On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

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1 Order 18: 49 60, Kluwer Academic Publishers. Printed in the Netherlands. 49 On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P ROZÁLIA SZ. MADARÁSZ Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Yugoslavia. BOŽA TASIĆ Department of Pure Mathematics, University of Waterloo, Waterloo ON N2L 3G1, Canada. (Received: 11 March 1999; accepted: 9 January 2001) Abstract. Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R(K) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties. Mathematics Subject Classifications (2000): 06F05, 08B99. Key words: class operators, operator of retraction, partially ordered semigroup and monoid. 1. Introduction The problem of describing semigroups generated by some class operators on algebras was posed in [4] under the number 24, though some partial solutions had been known before. Let σ be any set of monotone operators on classes of universal algebras. As the composition of class operators is associative we can consider the semigroup σ generated by σ.let denotes the usual partial ordering among class operators, i.e. O 1 O 2 iff O 1 (K) O 2 (K), for all classes K of algebras of the same type. By the definition of a partially ordered algebraic structure, ( σ, ) is a partially ordered semigroup (po-semigroup for short). We call it the po-semigroup of operators generated by σ. One of the well known results of this type is Pigozzi s description of the partially ordered semigroup generated by class operators I, H, S, P (see [8]). Some other semigroups of operators are considered in [6]. In the present paper the ordered semigroup generated by class operators I, R, H, S, P will be described.

2 50 R. SZ. MADARÁSZ AND B. TASIĆ 2. Preliminaries Let A = (A, F ) be an algebra. A subalgebra B A is a retract of algebra A if there is a homomorphism ϕ : A B such that ϕ is the identity on B (i.e. ϕ B = 1 B ). The homomorphism ϕ is often referred to as a retraction of A onto B. Further on, we will consider classes of algebras of the same type. For a class K of universal algebras, let R(K) be the class of all algebras isomorphic to a retract of a member of K and, of course, let R denote the corresponding class operator. By I, H, S, andp we denote the well-known universal algebraic class operators. If K is a class of algebras we shall write I(K), H(K) for the class of all isomorphic images, homomorphic images of algebras from K and S(K), P(K), respectively for the class of all algebras isomorphic to subalgebras, direct products of algebras in K. Let us denote by M r the partially ordered semigroup generated by {I,R,H,S,P} in respect to the usual order among class operators. Let us recall that an operator O is idempotent if O 2 = O, extensive if K O(K), for any class of algebras K, and monotone if K 1 K 2 implies O(K 1 ) O(K 2 ). It is easy to see that all operators I, R, H, S, P are idempotent, extensive and monotone. As operator I is the identity element in the semigroup I,R,H,S,P we can consider M r to be partially ordered monoid (po-monoid for short). In the sequel we will refer to M r as a po-monoid with I as the identity element. In general, a po-monoid G = (G,, 1, ) is positively ordered monoid if for all x,y G, x xy and y xy. The last condition is equivalent to: for all x G,1 x.ifx,y G are idempotent elements of a positively ordered monoid G, it is easy to see that x y xy = yx = y. So, the po-monoid M r satisfies the following conditions: R 2 = R, H 2 = H, S 2 = S, P 2 = P, (1) I R, I H, I S, I P, (2) R H, R S, (3) PR RP, SH HS, PH HP, PS SP. (4) We shall see that the structure of M r is completely determined by these relations. In fact, M r is isomorphic to the po-monoid F r generated by {r, h, s, p} with defining relations which are translations of (1) (4). 3. The Semigroup-Part of the po-monoid F r Let W be the set of all formal monoid words over the alphabet {r, h, s, p} and let be the equivalence relation on W induced by the following relations: hh = h, ss = s, pp = p, (5) 1 h, 1 s, 1 p, (6) sh hs, ph hp, ps sp, (7) rr = r, 1 r, r h, r s, pr rp, (8)

3 PO-SEMIGROUP OF OPERATORS I,R,H,S,P 51 Figure 1. Partial order of F. i.e., if w, w W then w w iff the identity w = w is a logical consequence of (5) (8) and the conditions defining a po-monoid. Let F r = W/ and let F r = (F r,, 1, ) be the corresponding po-monoid. The empty word will be denoted by 1 and we will consider a word and the element of F r it represents the same, for example we shall write h instead of h/. The structure of the po-monoid F generated by {h, s, p} with the defining relations (5) (7) is completely determined by Pigozzi in [8]. That po-monoid has exactly 18 elements and the corresponding partial order is shown on Figure 1. It is also proven in [8] that M = F,whereM is the po-monoid obtained from the po-semigroup I,H,S,P with I as the identity element. Let us call the elements of po-monoid F old words. So, the old words are: 1, h, s, p, sh, ph, ps, hs, hp, sp, psh, sph, phs, hps, sphs, shp, shps, hsp. The elements of F r which are not old we shall call new words. We shall prove that there are at most seven new words up to. Later, we shall see that these seven words represent really different elements in F r. Using relations (5) (8) and the conditions defining a positively ordered monoid, we can easily prove that in F r we have: rs = sr = s, rh = hr = h, (9) prp = rp, rpr = rp. (10) The following two lemmas and the results of [8] enable us to construct the multiplication table for F r.

