Type A ω 2 -semigroups

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1 ADVANCES IN MATHEMATICS(CHINA) Type A ω 2 -semigroups SHANG Yu 1,, WANG Limin 2, doi: /sxjz b (1. Department of Mathematics and Statistics, Puer University, Puer, Yunnan, , P. R. China; 2. School of Mathematics, South China Normal University, Guangzhou, Guangdong, , P. R. China) Abstract: In this paper, we study type A ω 2 -semigroups with a proper -kernel in which D = D and obtain their structure theorem. We also obtain an isomorphism theorem for such semigroups. Keywords: type A semigroup; ω 2 -semigroup; isomorphism MR(2010) Subject Classification: 20M10 / CLC number: O152.7 Document code: A 1 Introduction and Preliminaries In [1], a structure theorem of a -bisimple type A ω-semigroup is obtained. Type A ω- semigroups were studied and classified by Asibong-Ibe [2]. Warne investigated the bisimple ω n - semigroups, and proved in [15] that any bisimple ω n -semigroup has a structure as (G C n, ), where G is a group and C n is a 2n-cyclic semigroup, under a suitable multiplication. The studies of type A ω 2 -semigroups can be regarded as a natural next step beyond that of type A ω-semigroups. We study type A ω 2 -semigroups with a proper -kernel in which D = D in this paper. In this section we present some necessary notations and terminologies. Section 2 gives a structure theorem of these type A ω 2 -semigroups and an isomorphism theorem for such semigroups. For a semigroup S we denote by E(S) the set of idempotents of S. Let S be a semigroup such that E(S) is non-empty. We define a partial order on E(S) by the rule that e f if and only if ef = f = fe. Let N 0 denote the set of all non-negative integers and let N denote the set of all positive integers. If E(S) = {e i : i N 0 } and if the elements of E(S) form the chain e 0 > e 1 > e 2 >, then the semigroup S is called an ω-semigroup. We denote by C ω the set {e 0,e 1,e 2, }, with e 0 > e 1 > e 2 >. We define a partial order on N 0 N 0 in the following manner: if (m,n),(p,q) N 0 N 0, (m,n) (p,q) if and only if m > p or, m = p and n q. The set N 0 N 0 with the above partial order is called an ω 2 -chain, and is denoted by C ω 2. A partially ordered set order isomorphic to C ω 2 is also called an ω 2 -chain. We say that a semigroup Received date: Revised date: Foundation item: This research is supported by NSFC (No ) and the Science Foundation of the Department of Education of Yunnan Province (No. 2011Y478). shangyu503@hotmail.com; wanglm@scnu.edu.cn

2 2 S isanω 2 -semigroupifandonlyife(s)isorderisomorphictoc ω 2.Thus, ifs isanω 2 -semigroup, then we can write E(S) = {e m,n : m,n N 0 }, where e m,n e p,q if and only if (m,n) (p,q). In [11], Shang and Wang have shown that the set S = N 0 N 0 N 0 N 0 with an operation defined by (m,n q +max{q,b},d b+max{q,b},c), if p = a, (m,n,q,p)(a,b,d,c) = (m,n,q,c a+p), if p > a, (m p+a,b,d,c), if p < a is a bisimple ω 2 -semigroup and called it the quadrucyclic semigroup, which is denoted by B ω 2. Recall that S = α Y S α is a strong semilattice of the semigroups S α when Y is a semilattice, {S α : α Y} is a disjoint family of semigroups and for α,β Y with α β there are homomorphisms φ α,β : S α S β satisfying (i) φ α,α is the identity map for each α Y; (ii) φ α,β φ β,γ = φ α,γ for every α,β,γ in Y such that α β γ, and such that the product in S is given by ab = aφ α,αβ bφ β,αβ where a S α,b S β. We write S = S(Y;S α ;φ α,β ). Let S be a semigroup. Let a,b S such that for all x,y S 1, ax = ay if and only if bx = by. Then a,b are said to be L -equivalent and written as al b. Dually, ar b if for all x,y S 1, xa = ya if and only if xb = yb. If S has an idempotent e, the following characterization is known. Lemma 1.1 [4] Let S be a semigroup and e be an idempotent in S. Then the following are equivalent: (i) el a; (ii) ae = a and for all x,y S 1, ax = ay implies ex = ey. By duality, a similar condition holds for