Scientiae Mathematicae Vol. 3, No. 3(2000), CHARACTERIZATIONS OF 0-SIMPLE ORDERED SEMIGROUPS. Received June 16, 2000

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1 Scientiae Mathematicae Vol. 3, No. 3(2000), CHARACTERIZATIONS OF 0-SIMPLE ORDERED SEMIGROUPS NIOVI KEHAYOPULU AND MICHAEL TSINGELIS Received June 16, 2000 Abstract. We characterize the 0-simple ordered semigroups that is the ordered semigroups S having a zero element, denoted by 0,inwhich S 2 = f0g and f0g and S are the only ideals of S. When we speak about a simple ordered semigroup we assume that the ordered semigroup does not contain a zero element. Otherwise we speak about a 0-simple ordered semigroup. An ordered semigroup (without zero) is called simple if S is the only ideal of S. Following the terminology given by A. H. Clifford and G. B. Preston for algebraic semigroups in [2], an ordered semigroup (with 0) is called 0-simple if S 2 = f0g and if the sets f0g and S are the only ideals of S. An ordered semigroup is characterized in [4] as simple if (SaS] =S for every a 2 S. In this note a similar characterization of 0-simple ordered semigroups and some equivalent characterizations arising as corollaries are given. If (S; :;») is an ordered semigroup, a zero of S, denoted by 0,isanelement0ofS such that 0x = x0 =0and0» x for all x 2 S [1]. For an ordered semigroup S and A S, we denote (A] :=ft 2 S j t» a for some a 2 Ag: For A = fag (a 2 S), we write (a] instead of (fag]. If (S; :;») is an ordered semigroup, a non-empty subset I of S is called an ideal of S if 1) IS I and SI I. 2) a 2 I, S 3 b» a imply b 2 I [3; Definition 1]. Definition (cf. also [5]). Let S be an ordered semigroup with 0. S is called 0-simple if 1) S 2 6= f0g. 2) The only ideals of S are the sets f0g and S. As usual, we denote by S nf0g the complement off0g in S. Proposition 1 (cf. also [6]). 0-simple if and only if S 6= f0g and (SaS] =S for every a 2 S nf0g. Let (S; :;») be anordered semigroup with 0. Then S is Proof. =). Let S be 0-simple and let S = f0g. Then S 2 = f0g. Impossible. Let a 2nf0g (=) (SaS] =S?) The set (SaS] isanidealof S. Indeed: (SaS]S =(SaS](S] (SaS 2 ] (SaS], similarly S(SaS] (SaS]. If a 2 (SaS] and S 3 b» a, then b 2 (SaS]. Since S is 0- simple, we have (SaS] =f0g or (SaS] =S. Let (SaS] =f0g. We consider the set I := fx 2 S j (SxS] =f0gg Mathematics Subject Classification. 06F05. Key words and phrases. Simple, 0-simple ordered semigroups.

2 340 NIOVI KEHAYOPULU AND MICHAEL TSINGELIS I is an ideal of S. Indeed: ;6= I S (since a 2 I). Let y 2 S; x 2 I. Then yx 2 I. Indeed: Since x 2 I;(SxS] =f0g. Since SyxS SxS, we have (SyxS] (SxS] =f0g. Since ;6= SyxS (SyxS], we have (SyxS] 6= ;. Thus (SyxS] = f0g, andyx 2 I. Similarly IS I. Let x 2 I; S 3 y» x. Then y 2 I. Indeed: Since x 2 I;(SxS] =f0g. Since y» x, wehave (SyS] (SxS] =f0g. Since ;6= SyS (SyS], we have (SyS] 6= ;. Thus (SyS] =f0g, and y 2 I. Since S is 0-simple, we have I = f0g or I = S. Since a 2 I and a 6= 0 (a 2 S nf0g), we have I 6= f0g. Thus I = S. Then x 2 I for every x 2 S, and (SxS] =f0g for every x S 2 S:::::::::: (Λ) On the other hand, S 3 = SxS then, by (*), x2s (S 3 ]= [ x2s(sxs] =f0g :::::::: (ΛΛ) The set (S 2 ]isanidealof S, so (S 2 ]=f0g or (S 2 ] = S. If (S 2 ]=f0g then, since ;6= S 2 (S 2 ]=f0g, wehave S 2 = f0g. Impossible. Thus (S 2 ]=S. Then wehave S 2 = S(S 2 ]=(S](S 2 ] (S 3 ]=f0g by (**): Since S 2 6= ;, wehave S 2 = f0g. Impossible. (=. Suppose S 2 = f0g. Let a 2 S; a 6= 0(S 6= f0g). By hypothesis, (SaS] = S. Since ;6= SaS S 2 S = f0gs = f0g, wehave SaS = f0g, then(sas] =(0]=f0g, ands = f0g. Impossible. Let I be an ideal of S and I 6= f0g. Then I = S. Indeed: Let a 2 I; a 6= 0. Byhypothesis, (SaS] = S. Since a 2 I, wehave SaS SIS I, and (SaS] (I] =I, so S I, and I = S. 2 By Proposition 1, we have the Corollary 1. Let (S; :;») be anordered semigroup with 0 such that S 6= f0g. Then S is 0-simple if and only if (SaS] =S for every a 2 S nf0g. Corollary 2. Let (S; :;») be anordered semigroup with 0 such that S 6= f0g. Then S is 0-simple if and only if For every a; b 2 S nf0g there exist x; y 2 S such that b» xay. Proof. =). Let a; b 2 S nf0g. Since a 2 S nf0g and S is 0-simple, by Corollary 1, we have (SaS] =S. Since b 2 S, b 2 (SaS]. Then there exist x; y 2 S such that b» xay. (=. By Corollary 1, it is enough to prove thata 2 S nf0g implies (SaS] =S. Let a 2 S nf0g and b 2 S. If b = 0, then clearly b 2 (SaS]. Let b 2 S nf0g. Since a; b 2 S nf0g, by hypothesis, there exist x; y 2 S such that b» xay 2 SaS. Then b 2 (SaS]. Proposition 2. Let (S; :;») be anordered semigroup with 0 such that S 6= f0g. The 1) For every a; b 2 S nf0g there existx; y 2 S such that b» xay. 2) For every a 2 S nf0g and every b 2 S there exist x; y 2 S such that b» xay. Proof. 1) =) 2). Let a 2 S nf0g, b 2 S. If b = 0, then for the elements x = y =02 S, we have b» xay. If b 2 S nf0g then, by 1), there exist x; y 2 S such that b» xay. 2) =) 1). It is clear.

3 CHARACTERIZATIONS OF 0-SIMPLE ORDERED SEMIGROUPS 341 Proposition 3. Let (S; :;») be an ordered semigroup with 0 such that S 6= f0g. The 1) For every a; b 2 S nf0g there exist x; y 2 S such that b» xay. 2) For every a; b 2 S nf0g there exist x; y 2 S nf0g such that b» xay. Proof. 1) =) 2). Let a; b 2 S nf0g. By 1), there exist x; y 2 S such thatb» xay. If x =0ory = 0 then xay = 0, and b = 0. Impossible. Thus x; y 2 S nf0g. 2) =) 1).Itisclear. Proposition 4. Let (S; :;») be an ordered semigroup with 0 such that S 6= f0g. The 1) For every a 2 S nf0g and every b 2 S there existx; y 2 S such that b» xay. 2) For every a 2 S nf0g and every b 2 S there existx; y 2 S nf0g such that b» xay. Proof. 1) =) 2). Let a 2 S nf0g;b2 S. If b = 0, then for the elements x = y = a 2 S, we have 0» xay. Let b 2 S nf0g. Since a 2 S nf0g and b 2 S, byhypothesis, there exist x; y 2 S such that b» xay. If x =0ory =0,thenb = 0. Impossible. So x; y 2 S nf0g. 2) =) 1). It is clear. 2 By Corollary 1, Corollary 2, Proposition 2, Proposition 3 and Proposition 4, we have the following Theorem 1. Let (S; :;») be an ordered semigroup with 0 such that S 6= f0g. The 1) S is 0-simple. 