œ0;

d.ac/Dd.ac/C02Œd.ac/CdaDŒdaDŒ0

according to Lemma1(iii) and Lemma4(iv) showingaCb; ac2I. (ii) Clearly,0Dd 02d.Œ0/. Ifa; b2d.Œ0/then there existc; e2Œ0with

dcDaanddeDband hence

aDdcDdcC02ŒdcCcDŒcDŒ0;

aCbDdcCdeDd.cCe/2d.Œ0/

according to Lemma4(iii).

Theorem 6. Let RD.R;C;;⁰; 0; 1/ be a Łukasiewicz semiring,d 2DerR, as- sumed12BoolR, put

xWDx⁰^.d1/;

x\yWD.xCy/

for allx; y2d.R/and assumex2d.R/for allx2d.R/. Then.d.R/;C;\;; 0; d1/

is a bounded lattice with an antitone involution.

Proof. Sinced12BoolR, i.e..d1/^.d1/⁰D0, we have .x/D.x⁰^.d1//⁰^.d1/D.xC.d1/⁰/^.d1/

D.x^.d1//C..d1/⁰^.d1//DxC0Dx

for allx2d.R/. Obviously,is antitone. The rest of the proof is clear.

Theorem 7. LetRD.R;C;;⁰; 0; 1/andSD.S;C;;⁰; 0; 1/be Łukasiewicz semirings. ThenDer.RS/DDerRDerS.

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Proof. Clearly, DerRDerSDer.RS/. In order to prove the converse inclusion, letd 2Der.RS/. Then for each x2Rwe haved.x; 0/.x; 0/according to Lemma 4 (iii) and hence there exists some mapping d1 from R toR satisfying .d1x; 0/Dd.x; 0/for allx2R. Analogously, there exists some mappingd2fromS toSsatisfying.0; d2y/Dd.0; y/for ally2S. Now leta; b2Randc2S. Then

.d1.aCb/; 0/Dd.aCb; 0/Dd..a; 0/C.b; 0//Dd.a; 0/Cd.b; 0/D D.d1a; 0/C.d1b; 0/D.d1aCd1b; 0/;

i.e.d1.aCb/Dd1aCd1b. Moreover,

.d1.ab/; 0/Dd.ab; 0/Dd..a; 0/.b; 0//D.d.a; 0//.b; 0/C.a; 0/.d.b; 0//D D.d1a; 0/.b; 0/C.a; 0/.d1b; 0/D..d1a/b; 0/C.a.d1b/; 0/D D..d1a/bCa.d1b/; 0/;

i.e.d1.ab/D.d1a/bCa.d1b/. This showsd12DerR. Analogously,d22DerS.

Finally,

d.a; c/Dd..a; 0/C.0; c//Dd.a; 0/Cd.0; c/D.d1a; 0/C.0; d2c/D.d1a; d2c/

provingd 2DerRDerS. Hence Der.RS/DerRDerScompleting the proof

of the theorem.

Proposition 3. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring anda2BoolR.

Then.dNa.R/;C;; 0; a/is a commutative semiring satisfyingxCxxandxCa a.

Proof. Since dNa.xCy/D NdaxC Nday, dNa.xy/Da.xy/D.ax/.ay/D .dNax/.dNay/ for all x; y 2R and dNa0Da0D0 and dNa1 Da1 Da, dNa is a homomorphism from.R;C;; 0; 1/onto.dNa.R/;C;; 0; a/.

4. .f; g/-DERIVATIONS

The concept of an.f; g/-derivation is mentioned in the monograph [9]. For Łukasi- ewicz semirings, it is specified as follows:

Definition 4. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring andf; g2EndR.

An .f; g/-derivation on R is a mapping d from R to R satisfying the following identities:

d.xCy/dxCdy;

d.xy/.dx/f .y/Cg.x/.dy/:

Let Derf;gR denote the set of all.f; g/-derivations on R. Especially, Derid;idRD DerR.

The following example shows that there exist .f; g/-derivations on Łukasiewicz semirings which are not derivations.