R. A semigroup in which each L -class and each R -class contains an idempotent is called an abundant semigroup (see [4]). The join of the equivalence relations L and R is denoted by D and their intersection by H. Thus ah b if and only if al b and ar b. In general, L R R L and neither equals D. Basically, ad b if and only if there exist elements x 1,x 2,,x 2n 1 in S such that al x 1 R x 2 L L x 2n 1 R b. Let H be an H -class in a semigroup S with e H, where e is an idempotent in S. Then H is a cancellative monoid. Denote by R,L the left and right Green s relations respectively on S. We write H = L R and D = L R = L R = R L. It is well known that L L,R R,D D,H H for a semigroup S and if a,b are regular elements of S, then al b (ar b) if and only if alb (arb). To avoid ambiguity we sometimes denote a relation K on S by K(S). The following notations are used. An L -class containing an element a S is denoted by L a. Similarly R a is an R -class with an element a S. To avoid ambiguity we sometimes denote the L -class (resp. R -class) containing an element a S by L a(s) (resp. R a(s)). Let S be a semigroup and I be an ideal of S. Then I is called a -ideal if L a I and R a I for all a I. By a -kernel we mean a minimum -ideal. The smallest -ideal containing a is the principal -ideal generated by a and is denoted by J (a). For a,b in S, aj b if and only if J (a) = J (b). The relation J contains D. A semigroup S is said to be -simple if the only

3 , «: Type A ω 2 -semigroups 3 -ideal of S is itself. Clearly a semigroup is -simple if all its elements are J -related. Let S be a semigroup with a semilattice E of idempotents. Then S is called a right adequate semigroup if each L -class of S contains an idempotent. Dually, we have the notion of a left adequate semigroup. A semigroup which is both left and right adequate is called an adequate semigroup. In an adequate semigroup each L -class and each R -class contain a unique idempotent. Let x be an element of an adequate semigroup S, and let x (x + ) denote the unique idempotent in the L -class L x (R -class R x) of x. A right (left) adequate semigroup S is called a right (left) type A semigroup if ea = a(ea) (ae = (ae) + a) for all elements a in S and all idempotents e in S. An adequate semigroup S is type A if it is both right and left type A. A -subsemigroup of an adequate semigroup S is a subsemigroup U such that a U if and only if a,a + U. Lemma 1.2 [2] Let U be a -subsemigroup of an adequate semigroup S. Then L (U) = L (S) (U U) and R (U) = R (S) (U U). Remark Every -subsemigroup of an adequate (resp. type A) semigroup is itself adequate (resp. type A). Let S be a type A semigroupwith a semilattice ofidempotents E(S). Then S is called a type A ω 2 -semigroupif E(S) is an ω 2 -chain. Thus in type A ω 2 -semigroupe(s) = {e m,n : m,n N 0 } and e m,n e p,q if and only if (m,n) (p,q). In such a semigroup S, we denote by L m,n (resp. Rm,n) the L -class (resp. R -class) containing idempotent e m,n. That is Rm,n = {a S : ar e m,n }, L p,q = {a S : al e p,q }. Let H(m,n),(q,p) denote the R m,n L p,q. That is H (m,n),(q,p) = {a S : ar e m,n, al e p,q }. If H (m,n),(q,p), evidently H (m,n),(q,p) is an H -class of S. Clearly, if a H (m,n),(q,p), then a + = e m,n and a = e p,q. Lemma 1.3 [11] Let S be a type A ω 2 -semigroup. Then H (m,n),(q,p) H (a,b),(d,c) H (i,j),(l,k), where (m,n q +max{q,b},d b+max{q,b},c), if p = a, (i,j,l,k) = (m,n,q,p)(a,b,d,c) = (m,n,q,c a+p), if p > a, (m p+a,b,d,c), if p < a. Let S be an adequate semigroup, and let a,b S. The relation D is defined on S by a Db if and only if a Db and a + Db + for a,b,a +,b + E(S). D is an equivalence relation and satisfies the inclusion D D D on an adequate semigroup S (see [2]). Lemma 1.4 [2] Let S be an adequate semigroup. The following conditions are equivalent: (i) D = D; (ii) every nonempty H -class contains a regular element. Furthermore, if (i) and (ii) hold, then D = L R = R L.