2) For every a 2 S nf0g, wehave(sas] =S. 3) For every a; b 2 S nf0g there exist x; y 2 S such that b» xay. 4) For every a 2 S nf0g and every b 2 S there existx; y 2 S such that b» xay. 5) For every a; b 2 S nf0g there exist x; y 2 S nf0g such that b» xay. 6) For every a 2 S nf0g and every b 2 S there existx; y 2 S nf0g such that b» xay. 2 For an ordered semigroup S, we denote by I the equivalence relation on S defined by I := f(a; b) 2 S S j I(a) =I(b)g, where I(a) is the ideal of S generated by a (a 2 S). We have I(a) =(a [ Sa [ as [ SaS] (cf. [3]). As usual, we denote by (a) I the I- class containing a and by S=I the set of all (a) I, a 2 S. Theorem 2. Let (S; :;») be anordered semigroup with 0. Then S is 0-simple if and only if 1) S 2 6= f0g and 2) S=I = ff0g;snf0gg: Proof. =). Wehave S=I 3 (0) I = f0g. Indeed: Since 0 2 (0) I,wehave f0g (0) I : Let a 2 (0) I.Then(a; 0) 2 I, a 2 I(a) =I(0) = f0g, and a =0. We have S nf0g 2 S=I: Indeed: Since S is 0-simple, by Proposition 1, S 6= f0g, and f0gρs. Let a 2 S, a 6= 0. Then S=I 3 (a) I = S nf0g. Indeed:

4 342 NIOVI KEHAYOPULU AND MICHAEL TSINGELIS Let b 2 (a) I. Then (b; a) 2 I, a 2 I(a) = I(b). If b =0,thenI(b) =f0g, and a =0. Impossible. Thus b 2 S nf0g. If b 6= 0, then clearly b 2 S nf0g. Let b 2 S nf0g. Then b 2 (a) I. Indeed: Since I(b) is an ideal of S, byhypothesis, we have I(b) =f0g or I(b) =S. Since b 2 I(b), b 6= 0,wehave I(b) 6= f0g. Thus I(b) =S. In a similar way, since a 2 S n f0g, we have I(a) = S. Since I(a) = I(b), we have (a; b) 2 I, and b 2 (a) I. Let a 2 S. Then (a) I 2ff0g, S nf0gg. Indeed: If a =0,then(a) I =(0) I = f0g (cf. the proof above). If a 6= 0, then (a) I = S nf0g (cf. the proof above). (=. Let I be an ideal of S, I 6= f0g. Then I = S. Indeed: We have f0g ρi. Let a 2 I, a 6= 0. Since a 2 S, byhypothesis, we have (a) I = f0g or (a) I = S nf0g. If (a) I = f0g, then a 2 (a) I = f0g, a =0. Impossible. Thus (a) I = S nf0g:::::::::: (Λ) Let b 2 S. If b = 0, then clearly b 2 I. Let b 6= 0. Sinceb 2 S nf0g =(a) I (by (*)), we have (b; a) 2 I,andb 2 I(b) =I(a). Since a 2 I, wehave I(a) I. Then b 2 I. Remark 1. If S is an ordered semigroup with 0 and A := fi j I ideal of S; I 6= f0gg; then A 6= ; if and only if S 6= f0g. In fact: Let A 6= ;, and let I 2 A. Then I is an ideal of S and I 6= f0g. Since I is an ideal of S, wehave 02 I, andf0g I. Thenf0g ρi S, and S 6= f0g. If S 6= f0g, then clearly S 2 A. Remark 2. Let S is an ordered semigroup with 0 such that S 2 6= f0g. Then S 6= f0g, equivalently, jsj 2 (as usual, we denote by jsj the order of S). Remark 3. Let S is an ordered semigroup with 0 such thatthesetsf0g and S are the only ideals of S. If S 2 = f0g, thenjsj»2. In fact: Let jsj > 2 i.e jsj»3. Since 0 2 S, js nf0gj 2. Let a; b 2 S nf0g, a 6= b. The set (a] is an ideal of S. Indeed: (a]s =(a](s] (as] (S 2 ]=(0]=f0g (a] (S 2 = f0g) x 2 (a];s 3 y» x imply y 2 (a]: Thus (a] = f0g or (a] = S. If (a] = f0g, then a 2 (a] = f0g, a = 0. Impossible. Thus (a] = S. Similarly, since S 2 = f0g, the set (b] isanidealof S, then (b] = S. Then a 2 (a] =(b], and a» b. In a similar way, b» a, anda = b. Impossible. Remark 4. Let S is an ordered semigroup with 0 in which the sets f0g and S are the only ideals of S. Then S is 0-simple or we have S 2 = f0g and jsj»2. In fact: If S 2 6= f0g, then S is 0-simple. If S 2 = f0g then, by Remark3,jSj»2. 2 Note. An ordered semigroup (or groupoid) S without zero is called simple if S is the only ideal of S. If S is an ordered semigroup (or groupoid) with zero, this is the natural definition for S to be simple: An ordered semigroup (or groupoid) with zero is called simple (0-simple in the terminology of Clifford) if 1) S 6= f0g. 2) The only ideals of S are the sets f0g and S.