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Example 5. LetM be a set with jMj> 1and put RWD.2^M;[;\;⁰;¿; M /. If f 2EndRthenf 2Derf;f Rsince

f .A[B/Df .A/[f .B/;

f .A\B/Df .A\B/[f .A\B/D.f .A/\f .B//[.f .A/\f .B//

for allA; B22^M. Now leta2M andgdenote the mapping from2^M to2^M defined by

g.A/WD

M ifa2A;

¿ otherwise

(A22^M). Theng2EndR(and henceg2Derg;gR) as can be seen from the following table:

A B g.A/ g.B/ g.A[B/ g.A\B/ g.A⁰/ 3a 3a M M M M ¿ 3a 63a M ¿ M ¿ ¿ 63a 3a ¿ M M ¿ M 63a 63a ¿ ¿ ¿ ¿ M Butg…DerRsince

g.fag \ fxg/D¿¤ fxg D.g.fag/\ fxg/[.fag \g.fxg//

for allx2Mn fag.

Lemma 6. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,f; g2EndR,a; b2 Randd; d1; d22Derf;gR. Then

(i) dN02Derf;gR, (ii) d 0D0,

(iii) daf .a/^g.a/, (iv) ab impliesdadb,

(v) .f .a/Cg.a//.d1/da, (vi) d1D1if and only ifd Df Dg,

(vii) .d1.g.a///f .d2b/C.d1.f .b///g.d2a/.d1ıd2/.ab/,

(viii) if f ıf Df andgıgDg then d1ıd22Derf;gR if and only if for all x; y2Rwe have.d1.g.x///f .d2y/C.d1.f .y///g.d2x/..d1ıd2/x/

f .y/Cg.x/..d1ıd2/y/.

Proof.

(i) This is clear.

(ii) We haved 0Dd.00/D.d 0/f .0/Cg.0/.d 0/D.d 0/0C0.d 0/D0.

(iii) According to (ii) and Lemma1(ii) we have

.da/.f .a//⁰D.da/f .a⁰/d.aa⁰/Dd 0D0

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and hence.da/.f .a//⁰D0whencedaf .a/according to Lemma1(vi).

Analogously, according to (ii) and Lemma1(ii) we have .da/.g.a//⁰Dg.a⁰/.da/d.a⁰a/Dd 0D0

and hence.da/.g.a//⁰D0whencedag.a/according to Lemma1(vi).

(iv) can be proved exactly as Lemma4(iv).

(v) We have

.f .a/Cg.a//.d1/D.d1/f .a/Cg.a/.d1/d.1a/Cd.a1/DdaCdaDda:

(vi) Ifd1D1then

f .a/D.d1/f .a/d.1a/Ddaf .a/;

g.a/Dg.a/.d1/d.a1/Ddag.a/

according to (iii) showingd Df Dg. The converse is trivial.

(vii) We have

.d1.g.a///f .d2b/C.d1.f .b///g.d2a/

.d1.g.a///f .d2b/C.d1.f .b///g.d2a/C C..d1ıd2/a/f .f .b//Cg.g.a//..d1ıd2/b/D D.d1.d2a//f .f .b//Cg.d2a/.d1.f .b///C C.d1.g.a///f .d2b/Cg.g.a//.d1.d2b//D

Dd1..d2a/f .b//Cd1.g.a/.d2b//Dd1..d2a/f .b/Cg.a/.d2b//D Dd1.d2.ab//D.d1ıd2/.ab/:

(viii) Iff ıf Df andgıgDgthen according to the proof of (vii) we have .d1ıd2/.ab/D.d1.g.a///f .d2b/C.d1.f .b///g.d2a/C

C..d₁ıd₂/a/f .b/Cg.a/..d₁ıd₂/b/:

For the unary operation⁰, we can prove the following theorem:

Theorem 8. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,f; g2EndRand d 2Der_f;gR. Then

(i) dx⁰D.dx/⁰for allx2Rif and only ifdDf Dg,

(ii) .dx/.dx⁰/f .x/.dx⁰/f .x⁰/.dx/g.x/.dx⁰/g.x⁰/.dx/0 and hencedx⁰.dx/⁰for allx2R,

(iii) ifa2BoolRthendaCda⁰Dd1.

Proof.

