Nil extensions of Clifford ordered semigroup A. K. Bhuniya and K. Hansda 7 1 0 Department of Mathematics, Visva-Bharati University, 2 n Santiniketan-731235,West Bengal, India a J [email protected], [email protected] 5 2 January 26, 2017 ] A R . h Abstract t a m Inthispaperwedescribeall thoseorderedsemigroupswhicharethenilextension of Clifford, [ 1 left Clifford, group like, left group like ordered semigroups. v 5 8Key Words and phrases: nil extension, regular ordered semigroups, grouplike, left grouplike, Clifford 1 7 and left Clifford ordered semigroups. 0 . 12010 Mathematics subject Classification: 20M10; 06F05. 0 7 1 : v1 Introduction: i X r aNil extensions of a semigroup (without order), are precisely the ideal extensions by a nil semigroup. In 1984, S. Bogdanovic´ and S. Milic´ [2] characterized the semigroups (without order) which are nil extensions of completely simple semigroups, where as, a similar work was done by J. L Galbiati and M.L Veronesi [14] in 1980. Authors like S. Bogdanovic´, M. Ciric´, Beograd have investigated this type extensions for regular semigroup, group, periodic semigroup as well as completely regular semigroup(see [4], [5]). The notion of ideal extensions in ordered semigroups is actually introduced by N. Kehayopulu and M. Tsingelis [8]. In [11], they have worked on ordered semigroups which are nil extensions of Archimedean ordered semigroups. The concepts of nil extensions have been extended to ordered semigroups by Y. Cao [7], with characterizing ordered semigroups which are nil extensions of simple, 1 left simple, t-simple ordered semigroups. Further he described complete semilattices of nil extensions of such ordered semigroups. The aim of this paper is to describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups. This research originates from the research papers [4], [5]. Our paper organized as follows. The basic definitions and properties of ordered semigroups are presented in Section 2. Section 3 is devoted to characterizing the nil extensions of Clifford and left Clifford ordered semigroups. 2 Preliminaries: In this paper N will provide the set of all natural numbers. An ordered semigroup is a partiality ordered set S, and at the same time a semigroup (S,·) such that (forall a, b, x ∈ S) a ≤ b ⇒ xa ≤ xb and ax ≤ bx. It is denoted by (S,·,≤). For an ordered semigroup S and H ⊆ S, denote (H] := {t ∈ S : t ≤ h, for some h ∈ H}. An element z in S is called zero of S if z < x and zx = xz = z for every x ∈ S. Let I be a nonempty subset of an ordered semigroup S. I is a left (right) ideal of S, if SI ⊆ I (IS ⊆ I) and (I] = I. I is an ideal of S if I is both a left and a right ideal of S. An (left, right) ideal I ofS is proper if I 6= S. The intersection of all ideals of an ordered semigroup S, if nonempty, is called the kernel of S and is denoted by K(S). AnorderedsemigroupS iscalledagrouplikeorderedsemigroupifforalla,b ∈ S there are x,y ∈ S such that a ≤ xb and a ≤ by [1] and S left group like if for all a,b ∈ S there is x ∈ S such that a ≤ xb [1]. Kehayopulu [9] defined Greens relations on an ordered semigroup S as follows: aLb if L(a) = L(b), aRb if R(a) = R(b), aJb if I(a) = I(b), and H = L ∩ R. These four relations L, R, J and H are equivalence relations. In an ordered semigroup S, an equivalence relation ρ is called left (right) congruence if for a,b,c ∈ S a ρ b implies ca ρ cb (ac ρ bc). ρ is congruence if it is both left and right congruence. A congruence ρ on S is called semilattice if for all a,b ∈ S a ρ a2 and ab ρ ba. A semilattice congruence ρ on S is called complete if a ≤ b implies (a,ab) ∈ ρ. The ordered semigroup S is called complete semilattice of subsemigroup of type τ if there exists a complete semilattice congruence ρsuch that (x) is atype τ subsemigroup of S. Equivalently: ρ There exists a semilattice Y and a family of subsemigroups {S} of type τ of S such that: α∈Y 1. S ∩S = φ for any α, β ∈ Y with α 6= β, α β 2 2. S = S S , α∈Y α 3. S S ⊆ S for any α, β ∈ Y, α β αβ 4. S ∩(S ] 6= φ implies β (cid:22) α, where (cid:22) is the order of the semilattice Y defined by β α (cid:22):= {(α, β) | α = α β (β α)} [10]. Let S be an ordered semigroup with the zero 0. An element a ∈ S is called a nilpotent if an = 0 for some n ∈ N. The set of all nilpotents of S is denoted by Nil(S). S is called nil ordered semigroup (nilpotent) [7] if S = Nil(S). Due to Cao [7] the definition of nil extension of ordered semigroup is as follows. Definition 2.1. [7] Let I be an ideal of an ordered semigroup S. Then (S/I,·,(cid:22)) is called the Rees factor ordered semigroup of S modulo I, and S is called an ideal extension of I by ordered semigroup S/I. An ideal extension S of I is called a nil-extension of I if S/I is a nil ordered semigroup. Lemma 2.2. [7] Let S be an ordered semigroup and I an ideal of S. Thenthe followingare equivalent: (i) S is a nil-extension of I; (ii) (∀a ∈ S)(∃m ∈ N) am ∈ I. In [1], we have introduced the notion of Clifford and left Clifford ordered semigroups and char- acterized their structural representation. For the correspondences of the results of this paper we are stating some of them. Definition 2.3. [1] A regular ordered semigroup S is called Clifford ordered (left Clifford ordered ) semigroup if for all a,b ∈ S and ab ∈ (bSa] (ab ∈ (Sa]). Theorem 2.4. [1] Let S be a regular ordered semigroup. Then followings hold in S: 1. S is Clifford if and only if L = R. 2. L is a complete semilattice congruence if S is Clifford. 3. S is Clifford ordered semigroup if and only if it is a complete semilattice of group like ordered semigroups. Theorem 2.5. [1] Let S be a regular ordered semigroup. Then followings hold in S: 1. L is a complete semilattice congruence if S is left Clifford. 2. S is left Clifford ordered semigroup if and only if it is a complete semilattice of left group like ordered semigroups. 