Mitsch, Heinz

English

Mitsch Heinz

University of Szeged

Acta Sci. Math., 40 (1978), 301—283 Derivations and translations on 1-semigroups H. MITSCH Introduction. Functions of lattices into themselves have been studied in [4], [7], [9], [10]. With respect to pointwise intersection A and union V and to com-position of functions o the set (F(L), A, V, o) of all transformations of a lattice (L, A, V) forms a "right-lattice ordered semigroup" (rl-semigroup; see [3], [4]). This means a set S with three binary operations A, V and such that (S, •) is a semigroup, (S, A, V) is a lattice and (x\/y)z = (xz)V(vz), (xhy)z = (xz)A(yz) for all x,y,z£ S. Note that with respect to the order-relation induced by the lattice-operations multiplication satisfies: x s y ^ x z i y z for each S. Recently SzAsz [9], [10] started the investigation of special functions on lattices (L, A, V), so-called "derivations", motivated by the formal rules of derivations in rings, i.e. transformations cp of L which satisfy cp{x\ly) = <p(x)\J(p(y) a n d q>(xf\y) = [<p(jc) A y] V [jcAç>(>')] f o r al l x,y£L.-Since in F{L) also the composition of functions is defined, it is natural to con-sider transformations of the rl-semigroup (F(L), A, V, o) which satisfy also a formal chain rule: (pifog) = W(f)og]h<p{g) for all f,g£ F(L). For rings with a third operation o, so-called composition-rings, such derivations with chain-rule have been studied — especially for polynomial-rings — in [6], [8]. In the following we suppose S to be a right-lattice ordered semigroup and investigate transformations cp of (5, A, V, • ) — so-called C-derivations — which have the following properties: I. (p(x\/y) = <p(x)V(p(y) II. (p(xAy)=[(p(x)Ay]V[xA<p(y)] III. (P(xy) =[<p(x)y]Acp(y) for all x, S. Received July 20, in revised form December 3, 1977. 302 H. Mitsch Standard examples for S will be the rl-semigroups (F(L) , A, V, o) of all trans-formations of a lattice L, (¿[x], A, V, o) of all polynomials on L in the indetermi-nate x and (P(L) , A, V, o) of all polynomial-functions on L (see [3]). We shall use also the concept of lattice ordered semigroup (1-semigroup), which is defined as an rl-semigroup (S, A, V, •) satisfying also' the two left-distributive laws: x(yV z) = (xj>)V(xz) and x(yAz) = (xy)A(xz) for all x, y, z£ S. Note that now multiplication also satisfies: x S y = > z x ^ z y for each z € S ; for the general theory see [2]. 1. Reduction to translations The main purpose of this section is to show, that every C-derivation <p on an rl-semigroup S is a special meet-translation [9], i.e. <p(x) = xAa for all x£S,a£S fixed, if the lattice (S, A, V) has a greatest element or if the semigroup (S , •) has an identity. P r o p e r t i e s . Let cp be a C-derivation on S; then 0) (p{xAy) = (p{x)A(p(y) for all x , j € S ; <p(x)s=x for all x ^ S ([10]). 1) If S has a least element o, then cp(o)=o (by II). 2) If c is a left-zero of (S, •) with least element o, then cp(c) = o (tp(c) = cp(co) = =<p(c)oA<p(o)=o). 3) If S does not have a least element but admits a left-zero, then there is no C-derivation on S. (q>(c) = q>(cx) = cp(c)xA(p(x)^cp{x)^x for all x£S by 0): contradiction.) 4) If S has a right-identity e, then cp(x)^(p (e) for all xg .S". ((p(x) = (p(xe) = = (p(x)ei\(p(e)^q>(e) for all x£S.) 5) If S admits a right-identity e and a least element o, then (p (e) = cp (o) implies (p(x) = o for all x£S. 6) If (S , •) is 0-right-simple with o such that ox=o for all x£ S, then (p{a)=o for an element a?£o implies <p(x)=o for all x£S . (Since aS=S, hence for all x£S there exists a y£ S with ay=x; thus (p(x)=<p(ay) = q>(a)yA(p(y)= = oyAq>(y) = o for all x€S . ) E x a m p l e s . 1) If S admits a least element o, then (p(x)=o for all x£S is a C-derivation, the trivial C-derivation. 2) The identity-function (p(x)=x for all x£S is a C-derivation, iff xy^y for all x,y£S. 3) The constant function (p(x)—a for all x£S, a£S fixed, is a C-derivation, Derivations and translations on /-semigroups 303 iff a = o (if o£ S exists) (a = q>(xAx) — (p(x)Ax^x for all x). 