4 52 R. SZ. MADARÁSZ AND B. TASIĆ LEMMA 1. Every word in the po-monoid F r with exactly one r is equal either to an old word or to one of the following seven words: r, rp, pr, rph, rps, rphs, rpsh. Proof. There are four types of such words: (1) r, (2) rw,wherew is an old word, (3) wr, wherew is an old word, (4) w 1 rw 2,wherew 1,w 2 are old words. In case (1), r isanewone. In case (2), using (9), we see that rw is equal to w if w begins with s or h.ifw begins with p, there are five possible new words: rp, rph, rps, rphs, rpsh. In case (3), again by (9), if the last symbol in w is s or h then wr is equal to w. If the last symbol in w is p then wr is one of the following words: pr, hpr, spr, shpr, hspr. But,hp hpr hph hhp = hp so hpr = hp and sp spr sps ssp = sp so spr = sp. The last two equalities imply: shpr = shp and hspr = hsp. So, we have that pr is the only new word. In case (4), using case (3), we have that w 1 r is either pr or w 1 r = w 1.If w 1 r = w 1 then w 1 rw 2 = w 1 w 2, so it is old one. If w 1 r = pr then w 1 rw 2 = prw 2, but rw 2 could be one of these: rp, rph, rps, rphs, rpsh. So,w 1 rw 2 is one of the following words: pr, prp, prps, prph, prpsh, prphs. Using (10), we have prp = rp, prps = rps, prph = rph, prpsh = rpsh and prphs = rphs. So, we can conclude that every word with exactly one r is equal to one of the following seven words: r, rp, pr, rph, rps, rphs, rpsh. LEMMA 2. All words in the po-monoid F r with exactly two letters r are equal to words with one appearence of r. Proof. The new words with two appearence of r could be: (1) rw r, (2) w r r, (3) wrw r, (4) w r rw, where w r is some word with one r and w is an old word. In case (1), we have two possibilities: w r begins with r or p. Ifw r begins with r then using rr = r we have rw r = w r.ifw r begins with p then w r must be pr and we have rw r = rpr = rp by (10). In case (2), w r r could be one of the following words: rr, rpr, prr, rpsr, rphr, rpshr, rphsr which are respectively equal to r, rp, pr, rps, rph, rpsh, rphs. In case (3), using case (1), we have wrw r = ww r = w r or wrw r = wrp = w r, where w r is a new word with one r. In case (4), using case (2), we have w r rw = w r w = w r.

5 PO-SEMIGROUP OF OPERATORS I,R,H,S,P 53 So we have just proved that all words with exactly two r are equal to words with one appearence of r. Therefore we conclude that all possibly new words are among r, rp, pr, rph, rps, rphs, rpsh. As the consequence we establish the following: THEOREM 1. Every element of the free po-monoid F r generated by r, h, s, p with the defining relations (5) (8) is equal to one of the following words: 1,r,h,s,p,rp,pr,sh,hs,sp,ps,ph,hp,rph,rps,psh,phs, sph, shp, hps, rphs, rpsh, sphs, shps, hsp. It will follow immediately from the description of the partial order of F r (given later) that all these words are distinct. Then, the semigroup part of the structure F r will be completely determined, because the multiplication table for F r is easily constructed by application of (5) (8) and proofs of Lemmas 1, 2. (11) 4. The Structure of the po-monoid M r In order to prove that nothing else collapses from the list of elements (11) in Theorem 1 we proceed as follows. Let (P, ) be the poset of the 25 formal symbols from the list (11) with as depicted in Figure 2. We shall prove that (P, ) = (F r, ), and as a consequence that (M r, ) = (F r, ). Note the abuse of notation: the elements of P are just formal words w in the letters r, h, s, p while the elements of F r are w/, but we decided for the convinience to not write. We shall use the following results from [2] to prove the facts mentioned above and that will provide us with the minimal set of sufficient counterexamples. In a partially ordered set (A, ) a critical pair will mean a pair of elements (x, y) such that x is a minimal element of {p p y} and y is a maximal element of {q x q}. In [2] the following proposition is proven. PROPOSITION 1 (see [2, Proposition 4]). Let f : A B be an isotone map of partially ordered sets, and suppose that (A, ) has ascending and descending chain conditions. Then f is an order embedding if and only if every critical pair (x, y) of A satisfies f(x) f(y). The lemma below can help us to find critical pairs. LEMMA 3 (see [2, Lemma 5]). If (A, ) has both ascending and descending chain conditions, then pair (x, y) is a critical if and only if x is a minimal member of the class of join-irreducible elements of A that are y and y is a maximal member of the class of meet-irreducible elements of A that are x. THEOREM 2. Let F r = (F r,, 1, ) be the po-monoid generated by elements r, h, s, p satisfying relations (5) (8).ThenF r has 25 elements and (F r, ) = (P, ).

