In this paper we introduce the tensor product of partial algebras w.r.t. a quasi-primtive class of partial algebras, and we prove some of its main properties. This construction generalizes the well-known tensor product of total algebras w.r.t. varieties

Monserrat, M.

Rosselló, F.

Torrens. J.

Monserrat, Miquel

Rosselló, Francesc

Torrens, Joan

Diposit Digital de Documents de la UAB

Publicacions Matematiques, Vol 36 (1992), 773-,787.A bstractTENSOR PRODUCTSOF PARTIAL ALGEBRASM . MONSERRA,r, F . ROSSFLLÓ1 AND .) . TORRDNSIn this paper we introduce tlte tensor product of l artial algebrasw.r .C . a gttasi-printitive class of partial algebras, and we provesorna of its main properties . This construction generalizas Chesvell-known tensor product of total algebras w.r .t . varieties .t Partially supported by UPC PR901A .To Pere Mercal, i?a rnemoriamIn [Fra], the tensor product on a, variety is introduced, and its rrtainproperties are studied in [Gal] .The ma,in goal of this paper is to show the existente of the tensorproduct of two partial algebras with respect to a class of partial alge-bras (of the sarne type) closed under isomorphic images, direct productsand closed subalgebras . Then, the a,forernentioned tensor product on avariety becomes a special case of our construction .Moreover, we prove that one of the rrtain properties of the tensor prod-uct on a variety (Thm . 3 in [Gal]) still holds for tensor products of partialalgebras . Finally, we use soma general results from [BMRRT] to giveexplicit computations of Che tensor product w.r .t . a class M when suchM is either the class of all weak subalgebras of algebras in a variety K,or the class of al] partial algebras which satisfy weakly the equationaltheory of a variety K.Our interest in tensor products (for partial algebras) sterris from Chestudy of the behaviovir under it of some constructions and propertiesfor termal algebras (algebraic models for systems of inference roles onincompleta data; sea [BR] for a recent survey) . We will report on thisCopie elsewhere .774	M. MONSERRAT, F. ROSSELLÓ, J . TORRENSIn this section we gather some definitions and results which shall beused in this paper. For any other terminology not introduced in thissection we refer the reader to [Bur] (for partial algebras) and [BuS] (fortotal algebras) .Definition 1 . Let M be a non-empty class of partial algebras of thesame type .a) M is quasi-primitive iff it is closed under isomorplúc images,closed subalgebras and direct products .b) M is weakly quasi-primitive iff it is closed under isomorphicimages, weak subalgebras and direct products .For instante, if K is a variety of total algebras ; then K is quasi-primitive, while the classesW(K) ={A 1 A is isomorphic to a weak subalgebra of an algebra in K}and1 . PreliminariesE(K) = {A 1 A ~--, Egx(K)}(where Hti, denotes "weak satisfaction" in the sense of, for instante,[Ru], and Egx(K) denotes the equational theory of K over a (countablyinfinite) set of variables X) are weakly quasi-primitive . 