4) Concerning meet-translations we know by Corollary 3 of [10]: L e m m a 1.1. Let S be an rl-semigroup with greatest element i. Then every C-derivation q> on S has the form cp(x) = xAa for all x^S and a suitable element a£S. In order to determine the suitable elements a £ S we prove: L e m m a 1.2. Let S be an rl-semigroup. The function <p{x) = xAa for all x£S is a C-derivation, i f f 1) a is a neutral element of (S, A, V) and 2) xyAa^ayAy for all x, y£ S. P r o o f . If a£S satisfies 1), then cp(x)=xAa does 1 and II of the definition, too. If a also satisfies 2), then <p(xy)=xyAa=(xyAd)A(ayAy)=(xAa)yA(aAy) = = (p(x)yA<p(y) for all x,y£S. The converse is clear. Combining Lemmas 1.1 and 1.2 we get the following T h e o r e m 1.3. Let S be an rl-semigroup with greatest element. Then the C-deriv-ations on S are the functions (p of the form (p(x)=xAa with a fixed neutral element a of S such that xyAa^ayAy for each pair x,y of elements of S. Example . 5) Let S be an rl-semigroup with identity e admitting an invertible element a ^ e ; then <p(x)=xAe is not a C-derivation. (If (p is a C-derivation, then by Lemma 1.2: xyAe^y for all x, y£S; but since aa' = a'a — e for a'£S, this implies in particular e^a and e^a \ consequently we get a ^aa' =e, thus a' = a = e: contradiction.) T h e o r e m 1.4. Let S be an rl-semigroup with greatest element i, such that ix = i for all x£S. Then there is no C-derivation on S except the trivial one (if defined). Proof . By Theorem 1.3 every C-derivation on S has the form <p(x)=xAa such that xyAa^ayAy for all x, y£S. For x=i we get aSayAy^y for all y£S. If S has a least element o, then a = o and (p is the trivial C-derivation; if not, then we have a contradiction. The existence of i£S ensured that every C-derivation is a special meet-transla-tion. The same is true if an identity exists: L e m m a 1.5. Let S be an rl-semigroup with right-identity e. Then every C-deriva-tion on S has the form q>(x) = xAa for all x£S with a = (p(e). Proof . Since x = x\J(xAe), hence <p(x)^xA(p(e) for all x£S. But <?(*) = p(e) by 4) and (p(x)Sx for all x£S by 0); thus (p(x)^xAcp(e) and equality follows. C o r o l l a r y . If e is the identity of S and q>(x) = xAa is a C-derivation, then: 1) a^e; 2) a2 —a; 3) xy=y for all jSa^xSe. 304 H. Mitsch P r o o f . Since a — (p(e) = eAa, we have a^e; thus a2Sa, ay^y for all >>£ S. By Lemma 1.2: xyAa^ayAy=ay for all for x=e we get yAa^ay. For y = a we obtain a^a2, thus a2 = a; for y^a we conclude y^ay, so that ay=y for all y^a. Now if a^x^e, then y = aySxySy for all y^a and the assertion follows. R e m a r k . If S is an rl-semigroup with right-identity which is the least element of S, then there is only the trivial C-derivation on S. The same is true in the follow-ing case: L e m m a 1.6. Let S be a left-simple rl-semigroup with right-identity. Then there is no C-derivation on S except the trivial one (if defined). P r o o f . Again for every C-derivation on S we have: (p(x)=xAa with xyAa^y for all x, y{_ S. Since Sy=S for all y€ S, for each y£S there is an x£S with xy=e; thus by Corollary 1) of Lemma 1.5 we conclude a=eAa^y for all y£S and a=o (if o ^ S exists). C o r o l l a r y . Let S^{e} be an r¡-group; then there is no C-derivation on S. P r o o f . Since a semigroup 5 is a group iff 5 is left- and right-simple (see [1]), there is at most the trivial C-derivation cp(x)=o on S. But an rl-group cannot have a least element o.o^e implies o2 — o and since the only idempotent in S is e, we get o=e; thus e^a for all a£S implies so that a - 1 = e and a=e for all a£S. E x a m p l e . 