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(i) Ifdx⁰D.dx/⁰for allx2Rthen for allx2Rwe have f .x/D.f .x⁰//⁰.dx⁰/⁰Ddxf .x/;

g.x/D.g.x⁰//⁰.dx⁰/⁰Ddxg.x/

according to Lemma6(iii) and henced Df Dg.

(ii) Due to Lemma6(iii) and Lemma1(iii) and (ii) we have

.dx/.dx⁰/f .x/.dx⁰/f .x/f .x⁰/Df .xx⁰/Df .0/D0;

f .x⁰/.dx/f .x⁰/f .x/Df .xx⁰/Df .0/D0

for allx2R. The proof forg is analogous. By Lemma1(vi),dx⁰.dx/⁰ for allx2R.

(iii) can be proved exactly as Theorem3(iii).

Theorem 9. LetRD.R;C;;⁰; 0; 1/be a Łukasiewicz semiring,f; g2EndRand d 2Derf;gR. Then

(i) .Der_f;gR;C/is a join-semilattice with the least elementdN0, (ii) .Fixd /\d ¹.f0g/D f0g.

Proof.

(i) It is easy to see that Derf;gRis closed with respect toC. Hence.Derf;gR;C/ is a semilattice which is considered as a join-semilattice. We then haved1 d2if and only ifd1xd2xfor allx2R. Obviously,dN02Derf;gRanddN0

is the smallest element of.Der_f;gR;/.

(ii) can be proved exactly as Theorem4(iii).

It is clear that Proposition2, Theorem5(i) and Theorem6remain valid for.f; g/- derivations, too.

Theorem 10. Let RD.R;C;;⁰; 0; 1/ and SD.S;C;;⁰; 0; 1/ be Łukasiewicz semirings,f1; g12EndRandf2; g22EndS. Then.f1; f2/; .g1; g2/2End.RS/

and

Der.f1;f2/;.g1;g2/.RS/DDerf1;g1RDerf2;g2S:

Proof. Clearly,.f1; f2/; .g1; g2/2End.RS/and

Derf₁;g₁RDerf₂;g₂SDer.f₁;f₂/;.g₁;g₂/.RS/:

In order to prove the converse inclusion, letd 2Der_.f₁_;f₂_/;.g₁_;g₂_/.RS/. Then for eachx2Rwe haved.x; 0/.f1.x/; 0/according to Lemma6(iii) and hence there exists some mapping d1 from R toR satisfying.d1x; 0/Dd.x; 0/ for all x2R.

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Analogously, there exists some mappingd2fromStoSsatisfying.0; d2y/Dd.0; y/

for ally2S. Now leta; b2Randc2S. Then

.d1.aCb/; 0/Dd.aCb; 0/Dd..a; 0/C.b; 0//Dd.a; 0/Cd.b; 0/D D.d1a; 0/C.d1b; 0/D.d1aCd1b; 0/;

i.e.d1.aCb/Dd1aCd1b. Moreover,

.d1.ab/; 0/Dd.ab; 0/Dd..a; 0/.b; 0//D

D.d.a; 0//.f1.b/; 0/C.g1.a/; 0/.d.b; 0//D D.d1a; 0/.f1.b/; 0/C.g1.a/; 0/.d1b; 0/D D..d1a/f1.b/; 0/C.g1.a/.d1b/; 0/D D..d1a/f1.b/Cg1.a/.d1b/; 0/;

i.e.d1.ab/D.d1a/f1.b/Cg1.a/.d1b/. This showsd12Derf1;g1R. Analog- ously,d22Derf2;g2S. Finally,

d.a; c/Dd..a; 0/C.0; c//Dd.a; 0/Cd.0; c/D.d1a; 0/C.0; d2c/D.d1a; d2c/

provingd2Derf1;g1RDerf2;g2S. Hence

Der.f₁;f₂/;.g₁;g₂/.RS/Derf₁;g₁RDerf₂;g₂S

completing the proof of the theorem.

ACKNOWLEDGEMENT

We thank the referee for his valuable suggestions which made the paper better readable.

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Authors’ addresses

Ivan Chajda

Palack´y University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic

E-mail address:ivan.chajda@upol.cz

Helmut L¨anger

TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geo- metry, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria, and, Palack´y University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic

E-mail address:helmut.laenger@tuwien.ac.at