4Ve shall callthem the weak subalgebras class and the weak equational class ofK, respectively.The following result, due to J. Schmidt (see [Sch]) states that quasi-primitive classes admit "universal solutions" .Theorem 1 . Let A be an arbitrary partial algebra and M a quasi-primitive class of algebras of the same type . Then there exists an algebraB E M and a homomorphism h : A -> B satisfying the following uni-versal property :for any C E M and any homomorphism g : A -> C there exists aunique homomorphism g : B --> C such that g = gh:TENSOR PRODUCTS OI" PARCIAL ALGEBRAS	775It follows that such a homornorphism h. : A B is unique up toisomorphism .Definition 2. A homornorphism h : A --> B, with B E M, satisfyingthe universal property of the last theorem is called an AI-reflection ofA.As a byproduct of some results in [BMRRT] it turras out that suchara Ad-reflection can be explicitly computed for AJ a variety; the weaksubalgebras class of a variety and the weak equational class of a variety ;respectively. hor the conveniente of the reader, we recall llere (withoutproofs) the corresponding results .Let us fix henceforth a countably infinite set of variables X, and letT),(X) be the set of ternes of type .F over X .Theorem 2. Let A = (A, (fA)fE I-) be a partial algebra of type F =(F, 91) . Let K be a variety of total algebras of type .F and let E = Ecix(K)be the equational theory of K mr.t . X. Let llll = E(K) be the weakequational class of K .Consider the congruente ora ABA,M = BA,A1e xi>0zvhere BA'mr , i > 0, are defined recurrently as follows: 00 ,Al = idA and(a e) (a c ) (c b) ; (c b ) EOA,NI1 1 . . , 7n Tr+. 1 . i 1 . . . . TrL, 7n iP1, P2, ql, q2 E T.F(X) such that pl 'zd P2, ql ^ q2 E E,(a,, . . . , ano) E dorrl PA and pA(al, . . . , a,.,j = a,(bl, . . , b,n) E done ql and gA(bl, . . . , b,n) = b,(el, . . . , Crrt) E dom P2 n dorn q2 andP2(Cl . . . ., Grn) = q2 (el) . . , Can)Then the natural homomorphism nato, : A --> A/BA,n4 is ara M-reflectionof A.Theorem 3 . Let A = (A, (fA) fEr), K and E be as in the previoustheorem. Let M = W(K) be the weak subalgebras class of K . Let iA --> F(A) be the free completion ofA (see [Bur 1.5.4]) and let F(A)be the carrier of F(A) .776	M. MONSERRAT, P . ROSSELLÓ, J . TORRENSConsider the congruente on F(A)with 0F(A),K, i > 0, defned recurrently as follows : 0 (A),K = idA andF(al, cl), . . . , (am, cm), (Cl, bl), . . . , (Cm, bm) E 0t(A),KPi, P2,ql,q2 E TF(X) such that pl 'Zz! P2, ql ^ q2 E £Pl(A) (al, . . . , a,,) =a, ql(A) (bl, . . , b,n) = b,P2 (A)(el F(A) (Cl, . . . , Cm,),. . ., Crn) = q2Theni) The homomorphismis a K-reflection of A.ü) Let Bw ,M = Bé(A),K f1 A2 . Then the natural homomorphismis an M-reflection of A.BP(A),K =BF(A),K¡>onato, oí : A -> F(A)/0F(A),Knato_ : A -> Alo~,M2 . Tensor productsLet us start by introducing the notions of bihomomorphism and tensorproductDefinition 3. Civen partial algebras A = (A, (fA) fEF),B = (B, (fB )fEF) and C = (C, (f° ) fEF) of some type a bihomomorphism of A x B into C is a mapping h, : A x B C suchthat- for any b E B, h(-, b) : A -> C is a homomorphism frorrl A intoC, and- for any a E A, h(a, -) : B C is a homomorphism from B intoC.TENSOR PRODUCTS OP PARTIAL ALGEBRAS	777In other words,a) If f E F with rl(f) = 0 and dom fA =~ 0 then dom fc 9~ 0 too, andfor anybEBh(fA, b) = fcb) If f E F with 97(f) > 0 and (a,,. a,7 (f)) E dom fA for somef E F then for any b E Band in that case(h(al , b), . . . , h(aj(f), b)) E dom fcfc (h(ai, b), . . , h(an(f) , b)) = h(fA(al . . . . 