3 3 Main Results: Now we describe all those ordered semigroups which are the nil extensions of Clifford, left Clifford, group like, left group like ordered semigroups. We omit the proof of the following lemma as it is straightforward. Lemma 3.1. An ordered semigroup S is left group like ordered semigroup if and only if a ∈ (aSab] if for all a, b ∈ S. Theorem 3.2. An ordered semigroup S is a nil extension of a left group like ordered semigroup if and only if for every a,b ∈ S, there exists n ∈ N such that an ∈ (anSanb] and for every a ∈ S, b ∈ Reg (S), a ≤ ba implies a ≤ axb for some x ∈ S. ≤ Proof. Suppose that S is a nil extension of a left group like ordered semigroup K and a,b ∈ S. Then there is m ∈ N such that am ∈ K. Regularity of K implies that am ≤ amxam for some x ∈ K. Further, for xam,amb ∈ K; the left simplicity of K yields that xam ≤ yamb, for some y ∈ K. This gives that am ≤ amxam ≤ amyamb. Next let b ∈ Reg (S) and a ∈ S such that a ≤ ba. Since ≤ b ∈ Reg (S), there exists z ∈ S such that b ≤ b(zb)n for all n ∈ N. Then for some n ∈ N, ≤ 1 (zb)n1 ∈ K; whence b(zb)n1 ∈ K. Thus b ∈ K and so ba ∈ K. Thus a ∈ K. Since K is a left group like ordered semigroup, for a,ba ∈ K it follows that a ≤ asba for some s ∈ K, by Lemma 3.1. Thus the given conditions follow. Conversely, assume that given conditions hold in S. Let a ∈ S be arbitrary. Then by given condition we have am ≤ amxam+1, for some x ∈ S and m ∈ N. This implies am+1 ∈ Reg (S) and so ≤ Reg (S) 6= φ. Denote T = Reg (S). ≤ ≤ Let us choose s ∈ S and a ∈ T. Then the definition of T implies a ≤ a(xa)n, for all n ∈ N and some x ∈ S. Thus sa ≤ sa(xa)n, for all n ∈ N. Now for xa,sa ∈ S, there exists m ∈ N and t ∈ S such that 1 1 (xa)m1 ≤ (xa)m1t (xa)m1sa, by the first condition. 1 Thena ≤ a(xa)m1 impliessa ≤ sa(xa)m1 andhencesa ≤ sa(xa)m1t (xa)m1sa; where(xa)m1t (xa)m1 ∈ 1 1 S. So sa ∈ T. Also a ≤ (ax)na for all n ∈ N. Let m ∈ N be such that (ax)m2 ∈ T. Then 2 as ≤ (ax)m2as implies by the second condition that as ≤ ast (ax)m2 for some t ∈ S. (1) 2 2 4 Denote t = (ax)m2. Then the definition of T implies t ≤ t(zt)n for all n ∈ N for some z ∈ T. (2) Then using the first condition for zt,as ∈ S we have that (zt)m3 ≤ (zt)m3t (zt)m3as 3 for some m ∈ N and t ∈ S. That is t ≤ t(zt)m3t (zt)m3as, by (2). So from (1) we have as ≤ 3 3 3 ast t(zt)m3t (zt)m3as, and hence as ∈ T. 2 3 Next choose a ∈ S and b ∈ T such that a ≤ b. Since b ∈ T there is x ∈ S such that b ≤ b(xb)n for all n ∈ N. Now for xb,a ∈ S, it follows from the first condition that (xb)m4 ≤ (xb)m4t (xb)m4a for some m ∈ N and t ∈ S. (3) 4 4 4 So a ≤ b implies that a ≤ b(xb)m4t (xb)m4a 4 = bt a; where t = (xb)m4t (xb)m4. 5 5 4 Since b ∈ T, by above we have bt ∈ T. Say bt = t . Then for a ∈ S, and t ∈ T, a ≤ t a yields 5 5 6 6 6 that a ≤ at t for some t ∈ S, by second condition. Therefore 7 6 7 a ≤ at bt 7 5 = at b(xb)m4t (xb)m4 7 4 ≤ at b(xb)m4t (xb)m4t (xb)m4a, by (3). 