6) Concerning semigroup-left-translations (see [1]) we note the following: if S is a semigroup with left-identity e and <p a mapping of S into itself such that (p{xy)=(p{x)y for all x, y£ S, then for x=e one gets <p(y) = <p(e)y for all y£S and (p(x)=ax for all x£ S. L e m m a 1.7. Let S be an rl-semigroup with right-identity e. Then the mapping q>(x)=ax for all x£S, a£S fixed, is a C-derivation i f f 1) a£S is left-distributive with respect to V and 2) ab = aAb for all b£S. P r o o f . If a£S satisfies 1), then q>(x\/y) = a(x\/y)=ax\Jay = q>(x)V(p(y) for all x,y£S. If it also satisfies 2), then (p(xAy)=a(xAy)=aA(xAy)=[<p(x)Ay\\J V[xA</>0>)] for all x,y£S. Furthermore, since ax=aAx^a implies axy^ay for all x,y£S, it follows: <p (xy)=axy=(ax) y A ay=<p(x)yA<p(y) for all x, y£ S. Conversely, let (p(x)=ax be a C-derivation; then by I of the definition: a(x\Jy) = =axVay for all x, y£S; by Lemma 1.5 we have ax=(p(x)=xAcp{e)=xAa, that is ab—aAb for all b£S. Combining Lemmas 1.5 and 1.7 we get similarly to Theorem 1.3: Derivations and translations on /-semigroups 305 T h e o r e m 1.8. Let S be an rl-semigroup with right-identity. Then the C-deriva-tions on S are the functions (p of the form <p(x)—ax with a fixed element a£S which is left-distributive with respect to V such that ab = aAb for all b£S. R e m a r k . If 5 is an rl-semigroup with (right-identity e and) greatest element i, then (p(x)=ax such that ai—i is not a C-derivation except (p(x)=x (if possible). In fact: if q>(x) = ax = aAx for all x£S, then i=ai=q>{i) = aAi=a and <p(x)=x for all x£S. For 1-semigroups we have: T h e o r e m 1.9. Let S be an l-semigroup with identity e, which is the greatest element of S. Then the C-derivations on S are exactly the left-translations cp(x) = ax such that ab—ahb for all b£S. P r o o f . On an l-semigroup every function (p(x) = ax with ab = af\b for all b£S is a C-derivation by Lemma 1.7. Conversely, if <p is any C-derivation on S, then by Lemmas 1.2 and 1.5: (p(x) = aAx with xyAa^ay for all jc ,€5" . For x=e we get yAa^ay; but a,ySe implies ay^a and ay^y, thus ay^aAy and ay=aAy for all Consequently cp{x) = ax for all xiS with ab — aAb for all b£S. C o r o l l a r y . Let S be a Boolean l-semigroup with identity e (resp. a uniquely complemented l-semigronp with e as greatest element); then the C-derivations on S are exactly the left-translations of S. P r o o f . By the Corollary (resp. Remark) in § 6 of [4] we have in both cases e=i and xy=xAy for all x, y£S. Returning to general rl-semigroups with identity we show: L e m m a 1.10. Let S be an rl-semigroup with right-identity e (resp. with greatest element i). Then the set of all C-derivations on S is a commutative, idempotent semi-group with respect to composition of functions: ((po\p)(x) = (p[ij/(x)] for all S. I P r o o f . Let (p(x)=aAx, \J/(x)—bAx with a = <p(e), b=\j/(e) be arbitrary C-derivations on S (see Theorems 1.3 resp. 1.8). Then (q>oip)(x)=(aAb)Ax=cAx for all x£S with (<poijj)(e)=cAe=c, since by Corollary 1) to Lemma 1.5: a^e, bSe, hence c=aAbme. Furthermore, since a and b are neutral, c—aAb is neutral, too. Since xyAa^ayAy and xyAb^byAy for all ;t,j>£S, we get xyA(aAb)^ s(aAb)yAy for all x, y£ S and we can apply Lemma 1.2. Trivially we have ((po\l/)(x) = (il/o<p)(x) and (q>o<p)(x) = (p(x) for all xdS. The results deduced above show, that the class of rl-semigroups which admit non-trivial C-derivations is quite restricted. For concrete examples of rl-semigroups we can prove even more: 306 H. Mitsch T h e o r e m 1.11. Let (L, A, V) be an arbitrary lattice. Then on the rl-semigroups ( f ( L ) , A, V, o) resp. (P(L), A, V, o) there is no C-derivation except the trivial one (if o(LL exists). P r o o f . We give the proof for F(L). If a least element does not exist, then there is no C-derivation by Property 3. If a least element exists, then for the constant functions f„(x)=a,f0(x)-o for all x£L we have faof0=fa and f0ofa=f0 for all a£L. If <p is a C-derivation on F(L), then <?(/„)=[<p(/a)°/0]A<?(/,) = <?(/,,) and conversely <p(/0)S<p(/fl); thus q>(fa)=(p(f0) for all a£L. Since F(L) has an identity i d ( x ) = x for all x£S, with respect to o, we know by Lemma 1.5 that (p(/) =/A(p(id) for all f£F(L). Moreover, <p(id)^id by Property 0). Consequently: UP (id)] (a)=aA [<p (id)] (a) =fa (a) A [cp (id)] (a) = [cp(fa)] (a)=[<?