3an(f))~ b)c) Similar statements hold interchanging the róles of B and A.Definition 4 . Let A and B be two similar partial algebras and letM be a class of algebras of the same type . Assume that there exists apartial algebra T of the same type such that(i) TE M(ii) There exists a bihomomorphism 7r : A x B -> T(iii) Given any bihomomorphism h : A x B -> C into some C E M,there exists a unique homomorphism h : T -> C such that h7r = hAxB i Th,CThen .we shall call such a T a tensor product of A and B w.r.t .Al (or AI-tensor product of A and B) .Notice that an algebra, satisfying the properties above is unique up toisomorphism, so henceforth we shall talk about the AI-tensor product ofA and B, and we shall denote it by A (DA,, B . Moreover, we sliall callthe bihomomorphism 7r the tensor bihomomorphism .Examples .1) If A and B are discrete algebras, then their tensor product w.r .t .the class of * all partial algebras of the same type is the discrete algebraover the cartesian product of their carriers .2) If AI is a variety of total algebras and A, B E M, then A ®n,,, B canbe taken as the tensor producton a variety as defined in [Fral .778	M. MONSERRAT, F . ROSSELLÓ, J . TOR,RENS3) Let F = {fo , fi} with 77(fo) = 0 and 97(f i ) = 1 . Consider the partialalgebras of type (F, 97) A = ({al , a2}, foA, fA) and B = ({ b l , b2}, .fo , fi')given byB :fB.fóIt is not difficult to check that the tensor product of A and B w.r .t .the class M = {A, B} does not exist .This last example shows that, in general, given arbitrary partial alge-bras A and B and class M, such NI-tensor product need not exist, evenif A, B E 117 . Our main goal is to prove that if M is quasi-primitive thenA (qnr B always exists .Let A = (A, (f')fEF) and B = (B, (f')fEF) be two partial alge-bras of type F = (F, 7l) . Let us consider a partial algebra A x B =(A_X B, (fAxB)fEF) with carrier setA x B = A x B U {f((a l , b), . . . , (a j (f), b)) 1 f E F,(a,, . . . , aO(f)) E dom fA , b E B}U {f((a, bi), . . . , (a, bn(f))) 1 f E F;(bi, . . . , bn(f) ) E dom fB, a E A}U {f 1 f E F, 7l(f)=0and either dom fA or dom fB are non-empty}(so, A x B is a subset of the set TF (A x B) of terms of type .F overA x B) and operations fAxB , f E F, defined by- If 71(f) = 0 then dom fAx B 0 iff either dom fA or dom fB arenon-empty, and in that case fAxB = f E A x B- If 9j (f) > 0 thendom fAxB= {((ai, b), . . . . , (an(f) , b))1(a,, . . . , an(f)) Edom fA, b E B}U {((a, b l ), . .-. , (a, bn(f))) I (b,,.b,,(f)) E dom fB , a E A}and if ((al, b), . . . , (an(f), b)) E dom fAxB thenfAxB((al,b), . . . , ( an(f) , b)) = f((al, b), . . . , (an(f)7 b))TENSOR PRODUCTS OF PARTIAL ALCEBRAS	779while if ((a, bl), . . . , (a, b,,(f))) E domfAxB thenfAXB((a bl ) . . . , (a, b,(f))) = f((a, bj), . . . , (a, b,(f)))In other words, A x B is the relative subalgebra of the algebra T.F(AxB) of terms of type .F over A x B, the carrier set of which is the set A x Bdefined above .Now consider en A x B the following relationeo ={ (f «al, b), . . , (a ,7(f) , b)), (fA(al, . . . , a,,(f)) , b)) If E F, (al, . . . , an(f)) E dom fA, b E B)}" «f((a, b l ), . . . , (a, bn(f))) , (a, f'(b,, . . . , b,(f))))f E F, (b1, . . . ,b,,(f)) E dom fB, a E A)}U«(fA,b),f) I f E F, 77(f)=0, fA EA, bEB}U{((a,fB),f) I f E F, 77 (f) =0, fB E B, aEA}and let B be the least congruente en Á x B containing e0 . SetA®B=A x BloLemma 4. A ® B is the tensor product ofA and B w.r. t. the classof all partial algebras of type F .Proof.. Let us check that such A ®B satisfies conditions (i)-(iii) in thedefinition of tensor product w.r .t . the class of all partial algebras of thegiven type .