4 4 However, as an example in [1] shows, (i) and (ii) are not necessaryconditions for the equality L R = R L. 2 Type A ω 2 -semigroups In this section, we reduce the problem of determining the structure of type A ω 2 -semigroups in which D = D to that of determining the structure of -simple type A ω 2 -semigroups in which D = D. Let S be a type A ω 2 -semigroup with a semilattice E of idempotents. Then E(S) = {e m,n : m,n N 0 } and for e m,n,e p,q E, e m,n e p,q if and only if (m,n) (p,q). In such a semigroup S, the idempotent e 0,0 is an identity since e 0,0 e m,n = e m,n e 0,0 = e m,n for all idempotents e m,n, (m,n) N 0 N 0. Lemma 2.1 Let S be a type A ω 2 -semigroup and E be its chain of idempotents. If S i,j = e i,j Se i,j, then (i) S i,j is a type A ω 2 -semigroup with identity e i,j ; (ii) L (S i,j ) = L (S) (S i,j S i,j ) and R (S i,j ) = R (S) (S i,j S i,j ); (iii) H (i,j),(j,i) is the H -class of S i,j containing the identity of S i,j ; (iv) S i,j = {H(m,n),(q,p) : (m,n) (i,j), (p,q) (i,j)}. Proof (i) Certainly S i,j is a subsemigroup of S and e i,j is its identity. Since e i,j e k,l = e k,l e i,j = e k,l for all (i,j) (k,l), E(S i,j ) = {e k,l : (k,l) (i,j)}. Thus S i,j is an ω 2 -semigroup. To show that S i,j is type A, let x S i,j. Then x = e i,j xe i,j. So by [5, Proposition 1.6], x e i,j and x + e i,j. Thus x,x + S i,j. Conversely, if x,x + S i,j, then x = x + xx = (e i,j x + )x(x e i,j ) = e i,j xe i,j S i,j and so S i,j is a -subsemigroup of S. Hence, by Remark, since S is type A, then S i,j is also type A. (ii) This holds by Lemma 1.2. (iii) This is an immediate consequence of (ii). (iv) We observe that if x H(m,n),(q,p) where (m,n),(p,q) (i,j), then x = e m,n xe p,q = (e i,j e m,n )x(e p,q e i,j ) = e i,j (e m,n xe p,q )e i,j S i,j. Conversely, we have noted that for x S i,j we have x e i,j,x + e i,j so that x for some (m,n) (i,j),(p,q) (i,j). This completes the proof. H (m,n),(q,p) Lemma 2.2 Let S be a type A ω 2 -semigroup in which D = D. If S i = j,l=0 H (i,j),(l,i), B = i=0 S i, T p+j = H(p,j),(j,p) (j = 0,1,,q 1) and C = ( p 1 i=0 S i) ( q 1 j=0 T p+j) for p N and q N, then (i) B is an ω-chain of type A ω-semigroups in which D = D; (ii) B has no -kernel; (iii) C is a chain of S i (i = 0,1,,p 1) and T p+j (j = 0,1,,q 1) where S i is a type A ω-semigroup in which D = D and T p+j is a cancellative monoid. Proof (i) Let S be a type A ω 2 -semigroupin which D = D. Consider the set B = i=0 S i where S i = j,l=0 H (i,j),(l,i). Let E(S) = {e m,n : m,n N 0 }. For H(i,j),(l,i),H (k,m),(n,k) in B, we have by Lemma 1.3 that H (i,j),(l,i) H (k,m),(n,k) B, H (k,m),(n,k) H (i,j),(l,i) B.