6 54 R. SZ. MADARÁSZ AND B. TASIĆ Figure 2. Partial order of (P, ). Proof. Let us denote by θ : P F r the mapping defined by θ(w) = w (if we want to be totally precise it should be = w/, but since identification is in effect we write = w). In order to prove that θ is an isomorphism it suffices to show that θ is an embedding. It is easy to see using (5) (10) that θ is an isotone map from (P, ) onto (F r, ). By Proposition 1, in order to prove that θ is an embedding we have to find critical pairs (u, v) of (P, ), and check that for every critical pair θ(u) θ(v) holds, i.e., u v in F r. Using Lemma 3 and the diagram given by Figure 2 it turns out that (P, ) has 9 critical pairs. The critical pairs and the corresponding non-inclusions in F r are: (h, sp), h sp, (12) (s, hp), s hp, (13) (p, hs), p hs, (14) (hs, shp), hs shp, (15) (sp, hps), sp hps, (16) (hp, sphs), hp sphs, (17) (hsp, shps), hsp shps, (18) (r, p), r p, (19) (rp, phs), rp phs. (20)

7 PO-SEMIGROUP OF OPERATORS I,R,H,S,P 55 Let us note that the above non-inclusions are not independent. The non-inclusion (17) implies (12) and (14), the non-inclusion (18) implies (13), (15) and (16), and the non-inclusion (20) implies (19). So, we are left with having to show hp sphs, hsp shps and rp phs in F r. The po-monoid M r satifies relations (5) (8) defining F r.so,m r is a homomorphic image of F r. Now, it is enough to prove HP SPHS, HSP SHPS, (21) RP PHS. (22) The non-inclusions (21) are proven in [8] for commutative semigroups. Let us show that (22) holds for commutative semigroups as well. Let S = ({0, 1}, ) be the semilattice satisfying 0 1 = 1 0 = 0, and let K 3 = {S}. AsPHS(K 3 ) = P(K 3 ) it is easy to see that finite commutative semigroups belonging to PHS(K 3 ) are of the cardinality 2 n. Semigroup S 2 has a three element subsemigroup A = ({(0, 0), (0, 1), (1, 1)}, ) which is also a retract of S 2 under the retraction ϕ : S 2 A defined by ϕ = ( (0, 0) (0, 1) (1, 0) (1, 1) (0, 0) (0, 1) (0, 0) (1, 1) ). So, A RP(K 3 ) and A PHS(K 3 ). Hence θ is an embedding and we conclude that (P, ) = (F r, ). THEOREM 3. The po-monoid M r = (M r,,i, ) is isomorphic to the po-monoid F r. The partially ordered set (M r, ) is diagrammed on Figure 2 after relabeling 1,r,h,s,p with I,R,H,S,P. Proof. Let ϕ : F r M r be the homomorphic extension of the mapping ϕ 1 = ( h s p r ) H S P R.Thenϕ is isotone and onto. To prove that ϕ is an embedding we will make use of Proposition 1. Hence it is enough to prove that for every critical pair (x, y) of (F r, ) we have ϕ(x) ϕ(y) in (M r, ). As(F r, ) = (P, ), it follows that the critical pairs of (F r, ) are: (p, hs), (h, sp), (s, hp), (hs, shp), (sp, hps), (hp, sphs), (hsp, shps), (r, p), (rp, phs). Now, reasoning as in the proof of Theorem 2 (the part after the display (20)) we conclude that ϕ is an embedding which completes the proof. 5. Standard Semigroups of Operators I, R, H, S, P Let V be a variety and let σ be a set of monotone operators on classes of algebras. If V is closed under all operators from σ we can consider the po-semigroup of