(i) A ® B is a partial algebra of type .F by definition .(ii) The mapping 7r : A x B -> A ® B given by 7r(a, b) = [(a, b)] (theequivalente class of (a, b) with respect te B) for any a E A, b E B is abihomomorphism .Indeed, let (a,, . . . , a,,(f)) E dom fA and b E B . ThenUai, b), . . . ) (a?7(f) , b)) E domfAxB(by construction) and thus(7r(ai, b), . . . ,7r(a,7(f), b)) = ([(al, b)], . . . , [(a,,(f) , b)l) E domfAOB780	M. MONSERRAT, F. ROSSELLÓ, J. TORRENSby the definition of the quotient algebra . Moreover,7r(fA(a,, . . . , a,,(f)) , b) _ [(f'(a,, . . , an(f)), b)) =_ j((ar, b), . . , (an(f), b))] _= lfA®B([((al, b)], . . . , [(a~,(f) , b)]) _= fA®B(7r(ar,b), . . . ,7r(a,(f)5b))One proves in a similar way that if (b,.; bn(f)) E dom fB and a E Athenand(7r(a, b,),., -ff (a, b,,(f))) E domfAOB7r(a, fB (bi1 . . . , bn(f))) = fA®B(n(a, bi), . . . , -ff (a, b,,(J)))To check the bihomomorphism condition for the O-ary operations isstraightforward .(iii) Now, let g : A x B -> C be any bihomomorphism . First of all,we see that it can be extended in a unique way to a homomorphismg : A x B ---> C . Indeed ; consider the mapping g : A x B -> C given byg(a, b) = g(a, b) if a E A; b E B9(f«al, (a,)(f)- b))) = g(fA(ar, . . . , a,7(f)) , b)if f E F, (a,,., a,7 (f)) E dom fA ; b E B9(f((a, bl ), . . . , (a, b,,(f)))) = g(a, f'(br ; . . . ; b,,(f)))if f E F; (b,,., b,(f )) E dom fB, a E Aj(f) =fC iffEF,r7(f)=0, fEAxBn(f)(Notice that, since g is a bihomomorphism, if f E F, (a a) Er(J)dom fA and b. 1 6) E dom fs then (g(a, b), . . . , g(a, b)) E dom fCandg(fA(a, . . . , a), b) = fC(g(a, b), _ . . , g(a, b)) = .9(a, f'(b, . . . , b)) :and that if f is a O-ary operation symbol such that f E A x B then eitherdom fA or dom f$ are non-empty, and it implies that dorn fC 0 0 too,again because g is a bihomomorphism.)TENSOR, PRODUCTS Or PARTIAL ALGEBRAS	78 1It is not difficult to prove that g is a hornomorphism from A x B intoC . For instante, we check below the case of ((al, b), . . . , (a,j (f), b)) Edora fAxs, leaving the remaining ones to the reader .If ((al , b), . . . , (a, j (f) 1 b)) E dom fA,13 then (a,,.a,](f)) E dorn fAand thus, since g is a bihomomorphism, (g(a l , b), . . . , g(a,,(f), b))E dom fand9(fAxB((al., b), . . . , (a,(f) , b))) = 9(f«al , b), . . . , (av(f) , b)))= g(fA(al . . . . ; an(f)) , b)= fC(g(al, b), . . . , .q(a,,(f), b))= fC(9(ai, b), . . . , 9(an(f) , b))It is also obvious that such a g extending g is unique . Finally, it isstraightforward from the definition that 00 C_ ker bb . So it induces aunique homornorphism g : A ® B - C which makes the diagramto commute .AxB 7 A®BgCThis lemma allows us Lo prove that the tensor product w.r .t . quasi-primitive classes always exists, using Schmidt's theorem .Theorem 5. Let A and B be two parcial algebras of type .'F and letM be a quasi-primitive class of partial algebras of type F. Let A ® B bedefined as before and let, h, : A® B -> D be an M-reflection of A ® B .Then D is the M-tensor product of A arad B .Proof.