(/„)] (a) S f 0 (a)=o. There-fore [<p(id)](a)=o for all a£L. Thus <p(id) = 9, the zero-function on L and (p(f)=fA0=9 for all feF(L). The proof of this Theorem depends essentially on the constant functions on L, which are left-zeroes of the semigroup (F(L), o). We can generalize it to left-zero 1-semigroups with identity e, that means 1-semigroups S, such that xy—x for all x^e, y£S (see [1]) — for example the set of all constant functions on a lattice: L e m m a 1.12. Let S be a left-zero l-semigroup with identity e. Then there are no C-derivations on S except <p(x) = o and cp(x) = x for all x'£S (if defined). P r o o f . By Lemma 1.5, cp(x)=xA(p(e) for all S. If there is no least element in S, then by Property 3) there is no C-derivation on S. If there is o£S, then <p(x)=o for all x^e in S by Property 2) Thus we have to determine only <p(e): if q>(e)^e, (p(e) is a left-zero of Sand q>(e)=(p[(p(e)]=o by Lemma 1.10; if (p(e)~e, we have for any x^e: o=(p(x)—xAe. If e is not the greatest element, then there is an x>e and o=xAe=e=(p(e); if e is the greatest element, then o=xAe=x for all x^e in S, 5={o , e) and <p(x) = x for all 5". 2. Derivations with dual chain-rule As mentioned above, a large class of rl-semigroups admits only the trivial C-derivation (if defined). Even the standard examples of mappings resp. polynomial-functions on lattices belong to this class. Therefore the abstraction of derivation of functions, which formalizes the rules of differentiating a sum, a product and the composite of functions, turns out to.be not very useful. Also if axiom III of a C-derivation is replaced by its dual: III'. (p(xy) = (p{x)yW<p(y) for all x,y£S we get nothing new. We can show even more: Derivations and translations on /-semigroups 307 T h e o r e m 2.1. Let S be an rl-semigroup with identity e resp. o = ox for all x6 S (if o^S exists). Then there is no derivation with dual chain-rule except the trivial one (if defined). P r o o f . If S admits no least element and if cp is any mapping satisfying I, II and III', then (p(x) = q>(xe) = tp(x)eV <p(e)^tp(e) for all x£ S; but cp(;c)Sx for all x£S by Property 0) (valid also in this case) and thus cp(e) is the least element of S: contradiction. If S admits o with ox=o for all x£S, then cp(o) = (p(ox) = =(p(o)x\/ cp (x) ==" cp (x) for all x£ S; by Axiom I the mapping <p is order-preserving, hence (p(o)^<p(x) for all x £ S and rp(x)=q>(o)=a with some af_S, for all xf^S; by Axiom II we have a=(p(x) — <p(xAx) = (p(x)Ax^x for all x£S and a=o; thus <p(x) = o for all x£S . R e m a r k . Motivated by the properties of "derivations of formal languages", which are in short lattice-endomorphisms of the 1-semigroup of all formal languages on an alphabet X satisfying the dual chain rule III', the Axiom II of a derivation finally may be replaced by II'. (p(xAy) = (p(x)A<p(y) for all x,y£S. Such "derivations" are studied in [5]. References [1] A. H. CLIFFORD and G. B. PRESTON, The algebraic theory of semigroups, Amer. Math. Soc. Math. Surveys 7, Vol. I (Providence, 1961). [2] L . FUCHS, Partially ordered algebraic systems. Monographs in Pure and Applied Math. 2 8 , Pergamon Press (1963). [3] H. LAUSCH and W. NÖBAUER, Algebra of polynomials, North-Holland Math. Library, Vol. 5 (Amsterdam—New York, 1973). [4] H. MITSCH, Rechtsverbandshalbgruppen, J. reine und angew. Math., 264 (1973), 172—181. [5] H. MITSCH, Derivations on 1-semigroups and formal languages, J. Pure and Appl. Algebra (submitted). [6] W. MÜLLER, Eindeutige Abbildungen mit Summen-, Produkt- und Kettenregel, Monatsh. Math., 7 3 ( 1 9 6 9 ) , 3 5 4 — 3 6 7 . [7 ] J. NIEMINEN, Derivations and translations on lattices, Acta Sei. Math., 3 8 ( 1 9 7 6 ) , 3 5 9 — 3 6 3 . [ 8 ] W . NÖBAUER, Derivationssysteme mit Kettenregel, Monatsh. Math., 6 7 ( 1 9 6 3 ) , 3 6 — 4 9 . [9] G. SzÄsz, Die Translationen der Halbverbände, Acta Sei. Math., 17 (1956), 165—169. [10] G. SzÄsz, Derivations on lattices, Acta Sei. Math., 37 (1975), 358—363. MATH. INSTITUT, UNIVERSITÄT W IEN STRUDLHOFGASSE 4 1090 WIEN, AUSTRIA 7 

Derivations and translations on l-semigroups

http://acta.bibl.u-szeged.hu/14685/1/math_040_fasc_003_004_301-307.pdf

Derivations and translations on l-semigroups

Abstract

Similar works

Full text

Available Versions

University of Szeged