7 4 4 Clearly t b(xb)m4t (xb)m4t (xb)m4 ∈ T, as b ∈ T. Thus a ∈ T, which shows that T is an ideal of S. 7 4 4 Finally let c,d ∈ T. Then there is v ∈ S such that c ≤ c(vc)n for all n ∈ N. Now for vc,d ∈ S, there exists t ∈ S such that c ≤ c(vc)m5t (vc)m5d for some m ∈ N. Since 8 8 5 c ∈ T, c(vc)m5t (vc)m5 ∈ T. Hence T is left simple. Thus T is left group like ordered semigroup such 8 that for every a ∈ S there is m ∈ N, am ∈ T. Hence S is nil extension of a left group like ordered semigroup T. In the following result we provide an independent proof of Corollary 5.2 of [7]. Theorem 3.3. An ordered semigroup S is a nil extension of a group like ordered semigroup if and only if for all a,b ∈ S, there exists n ∈ N such that an ∈ (bnSbn]. 5 Proof. Suppose that S is a nil extension of a group like ordered semigroup G and a,b ∈ S. Then there exists n ∈ N such that an,bn ∈ G. Since G is a group like ordered semigroup, there exists u ∈ G such that an ≤ bnu. Also for u,bn ∈ G there exists x ∈ G such that u ≤ xbn. This implies bnu ≤ bnxbn. Thus an ≤ bnxbn and hence an ∈ (bnSbn]. Conversely, let us assume that given condition holds in S. Choose a ∈ S. Then for some m ∈ N and x ∈ S, am ≤ amxam. Thus Reg (S) 6= φ. Say G = Reg (S). So for every a ∈ ≤ ≤ S, there exists m ∈ N such that am ∈ G. Let us consider b ∈ G and s ∈ S. Then for all n ∈ N bs ≤ bybs ≤ (by)nbs for y ∈ S. (4) Using the given condition for bs,by ∈ S, we obtain (by)mbs ≤ (bs)mz(bs)m+1 for z ∈ S and m ∈ N. This yields that bs ≤ (bs)mz(bs)m+1 ≤ bstbs; where t = bsm−1zbsm. Thus bs ∈ G. Similarly sb ∈ G. Next let a ∈ S and b ∈ G be such that a ≤ b. Since b ∈ G there exists x ∈ S such that b ≤ bxb and hence b ≤ (bx)nb(xb)n for all n ∈ N, which implies that a ≤ am(z ambanz)an, for some m,n ∈ N and z,z ∈ S 1 1 = ata, where t = am−1z ambanzan−1. 1 So a ∈ G. Hence G is an ideal of S. Finally, consider a,b ∈ G. Then there exists x ∈ S such that a ≤ (ax)na for all n ∈ N, and so by the given condition it follows that a ≤ bm′z′bm′a for some m′ ∈ N and z′ ∈ S. This gives that a ≤ bu for some u = bm′−1z′bm′a ∈ G. Similarly there is some v ∈ S such that a ≤ vb. This shows that G is a group like ordered semigroup. Hence S is a nil extension of a group like ordered semigroup G. Theorem 3.4. An ordered semigroup S is a nil extension of a Clifford ordered semigroup if and only if for every x,a,y ∈ S, there exists n ∈ N such that xany ∈ (xanySyanx]∩(yanxSxany] and a ∈ S, b ∈ Reg (S) such that a ≤ b, implies a ∈ (Sab]. ≤ 6 Proof. First suppose that S is a nil extension of a Clifford ordered semigroup K. Let x,a,y ∈ S. Then there is m ∈ N such that am ∈ K. Since K is an ideal of S, xamy ∈ K. Since K is a regular, there exists z ∈ K such that 1 xamy ≤ xamyz xamy. (5) 1 Now z x,amy ∈ K implies that 1 z xamy ≤ (amy)u (z x), for some u ∈ K, since S Clifford. (6) 1 1 1 1 Similarly for am,(yu z ) ∈ K there is u ∈ S such that 1 1 2 am(yu z ) ≤ (yu z )u am. (7) 1 1 1 1 2 Therefore xamy ≤ xamyz xamy 1 ≤ xamyam(yu z )x, by (6) 1 1 ≤ xamy2u z u amx, by (7). 