5 CHARACTERIZATIONS OF 0-SIMPLE ORDERED SEMIGROUPS 343 In this respect, we note the following: If (S; :;») is an ordered semigroup (or groupoid) with 0 such that S 2 = f0g then, for a 2 S, theset(a] :=fx 2 S j x» ag is an ideal of S::::::::(Λ) Indeed: ; 6= (a] S (since a 2 (a]). ; 6= (a]s S 2 = f0g (since a 2 2 (a]s), so (a]s = f0g (a] (since 0» a). Similarly S(a] (a]. If x 2 (a] and S 3 y» x, then y» a, so y 2 (a]. Remark A. If (S; :;») is an ordered semigroup (or groupoid) with 0, jsj 3 and S 2 = f0g, then there exists an ideal I of S such that I 6= f0g and I 6= S. In fact: Let a; b 2 S, a 6= b, a 6= 0,b 6= 0. Then: i ) Let b 2 (a]. By (*), the set (b] is an ideal of S. We have (b] 6= f0g and (b] 6= S. Indeed b 2 (b] andb 6= 0;a 2 S and a 62 (b] (sincea 2 (b] implies a» b. Moreover, since b 2 (a], we have b» a, then b = a. Impossible). ii) Let b 2 S nfag. By(*),(a] is an ideal of S. Wehave (a] 6= f0g and (a] 6= S. Indeed: a 2 (a] and a 62 f0g; b 2 S and b 62 (a]. Remark B. If (S; :;») is an ordered semigroup (or groupoid) with 0 and jsj = 2, then the only ideals of S are the sets f0g and S. In fact: Let S = f0;ag, a 6= 0. Let I be an ideal of S. Since;6= I S, wehave I = f0g or I = fag or I = S. If I = f0g, then I is an ideal of S. If I = S, then I is an ideal of S. Let I = fag. Then IS = fagf0;ag = f0;a 2 g6 fag (since f0;a 2 g fag implies 0 = a. Impossible). Thus the set fag is not an ideal of S. 2 We moreover remark that: If (S; :;») is an ordered semigroup with 0 and jsj =2,that is, if S = f0;ag, a 6= 0, then S is one of the following: A) S is the ordered semigroup with the multiplication and the order defined by: : 0 a a 0 0» = f(0; 0)g which is not 0-simple (in the terminology of Clifford) since S 2 = f0g. B) S is the ordered semigroup with the multiplication and the order defined by: : 0 a a 0 a» = f(0; 0); (0;a); (a; a)g: which is 0-simple (in the terminology of Clifford) since S 2 6= f0g, andforevery ideal I of S, wehave I = f0g or I = S. References [1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll. Publ. Vol. XXV, Providence, Rh. Island, [2] A. H. Clifford and G. B. Preston, The Algebraic TheoryofSemigroups Vol. I, Amer. Math. Soc., Math. Surveys 7, Providence, Rh. Island, [3] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Mathematica Japonica, 35 (No. 6) (1990),

6 344 NIOVI KEHAYOPULU AND MICHAEL TSINGELIS [4] N. Kehayopulu, Note on Green's relations in ordered semigroups, Mathematica Japonica, 36 (No. 2) (1991), [5] N. Kehayopulu and M. Tsingelis, On right simple and right 0-simple ordered groupoids- semigroups, Scientiae Mathematicae, to appear. [6] M. Tsingelis, Contribution to the structure theory of ordered semigroups, Doctoral Dissertation, University ofathens, University of Athens, Department of Mathematics Mailinig (home) address: Nikomidias 18, Kesariani, Greece