5 , «: Type A ω 2 -semigroups 5 Thus B is a subsemigroup of S. Evidently, S i,i N 0 is a subsemigroup of S and e i,0 is its identity. Since E(S i ) = {e i,j j N 0 }, S i is an ω-semigroup. To show that S i is a type A semigroup in which D = D, let x S i. Then x H (i,j),(l,i) for some j N0 and some l N 0 and so x = e i,l and x + = e i,j. Thus, x,x + S i. Conversely, if x,x + S i, then x = e i,n and x + = e i,m for some n N 0 and some m N 0 and so x H (i,m),(n,i) S i. Thus S i is a -subsemigroup of S. Since S is type A, so S i is also type A by Remark. By Lemmas 1.2 and 1.4, since S is a type A semigroup in which D = D, so also is S i. Define γ i : S i S i+1 by xγ i = xe i+1,0. Since xe i+1,0,e i+1,0 x H (i+1,0),(0,i+1) S i+1, then (xy)γ i = xye i+1,0 = x(e i+1,0 ye i+1,0 ) = (xe i+1,0 )(ye i+1,0 ) = xγ i yγ i, showing that γ i is a homomorphism with Imγ i H (i+1,0),(0,i+1). For i,j N0 such that i < j, let α i,j = γ i γ i+1 γ j 1 ; and for each i N 0, let α i,i denote the identity automorphism of S i. Then Imα i,j H (j,0),(0,j) for i < j. Define a multiplication on B by the rule that a i b j = (a i α i,t )(b j α j,t ), a i S i,b j S j, where t = max{i,j}. If a i S i and b j S j, then a i b j S t, and that a i b j = (a i e t,0 )(b j e t,0 ). Thus B is an ω-chain of type A ω-semigroups in which D = D. (ii) Write T n = i=n S i (n N 0 ). Then it is clear from the law of multiplication that T n is a -ideal of B for all n N 0. Moreover, T n =. n=0 Hence, B has no -kernel. Thus we have proved (ii). (iii) Now, let p N and q N. We consider the set C = ( p 1 i=0 S i) ( q 1 j=0 T p+j), where T p+j = H(p,j),(j,p) (j = 0,1,,q 1). By Lemma 1.3, C is a subsemigroup of S. Evidently, T p+j (j = 0,1,,q 1) is a cancellative monoid, with identity e p,j. Define γ i : S i S i+1 by xγ i = xe i+1,0 (i = 0,1,,p 2), γ p 1 : S p 1 T p by xγ p 1 = xe p,0 and γ p+j : T p+j T p+j+1 by xγ p+j = xe p,j+1 (j = 0,1,,q 1). Evidently, γ i (i = 0,1,,p 1) is a homomorphism with Imγ i H(i+1,0),(0,i+1) and γ p+j (j = 0,1,,q 1) is a homomorphism. For i,j {0,1,,p+q 1} such that i < j, let α i,j = γ i γ i+1 γ j 1 ; and for each i {0,1,,p 1}, let α i,i denote the identity automorphism of S i ; and for each j {0,1,,q 1}, let α p+j,p+j denote the identity automorphism of T p+j. Define a multiplication on C as follows: (1) a i b j = (a i α i,t )(b j α j,t ); (2) a i x p+j = (a i α i,p+j )x p+j, x p+j a i = x p+j (a i α i,p+j ); (3) x p+i y p+j = (x p+i α p+i,p+t )(y p+j α p+j,p+t ), where a i S i (0 i p 1), b j S j (0 j p 1), x p+i T p+i (0 i q 1), x p+j,