1 1 2 Thus xamy ≤ xamyz xamy2u z u amx. (8) 1 1 1 2 Also, for amy2,u z u ∈ K there exists u ∈ K such that amy2u z u ≤ u z u u amy2. Then from 1 1 2 3 1 1 2 1 1 2 3 (8), we obtain that xamy ≤ xamy(z xu z u u amy)yamx 1 1 1 2 3 ≤ xamysyamx; where s = z xu z u u amy. 1 1 1 2 3 Therefore xamy ∈ (xamySyamx]. Similarly xamy ∈ (yamxSxamy]. NowK ⊆ Reg (S)impliesthatReg (S) 6= φ. Consider b ∈ Reg (S) and a ∈ S such that a ≤ b. ≤ ≤ ≤ Since b ∈ Reg (S), there exists z ∈ S such that b ≤ (bz)nb for all n ∈ N. Since S is a nil extension ≤ of K, there exists n ∈ N such that (bz)n1 ∈ K. This gives (bz)n1b ∈ K, which gives b ∈ K and so 1 a ∈ K, since K is an ideal of S. Since K is a Clifford ordered semigroup, by Theorem 2.4(ii) L is a congruence on S. Since a,b ∈ K we have aLab and hence a ⊆ (Sab]. Conversely, let us assume that given conditions hold in S. Let a ∈ S be arbitrary. Then by the first condition there exists n ∈ N such that an+2 ≤ an+2xan+2, 7 for some x ∈ S. Thus Reg (S) 6= φ. Say T = Reg (S). It is now clear that for each a ∈ ≤ ≤ S, there exists m ∈ N such that am ∈ T. Let s ∈ S and x ∈ T. Then for all n ∈ N and for some t ∈ S, x ≤ (xt)nx which implies that sx ≤ sx(tx)n−1tx, for all n ∈ N. By first condition there are s ∈ S and m ∈ N such that 1 1 sx ≤ sx(tx)m1txs tx(tx)m1sx and thus sx ≤ sxpsx; where p = (tx)m1txs tx(tx)m1. Also for every 1 1 n ∈ N, xs ≤ x(tx)ns ≤ xt(xt)n−1xs. So there is m ∈ N such that 2 xs ≤ xs(xt)m2xts xt(xt)m2xs 2 ≤ xsqxs; where q = (xt)m2xts xt(xt)m2. 2 Thus sx,sx ∈ T. To show T, a Clifford ordered semigroup, choose a,b ∈ T. Then there is r ∈ S such that ab ≤ abrab ≤ (abra)(bra)n−1b, for all n ∈ N. (9) Now for abra,bra,b ∈ S, the first condition yields that abra(bra)m3b ≤ b(bra)m3abrap abra(bra)m3b for some p ∈ S and m ∈ N. 1 1 3 Therefore from (9) ab ≤ b(bra)m3abrap abra(bra)m3b 1 ≤ b(bra)m3abrap abra(bra)m3−1brab 1 ≤ bgab; where g = (bra)m3abrap abra(bra)m3−1br ∈ T. (10) 1 Similarly there are m ∈ N and p ∈ S such that ab ≤ abra(bra)m4bp b(bra)m4abra. So from (10), 4 2 2 ab ≤ b(gabra(bra)m4bp b(bra)m4)abra = bua; where u = gabra(bra)m4bp b(bra)m4abr ∈ T. 2 2 Hence T is Clifford ordered semigroup. 8 Now let a ≤ b for some a ∈ S and b ∈ T. Then by the second condition, there is z ∈ S such that a ≤ zab, that is, a ≤ zabtab for some t ∈ S. (11) Since T is Clifford ordered semigroup, for zabt,ab ∈ T it follows that zabtab ≤ abp zabt for some p ∈ S. 3 3 Similarly for bp za,bt ∈ T, we have bp zabt ≤ btp bp za for some p ∈ S. 3 3 4 3 4 The last two inequalities together with (11) yields that a ≤ aha, where h = btp bp z. Thus 4 3 a ∈ T and so T is an ideal of S. Hence S is a nil extension of a Clifford ordered semigroup T. Theorem 3.5. An ordered semigroup S is a nil extension of a left Clifford ordered semigroup if and only if for every x,a,y ∈ S, there