8 56 R. SZ. MADARÁSZ AND B. TASIĆ operators generated by σ such that we restrict domains of operators to be subclasses of V. In such po-semigroup, two operators O 1 and O 2 are consider to be equal if and only if for every K V we have O 1 (K) = O 2 (K). In spite of [8] we call these po-semigroups the standard po-semigroups of operators σ of the variety V. Of course, the standard po-semigroup of σ of variety V is a homomorphic image of the corresponding po-semigroup of operators generated by σ. If these two semigroups are isomorphic we say that the variety V has the full standard po-semigroup of operators σ. We denote by M(V) the standard po-semigroup of operators I, H, S, P of the variety V. Paper [3] surveys results on M(V) for the following varieties: Boolean algebras, distributive lattices, lattices, Abelian groups, groups, commutative semigroups and unary algebras. Given a variety V we denote by M r (V) the standard po-semigroup of operators I, R, H, S, P of the variety V. As a consequence of Theorems 2, 3 the following result can be established. LEMMA 4. Let V be any variety. A necessary and sufficient condition for V to have the full standard po-semigroup M r (V) is that there exist classes K 1, K 2, K 3 V satisfying the following non-inclusions: HSP(K 1 ) SHPS(K 1 ), (23) HP(K 2 ) SPHS(K 2 ), (24) RP(K 3 ) PHS(K 3 ). (25) Pigozzi gives in [8] examples of classes K 1, K 2 of commutative semigroups satisfying (23) and (24) respectively. So, the variety C of commutative semigroups has the full standard po-semigroup of operators M(C). As we have given an example of the class of commutative semigroups satisfying (25) in the proof of Theorem 2 we can formulate the following result. THEOREM 4. The variety C of commutative semigroups has the full standard po-semigroup of operators M r (C). The corresponding partial order is given by Figure 2. There are properties of varieties which imply some relations between operators. For example, if V is a variety satisfying congruence extension property (CEP), then HS(K) = SH(K) for every class K V, i.e., HS = SH on V. The varieties U of monounary algebras and Ab of Abelian groups have CEP and hence satisfy HS = SH. Let us see what happens with M r if HS = SH. Let M r [HS = SH] denote a homomorphic image of M r corresponding to the congruence relation of M r generated by {(HS,SH)}. HS = SH implies the following relations among the elements of M r : HSP = SHPS = SHP, (26)

9 PO-SEMIGROUP OF OPERATORS I,R,H,S,P 57 Figure 3. Partial order of M r [HS = SH]. RPHS = RPSH, (27) SPHS = SPH, (28) PHS = PSH, (29) SH = HS. (30) After the reduction by SH = HS we are left with the set T 1 ={I,R,H,S,P,PR,RP,SH,PH,PS,RPH,RPS,SP,PSH, RPSH,HP,HPS,SPH,HSP} of reduced words. We want to prove that no further reduction is possible using HS = SH. Colapsing points of the diagram of M r given by Figure 2 modulo relations (26) (30) we can obtain a diagram of the partially ordered set (T 1, ) (see Figure 3). As the variety U of monounary algebras satisfies HS = SH, M r (U) is a homomorphic image of M r [HS = SH]. We know that M r [HS = SH] 19. In order to prove that M r [HS = SH] has 19 elements we will prove that (T 1, ) is isomorphic to the partial order of M r (U). THEOREM 5. The standard po-semigroup of operators M r (U) of the variety of monounary algebras has 19 elements and the corresponding partial order is given by Figure 3. Proof. M r (U) as a homomorphic image of M r [HS = SH] has also at most 19 elements. The identity map from T 1 to M r (U) is an isotone map. Now, by Proposition 1 in order to prove that (T 1, ) = (M r (U), ) it suffices to show that