6 6 y p+j T p+j (0 j q 1), and t = max{i,j}. Then C is a chain of S i (i = 0,1,,p 1) and T p+j (i = 0,1,,q 1). If a i S i and b j S j, then a i b j S t, and that a i b j = (a i e t,0 )(b j e t,0 ). Thus we have proved (iii). Lemma 2.3 Let S be a type A ω 2 -semigroup in which D = D such that Rm,n j=0 H (m,n),(j,m) for some (m,n) N0 N 0, and suppose that (p,q) is the maximal pair (m,n). Then H(p,i),(j,p) = for all i,j N0 such that i < q and i j. Proof As Rp,q j=0 H (p,q),(j,p), there are two non-negative integers k and l where k p such that H(p,q),(l,k). Suppose that a H (p,q),(l,k). If H (p,i),(j,p) for some i N0 and some j N 0 such that i < q and i j, let b H(p,i),(j,p). Now, we consider the following three cases. Case I i < q j. In this case, by Lemma 1.3, ba H(p,i),(j,p) H (p,q),(l,k) H (p,i),(l q+j,k) and so Rp,i l=0 H (p,i),(l,p), which contradicts the maximality of (p,q). Case II i < j < q. In this case, by Lemma 1.3, ba H(p,i),(j,p) H (p,q),(l,k) H (p,i j+q),(l,k) and so Rp,i j+q l=0 H (p,i j+q),(l,p), which contradicts the maximality of (p,q). Case III j < i < q. Since H(p,i),(j,p) contains a regular element s, e p,ir sl e p,j so that e p,i De p,j and e p,i L tr e p,j for some t S. Thus H(p,j),(i,p). Let c H (p,j),(i,p). Then, by Lemma 1.3, ca H (p,j),(i,p) H (p,q),(l,k) H (p,j i+q),(l,k) and so Rp,j i+q l=0 H (p,j i+q),(l,p), again contradicting the maximality of (p,q). Summing up all the above facts, we have the following result. Lemma 2.4 Let S be a type A ω 2 -semigroup in which D = D such that R m,n j=0 H (m,n),(j,m) for some (m,n) N0 N 0 and suppose that (p,q) is the maximal pair (m,n). Then S p,q is the -kernel of S and is a -simple type A ω 2 -semigroup in which D = D. If p > 0 and q > 0, then S = A S p,q, A S p,q =, where A is the subsemigroup ( p 1 i=0 ( j,l=0 H (i,j),(l,i) )) ( q 1 j=0 H (p,j),(j,p)) of S. Proof By Lemma 2.1, S p,q = {H(a,b),(d,c) : (a,b) (p,q),(c,d) (p,q)}. Suppose that (i,j) > (p,q), i m and H(i,j),(n,m) for some n N0. Then Ri,j l=0 H (i,j),(l,i) which contradicts the maximality of (p,q). Suppose that j < q, j l. Then, H(p,j),(l,p) contradicts Lemma 2.3. On the other hand, if (r,s) > (p,q), r m and H(m,n),(s,r), then since H(m,n),(s,r) contains a regular element t, e m,nr tl e r,s so that e m,n De r,s and e r,s L wr e m,n for some w S. Thus H(r,s),(n,m) and so R r,s l=0 H (r,s),(l,r), again contradicting the maximality of (p,q). If s < q, s n and H(p,n),(s,p), then since H (p,n),(s,p) contains a regular element v, e p,n R vl e p,s so that e p,n De p,s and e p,n L zr e p,s for some z S. Thus H(p,s),(n,p), again contradicting Lemma 2.3. Therefore, if (i,j) (p,q), Similarly, L i,j S p,q for (i,j) (p,q). R i,j = {H (i,j),(l,k) : (k,l) (p,q)} S p,q. We first show that S p,q is an ideal of S. Let t H(a,b),(d,c) for some (a,b) (p,q),(c,d) (p,q) and let x S. Then x H (k,l),(j,i) for some (k,l) N0 N 0,(j,i) N 0 N 0 and tx R m,n