- Let h : A ® B -> D be a,n A7-reflection of A ® B. Then onone hand h7r : A x B -> D is a bihomomorphism and D E AL So weonly have to check that h7r satisfies che universal property (iii) in thedefinition of AI-tensor product .Let g : A x B --> C be any bihomomorphism into an algebra C E Al .By che previous lemma, there exists a unique homomorpl -iism g : A®B ->C making cominutative the following diagram782	M. MONSERRAT, F. ROSSELLÓ, J . TORRENSby Thm. 3.(i) .A x B 7r5 AOBCNow, by the universal property of M-reflections (see Theorem 1) thereexists a unique g : D -> C making commutative the diagrainAxB - > AOBh) D9CFrom this last diagram we get that there exists a homomorphism fromD into C ; namely g, such that g = gh7r .On the other side, it is not difficult to check (from the "uniqueness',of g given g, and of g given b) that q is the unique homomorphism withthis property . This ends the proof .Examples. Let A and B be two partial algebras of type .F, let A (D Bbe as defined above and let AO B be the carrier set of A OB.1) The free completion F(A ® B) of A O B is the tensor product ofA and B w.r .t . the class of all total algebras of type F.2) Let K be a variety of total algebras of type F. Let BF(AOB),K bethe congruente on F(A O B) introduced in Thm . 3 . ThenA Ox B - F(A (9 B)/Or(A®B),x3) Let M = W(K), with K as before, and let BA®B,n1(A (9 B)2 . ThenA OW(x) B _ A (9 B/aA®B,Mby Thm. 3.(ii) .4) Let M = E(K), with K as before, and let BA®B,°l be the congru-ence on A O B introduced in Thm. 2 . ThenA OE(x) B - A 0 B/BA®B.Ní_ eF(A®B),K nby Thm . 2 .TENSOR PRODUCTS OP PARTIAL ALCEBRAS	7835) More in general, Prop . 4 in [BMRRT] implies that if M is a weaklyquasi-primitive class of partial algebras then A®MB is isornorphic to thequotient of A ® B by the least congruente on it such that the quotientbelongs to M .Let now A, B be two partial algebras of type Y and let Mi C 11/12 betwo quasi-primitive classes of partial algebras of the same type . SinceA ®M, B E Ml C_ M2, there exists a (unique) homomorphism h, : A ®M2B -> A ®M, B making the following diagram'ír2AxB --% A®M2 BA ®nr, Bto commute (where 7r l and 72 denote the corresponding tensor bihomo-morphisms) .Proposition 6. With these notations, h : A ®M2 B ---> A ®M, B isan Ml-refection.Proof. Let C E Ml and let g : A ®M2 B -> C be a homomorphism .Then g7F2 : A x B -> C is a bihomomorphism, and therefore there existsa unique homomorphism ~l : A ®M, B --> C which makes the followingdiagram to commuteCSince 7rl = h7r2, we get that both g and gh make commutative thediagram784	M. MONSERRAT, F. ROSSELLÓ, J . TORRENS' AxB '7r2 1Aom,B972Cand thus, since C E Mi C M2, by the universal property of the tensorproduct w.r .t . M2 we get g = ghA ®Af, B A &m, BCOn the other hand, it is not difficult to prove that such a g with g = ghhas to be unique . This shows that h : AOm, B --> A ®M, B satisfles theconditions of Ml-reflections .Remark. This result generalices Thm. 3 in [Gall to the partial case .A similar result was also proved in [Gril taking A and B commutativesemigroups, M2 the variety of all semigroups and 11/11 the subvariety ofall commutative semigroups .As a particular case of the last Proposition, if K is a variety then theinclusionW (K) C E(K)implies that there are natural mappingsA ® B ~ A ®C(K) B - A OW(K) Bwhich are in fact reflections w.r .t . the corresponding classes . In general,all there tensor products are different (Le . non-isomorphic), even forA, B E W(K), as the following example shows .Example . Let F = {f,, 1 n E X+} and «fn) = 1 for all n . Let K bethe variety of all total algebras of type (F, 97) satisfying all equationsf, ( . . . (f,~,> (x)) . . . ) r^'v f.] ( . . . (f L, (x)) - . . ), n 1 -1- . . .