10 58 R. SZ. MADARÁSZ AND B. TASIĆ for every critical pair (x, y) of (T 1, ) we have x y in (M r (U), ). Critical pairs of (T 1, ) and the corresponding non-inclusions in (M r (U), ) are listed below. (R, P ), R P, (31) (H, SP ), H SP, (32) (S, H P ), S HP, (33) (P, SH ), P SH, (34) (RP, P H S), RP PHS, (35) (SP, H P S), SP HPS, (36) (H P, SP H ), H P SPH. (37) It is easy to see that these non-inclusions are not independent. The non-inclusion (36) implies (33), (34), the non-inclusion (37) implies (32) and the non-inclusion (35) implies (31). So, we are left with showing (35), (36) and (37) in M r (U).That non-inclusions (36) and (37) hold in M(U) is noted in [3]. We give an example of a monounary algebra A such that RP({A}) PHS({A}). Let A = ({0, 1, 2},f)be the monounary algebra defined by f(0) = 0, f(1) = 2, f(2) = 1. Algebra A 2 has a five element retract B = ({(0, 0), (0, 1), (0, 2), (1, 0), (2, 0)},f). As all finite algebras from PHS(A) either have cardinality which is a multiple of 2 or 3, or they are one element algebras, it follows that B PHS(A) and B RP(A). As the identity map is an embedding we found that (T 1, ) = (M r (U), ).So, M r (U) has 19 elements and M r [HS = SH] = M r (U). The variety Ab of Abelian groups also satisfies HS = SH. Hence to show that M r [HS = SH] = M r (Ab) by the same reasoning as in Theorem 5 it is enough to find classes K 1, K 2 and K 3 of Abelian groups such that HP(K 1 ) SPH(K 1 ), (38) SP(K 2 ) HPS(K 2 ), (39) RP(K 3 ) PHS(K 3 ). (40) THEOREM 6. The standard po-semigroup of operators M r (Ab) of the variety of Abelian groups has 19 elements and the corresponding partial order is given by Figure 3. Proof. We recall again [3] where it is noted that the standard monoid of operators M(Ab) of the variety of Abelian groups has 13 elements. This implies (38) and (39). To prove (40) let K ={Z p p P } where P is the set of all primes and Z p is the Prüfer group for every p P. It is easy to see that PHS(K) = P(K {Z p n n 1,p P }). So, every group from PHS(K) has elements of finite order different from the identity element. The group p P Z p is divisible as a product of divisible groups.

11 PO-SEMIGROUP OF OPERATORS I,R,H,S,P 59 By the theorem on the decomposition of divisible Abelian groups we have Z p = Q Z p, j J p P i I where both I and J are uncountable. As every direct sumand of an Abelian group is its retract we found that i I Q RP(K). i I Q is a torsion free group and hence it cannot be in PHS(K) which completes the proof. It is proven in [7] and [1] that the variety G of groups has the full standard posemigroup of operators M(G) (i.e., non-inclusions (23) and (24) hold). Using the example for (40) from the proof of Theorem 6 the following result follows by Lemma 4. THEOREM 7. The variety G of groups has the full standard po-semigroup of operators M r (G). The corresponding partial order is given by Figure 2. At the end we give a few remarks on the standard po-semigroups of the variety of lattices. We are able to give only a partial solution since the folowing is still an OPEN PROBLEM. Does HSP = SHPS for every class K of lattices? It is noted in [3] that for the variety L of lattices the standard po-semigroup of operators M(L) has 17 or 18 elements depending on the solution of the problem above. We give an example of a class K of lattices satisfying RP(K) PHS(K). As the consequence we get that M r (L) has 24 or 25 elements again depending on the problem above. Let 2 B = ({0, 1},,,, 0, 1) be Boolean algebra and 2 L be its lattice reduct. It is easy to see that Boolean algebra B PHS({2 B }) if and only if B = 2 I B = P (I) for some set I and hence it is complete and atomic. Also, it is known that RP({2 B }) is exactly the class of complete Boolean algebras. If D = (D, ) is a dense linear ordering, then the regular open algebra Ro(D) is complete and atomless Boolean algebra, and hence RP({2 B }) PHS({2 B }). Let us note that lattice reducts of retracts of Boolean algebras are retracts of lattice reducts of Boolean algebras. In terms of operators we have Rd L R RRd L. Since Rd L P = PRd L we have Rd L RP({2 B }) RP(Rd L ({2 B })) = RP({2 L }). So, we conclude that lattice reduct of Boolean algebra Ro(D) belongs to RP({2 L }) and does not belong to PHS({2 L }), i.e., RP({2 L }) PHS({2 L }). References 1. Bergman, G. M. (1989) SHPS HSP for metabelian groups, and related results, Algebra Universalis 26, Bergman, G. M. (1994) Partially ordered sets, and minimal systems of counterexamples, Algebra Universalis 32,

12 60 R. SZ. MADARÁSZ AND B. TASIĆ 3. Comer, S. D. and Johnson, J. S. (1972) The standard semigroup of operators of a variety, Algebra Universalis 2, Grätzer, G. (1979) Universal Algebra, 2nd edn, Springer-Verlag, New York. 5. Höft, H. (1974) A normal form for some semigroups generated by idempotents, Fund. Math. 84, Nelson, E. (1967) Finiteness of semigroups of operators in universal algebra, Canadian J. Math. 19, Neumann, P. M. (1970) The inequality of SQPS and QSP as operators on classes of groups, Bull. Amer. Math. Soc. 76, Pigozzi, D. (1972) On some operators on classes of algebras, Algebra Universalis 2,