7 , «: Type A ω 2 -semigroups 7 for some (m,n) N 0 N 0. Since each H -class contains a regular element, there exist regular elements t and x such that t H(a,b),(d,c) and x H(k,l),(j,i). By Lemma 1.3, we have that t x H tx and so t x R tx. Thus t x R e m,n. It can readily be shown that t x is regular and so t x Re m,n. Similarly t Re a,b. It then follows easily that (m,n) (a,b). Thus tx Rm,n where (m,n) (a,b). Since (a,b) (p,q), it follows that tx S p,q. Similarly, xt S p,q. Summing up all the above facts, we have that S p,q is a -ideal. To show that S p,q is -simple, suppose that x S p,q and xl e h,k where (h,k) (p,q). Then to show that J (x) = S p,q, it is sufficient to prove that e p,q J (e h,k ). As Rp,q j=0 H (p,q),(j,p), there aretwo non-negativeintegersiand j where i p such that H(p,q),(j,i). If i < p, then (i,j) > (p,q) and H(i,j),(q,p) and so R i,j l=0 H (i,j),(l,i) which contradicts the maximality of (p,q). Hence, i > p. Put h = p+u, i = p+l where u N 0 and l N. Suppose that b H(p,q),(j,i). If m is an integer such that ml > u, certainly e p+ml,j J (e p+u,k ) so that L p+ml,j J (e p+u,k ). Since b m H(p,q),(j,p+ml), then bm J (e p+u,k ). But b m R e p,q in S p,q, and hence e p,q J (e h,k ) as required. If I is a -ideal of S, then I S p,q is a -ideal of S p,q. Hence, I S p,q = S p,q so that S p,q I, that is, S p,q is the -kernel of S. It is immediate that S p,q is a type A ω 2 -semigroup in which D = D. Forp > 0andq > 0, leta = S\S p,q. ThenA = ( p 1 i=0 ( j,l=0 H (i,j),(l,i) )) ( q 1 j=0 H (p,j),(j,p) ) which is a subsemigroup by Lemma 1.3. Hence, S = A S p,q, A S p,q = and the proof is complete. As an immediate corollary we have Corollary 2.1 S is -simple if and only if R (0,0) j=0 H (0,0),(j,0). From the above we obtain the following analogue of Munn s [9, Theorem 2.6] for type A ω 2 -semigroups without -kernels, which satisfy the condition D = D. Theorem 2.1 Let S be a type A ω 2 -semigroupin which D = D. The followingconditions are equivalent: (i) S has no -kernel; (ii) S is an ω-chain of type A ω-semigroups in which D = D. Proof If (i) holds, then by Lemma 2.4, Rm,n = j=0 H (m,n),(j,m) for all (m,n) N0 N 0. Hence, S = m=0 ( n,j=0 H (m,n),(j,m)) and is an ω-chain of type A ω-semigroups in which D = D by Lemma 2.2. Conversely, as was shown in Lemma 2.2, an ω-chain of type A ω- semigroups in which D = D has no -kernel. Theorem 2.2 Let T = p+q 1 i=0 T i be a chain of pairwise-disjoint semigroups T 0,T 1,, T p+q 1, where {T 0,T 1,,T p 1 } is a set of type A ω-semigroups in which D = D for some p > 0 and {T p,t p+1,,t p+q 1 } is a set of cancellative monoids for some q > 0 with linking homomorphisms α i,j (0 i < j p + q 1) where Imα i,j H (j,0),(0,j) for 0 i < j p. Let K be a -simple type A ω 2 -semigroup with D = D, and α p+q 1,p+q : T p+q 1 K be a homomorphism with image Imα p+q 1,p+q H 1, where H 1 is the H -class which contains the identity element of K. Then the chain S = T K of semigroups with a linking homomorphism