exists n ∈ N such that xany ∈ (xanySyanx]∩(xanySxany] and a ∈ S, b ∈ Reg (S) such that a ≤ b implies a ≤ azab for some z ∈ S. ≤ Proof. Let S be a nil extension of a left Clifford ordered semigroup K. Choose x,a,y ∈ S. Then there exists m ∈ N such that am ∈ K. Since K is an ideal of S, xamy ∈ K. Also the regularity of K yields that xamy ≤ xamyz xamy, for somez ∈ K. (12) 1 1 Since K is a left Clifford ordered semigroup and z x,amy ∈ K, by Theorem 2.5 it follows that 1 z xamy ≤ z (z x) for some z ∈ K. Therefore 1 2 1 2 xamy ≤ xamyz xamy 1 ≤ xamyz xamyz xamy 1 1 ≤ xamyz xamyz (z x). (13) 1 2 1 Similarly, for am,yz z ∈ K there is z ∈ K such that 2 1 3 amyz z ≤ z am, (14) 2 1 3 and for am,z xz ∈ K there is z ∈ K such that 1 3 4 amyz xz ≤ z amy. (15) 1 3 4 Thus from (13) we obtain that xamy ≤ xamyz xamyz z x 1 2 1 ≤ xamyz xamz amx, from (14) 1 3 ≤ xamyz amyamx, from (15). 4 9 Hence xamy ∈ (xamySyamx] and so xamy ∈ (xamySyamx]∩(xamySxamy], from (12). To show the second condition choose a ∈ S and b ∈ Reg (S) be such that a ≤ b. By the ≤ regularity of b yields that b ≤ b(tb)n for some t ∈ S and for all n ∈ N. Then there is r ∈ N such that (tb)r ∈ K. Since K ia an ideal b(tb)r ∈ K and so b ∈ K. Thus a ∈ K. Since K is left Clifford ordered semigroup, L is congruence on S, by Theorem 2.5(i). Thus aLab. Then a,ab are in Theorem 3.1, a ≤ azab for some z ∈ S. This proves the necessary condition. Conversely, suppose that given conditionsholdinS. Let a ∈ S. Then by thefirst conditionthere exists n ∈ N such that an+2 ≤ an+2xan+2, for some x ∈ S. Thus Reg (S) 6= φ. Say T = Reg (S). ≤ ≤ Now for each a ∈ S, there exists m ∈ N such that am ∈ T. Let s ∈ S and x ∈ T. Then for all n ∈ N and for some t ∈ S, x ≤ (xt)nx. This implies sx ≤ sx(tx)n−1tx, for all n ∈ N. Then by the first condition there are s ∈ S and m ∈ N such that 1 1 sx ≤ sx(tx)m1txs tx(tx)m1sx 1 ≤ sxpsx; where p = (tx)m1txs tx(tx)m1. 1 Therefore sx ∈ T. We now show that T is a left Clifford ordered semigroup. For this let us assume that a,b ∈ T. Then there is t ∈ S such that 1 ab ≤ abt ab ≤ a(bt a)nbt ab for all n ∈ N. 1 1 1 Then by first condition there are t ∈ S and m′ ∈ N such that 2 ab ≤ a(bt a)m′bt ab 1 1 ≤ a(bt a)m′bt abt bt ab(bt a)m′a 1 1 2 1 1 Therefore ab ≤ t′a; where t′ = a(bt a)m′bt abt bt ab(bt a)m′. Since a ∈ T we have t′ ∈ T. Hence T 1 1 1 1 2 1 1 1 is left Clifford ordered semigroup. Next let a ∈ S and b ∈ T such that a ≤ b. Using second condition we have a ≤ azab, for some z ∈ S. Then for all n ∈ N, a ≤ a(za)nbn. So for some m′′ ∈ N, (za)m′′ ∈ T. Since T is a left Clifford, (za)m′′bm′′ ∈ (Ta]. So a ∈ (aSa], that is a ∈ T. Finally to show T, an ideal of S we need only to show that xs ∈ T. The regularity of x yields that xs ≤ x(tx)ns for all n ∈ N. Also for some l ∈ N, x(tx)ls ∈ T. Then xs ≤ x(tx)ls implies that xs ∈ T by above. Thus T is an ideal of S. Hence S is a nil extension of a left Clifford ordered semigroup T. 10