Dwars, Martin

Strecker, Reinhard

English

Dwars Martin

Strecker Reinhard

University of Szeged

Acta Sei. Math., 49 (1985), 89—129 On homomorphic images of normal complexes in varieties of semigroups MARTIN DWARS and REINHARD STRECKER Following LJAPIN [2] we call a subset T of a semigroup S a normal complex if T satisfies the condition: xhydT o xt2y£T for all x ^ C S 1 and t^t^T. A subset of a semigroup is a normal complex if and only if it is a congruence class for some congruence relation. Under a homomorphism of S, the image of a normal complex need not be a normal complex of the image of S. In connection with the investigation of M-radicals introduced by HOEHNKE [1] , the question arises, which subsemigroups T of a semigroup S are homomorphic images of normal complexes under homomorphisms of semigroups from the given class onto S (STRECKER [4] ) . In the class of all semigroups it is easily seen that there are a semigroup S', a normal complex T' of S' and a surjection cp of S' onto S with (p(T')=T, where T is an arbitrary subsemigroup of S. In the present paper we consider an arbitrary semi-group variety V and describe, which subsemigroups are homomorphic images of normal complexes of semigroups from V. The result we obtain generalizes the cor-responding one on monoids (see [3]). The authors wish to express their thanks to Dr. L. Márki for some valuable comments in preparing this paper. The referee has provided us with much assistance for which we are very grateful. Let V be a variety of semigroups. A subsemigroup J of a semigroup S€ V is called F-normal if there are a semigroup SW with a normal complex T' and a surjection cp: S'—S such that <p{T')= T. T h e o r e m . Let V be a variety of semigroups. IfV consists of completely simple semigroups, then the V-normal subsemigroups are exactly the normal subsemigroups; otherwise every subsemigroup of semigroups in V is V-normal. Received February 24, 1983. 90 M. Dwars and R. Strecker R e m a r k . For some varieties, every subsemigroup is normal; this is the case e.g. if V is the variety of zero semigroups or a variety of rectangular bands, but not if V contains the variety of semilattices. The Theorem will be proved in three steps. P r o p o s i t i o n 1. IfV contains the two element zero semigroup (hence all zero semigroups), or the two element semilattice (hence all semilattices), then every sub-semigroup T of a semigroup S€ V is V-normal. P r o o f . Denote by FV{T) and FV(S) the free semigroups in V generated by the underlying sets ET of T and Es of S, respectively. Suppose that V contains the zero semigroups and denote by FZ(S) the free zero semigroup generated by S. No > equality of the form wtw'=t' can ihold in FV(S), where >v, i/£jFV(S) or empty, but not both empty, /, t'£_ET ..For, if w.tw'=t' then let q> be the natural homomorphism of FV(S) onto FZ(S). It follows <p(w)<p{t)(p(\v')=q>(t'). Here the left hand side is equal to zero, the right hand side is not, a contradiction. Therefore ET is a normal complex of FV(S), and the natural homomorphism from FV(S) onto S maps ET onto T. Suppose now that V contains the semilattices and denote by FW(T) and FW(S) the free semilattices generated by ET and Es, respectively. FW{T) is a normal com-plex in FW(S). Consider the natural homomorphism (p of FV(S) onto Ftv(S). For any word w£Fv(S), <p(w) contains all the letters from w. Therefore <p~1(FH,(7,))= — FV(T), hence the latter is a normal complex in FV(S). Now the natural homo-morphism from FV(S) onto S maps FV(T) onto T. P r o p o s i t i o n 2. Let V be a variety of semigroups containing neither zero semi-groups nor semilattices. Then every semigroup in V is completely simple. Proof . Since the free cyclic semigroup in V admits no non-trivial zero semi-group as a homomorphic image, it must be a (finite) group. Thus all semigroups in V are unions of groups. Such a semigroup is a semilattice of completely simple semigroups, but V contains no semilattices either, hence every semigroup in V is completely simple. P r o p o s i t i o n 3. The homomorphic images of normal complexes of completely simple semigroups are normal complexes of the images. P r o o f , (i) In a completely simple semigroup S, if xy=e and e is an idem-potent then ex=x and ye=y, for xSt^eS, xS is a minimal right ideal, therefore x£eS. (ii) Let TczS be a normal complex of S, x,y£S, t, u£T. By w - 1 we denote the inverse of u in the maximal subgroup of S containing u. Let q> be a homomorphism Homomorphic images of normal complexes 91 of S onto S. S is also completely simple. From <p(x)<p(t)(p(y)=(p(u)£eSe, e idem-potent, it follows that e = ((p(u))-L<p(x)cp(t)(p(y) = (p(x)q>(l)<p(y)(<p(u))-\ and by (i) we have ecp(x)=q>(x) and (p(y)e=cp(y). Let xtyu~1=g£eSe. Again by (i), ex=x, u~1e=u~1, further q>{g)=e and (q>(g))~l = e=(p(e). Let u^e'Se', then cp{e')=e and from u~1e=u~1 it follows e'e=e'. Now we have g~1xtyu~1=e and therefore g~xxtye'=eu and e'g~1xtye'=e'eu=e'u=u. Since T is a normal complex, it follows e'g~^xt'ye'd.T for all t'£T. Applying q> we obtain cp(x)(f>{t')q>{y) = q>(xt'y) = eq>(xt'y)e = <p(e' g^xt' ye%(p(T). References [1] H.-J. HOEHNKE, Radikale in allgemeinen Algebren, Math. Nachr., 32 (1966), 347—383. [2] E. S. LJAPIN, Semigroups, Gos. Izd. Fiz.-Mat. Lit. (Moscow, 1960). (Russian) [3] L . MÄRKI, R . MUTZ, R . STRECKER, Strict radicals of monoids, Semigroup Forum, 21 (1980), 27—66. [4] R. STRECKER, Af-Radikale von universellen Algebren, Puhl. Math. Debrecen, 26 (1979), 245— 254. PÄDAGOGISCHE HOCHSCHULE "LISELOTTE HERRMANN" SEKTION MATHEMATIK GOLDBERGER STR. 12 2600 GÜSTROW, GDR 

On homomorphic images of normal complexes in varieties of semigroups

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On homomorphic images of normal complexes in varieties of semigroups

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