+np = mi.+ . . .+mqLet E(K) and W (K) be the equational class and the weak subalgebrasclass of K. It turns out that E(K) 7~ W(K) . (Actually, the class E(K)TENSOR PRODUCTS OF PARTIAL ALCEBRAS	785can be identified with the class of all monounary termal algebras, andW(K) with the class of all consistent monounary termal algebras ; see forinstance [BR], where in particular it is showed that these two classes aredifferent .)Let A, B E W(K) be the (F, 91)-pa,rtial algebras given by the diagramBNotice that both A, B E W (K) .Applying the "recipes" in the examples after Thm. 5 it is not difficultto prove thata) A ® B is given by the diagram (where, for simplicity, we skip theoperations' superscripts and we use a single arrow to denote several op-erations if they give the same result)al f3, f4, a2fl, f2 fl, f2ce3 f3, f4 a4with the tensor homomorphism 7r : A x B -> A ® B given by7r(al, bl) = al,7r(al., b2) = a2,7r(a2, bl) = a3, 7r(a2, b2)Notice that A ® B ¢ E(K) .b) A ®E(K) B is given by the diagram21 fl,f2>(32f3, f\ If3, f4fl, f2with the tensor homomorphism 7r : A x B --> A ®E(K) B given by7r(al,b1) = Nl, 7r(al,b2) = ~3 2, 7r(a2,b1) = 7r(a2,b2) = R3-It is not difficult to check that A® E(K) B 0 W(K).= a4 .786	M. MONSERRAT, F . RosSELLó, J . TORRENSc) A ®W(K) B is given by the diagramyl`y2fl, f2, f3, f4with the tensor homomorphism 7r : A x B -> A ®w(K) B given by7r(al,bl) = -%1,ir(a2,bl) = 7r(al ;b2) = 7r(a2,b2) = 73-In particular, from this example (and with the notations of [BR]) wededuce that the tensor product of consistent termal algebras w.r .t . theclass of termal algebras of the same type need not be consistent .Referentes[BMRRT] BARTOL, W ., MONSERRAT, M., ROSSELLó, F., RUDAK,L. AND ToRRENS, J., A note on least congruentes related to termalalgebras, Preprint (1991) .[BR] BARTOL, W . AND ROSSELLó, F ., Termal algebras : a guidedtour, Preprint (1991) .[Bur] BURMEISTER, P., "A model-theoretic approach to partial alge-bras," Mathematical Research Band 32 ; Akademie-Verla,g, Berlin,1986 .[BuS] BURRIS, S . AND SANKAPPANAVAR, H ., "A Course in UniversalAlgebra," Grad . Texts in Math. 78, Springer-Verlag, 1981 .[Fra] FRASER, G . A ., Tensor product of semilattices and distributivelattices, Semigroup Forum 13 (1976), 178-184 .[Gal] GALANOVÁ, J., The tensor product on a variety, in "Univer-sal and Applied Algebra; Proc . of the V Universal Algebra Syrnp.,Tarawa 1988;" (H . Halkowska, B . Stawski eds .), World Scientific,1989, pp . 92-98 .[Gri] GRILLET, P . A ., The tensor product of commutative semigroups,7~rans . Am . Math . Soc. 138 (1969), 281-293 .[Ru] RUDAK, L ., A completeness theorem for weak equational logic,Alg. Univ . 16 (1983) ; 331-337 .TENSOR PRODUCTS OP PARTIAL ALCEBRAS	787(Sch] SCHMIDT, J ., A general existente theorem on partial algebrasand its special cases ; Colloq. Math . 14 (1966), 73-87 .Departament de Ciéncies Matemátiques i InformáticaUniversitat de les Illes Balears07071 Palma de MallorcaSPAINRebut el 7 de Gener de 1992

Tensor products of partial algebras

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