8 8 θ on T K given by xθ = xα i,p+q 1 α p+q 1,p+q for x T i is a type A ω 2 -semigroup with the -kernel K and in which D = D. Conversely, if S is a type A ω 2 -semigroup containing a -kernel K S and D = D on S, then S is isomorphic to a semigroup constructed as above. Proof It is routine to verify that θ is a homomorphism and so S is a semigroup with K as an ideal. Let E(T i ) = {e i,j : j N 0 } (i = 0,1,,p 1) and E(T p+j ) = {e p,j } (j = 0,1,,q 1). Now α p+q 1,p+q maps T p+q 1 into an H -class of the identity of K, therefore if e i,j is the idempotent in T, then e i,j θ = e i,j α i,p+q 1 α p+q 1,p+q = e p,q 1 α p+q 1,p+q = f 0,0 is the identity of K. Thus e i,j f 0,0 = f 0,0 = f 0,0 e i,j and hence if then E(S) consists of the ω 2 -chain E(K) = {f i,j : i,j N 0 }, e 0,0 e 0,1 e 0,2 e p 1,0 e p 1,1 e p 1,2 e p,0 e p,1 e p,q 1 f 0,0 f 0,1 f 0,2 f 1,0 f 1,1 f 0,2, showing that S is an ω 2 -semigroup. If x K and y T, then since xf 0,0 = x(e 0,0 θ) = xe 0,0, whereas ye 0,0 = y yf 0,0, we have that (x,y) / L. This shows that L x K. Similarly we have R x K and thus K is a - ideal. Thusboth K andt areunionsofl andr -classesofs. Infact, L (U) = L (S) (U U), R (U) = R (S) (U U) for U = T and U = K. Suppose that a,b K, ar (K)b and xa = ya where x,y S 1. Then x(f 0,0 a) = y(f 0,0 a), so (xf 0,0 )a = (yf 0,0 )a. Since ar (K)b, xf 0,0 K and yf 0,0 K, (xf 0,0 )b = (yf 0,0 )b. Thus xb = yb and it follows that ar (S)b. On the other hand, if c,d T, cr (T)d and hc = kc where h,k S 1, then either h,k T 1 in which case certainly hd = kd, or h,k K. In the latter case it is easy to see that h = k and so hd = kd. It follows that cr (S)d. The dual arguments give the results for the relation L. As [4, Proposition 6.9] now applies and the idempotent of S commutes, S is an adequate semigroup. That S is a type A semigroup follows in a straightforward manner from the fact that T and K are both type A semigroups. Suppose that I is a -ideal of S. Then I contains an idempotent element e i,j S and since e i,j e m,n = e m,n e i,j = e m,n for all e m,n S,(m,n) (i,j), we have that I K. Thus I K is a -ideal of K and as K is -simple we have that I K = K. Thus K I and K is the -kernel of S. Finally if H is an H -class of S, either H is an H -class of T i, 0 i p+q 1 or H is an H -class of K, in which case each H -class contains a regular element. Hence, D (S) = D(S). To establish the converse, consider a type A ω 2 -semigroup with the -kernel K S and D (S) = D(S). Let E(S)betheω 2 -chainofidempotents ins andlete(s) = {e m,n : m,n N 0 }.

9 , «: Type A ω 2 -semigroups 9 Without loss of generality, let Ri,j h=0 H (i,j),(h,i) for some (i,j) N0 N 0, and (p,q) be the maximal pair (i,j). By Lemma 2.4, K = S p,q is a -simple type A ω 2 -semigroup, in which D (K) = D(K). Since (p,q) < (0,0), Lemma 2.4 shows that S = A K, A K =, where A is the subsemigroup p 1 i=0 ( j,l=0 H (i,j),(l,i) ) ( q 1 j=0 H (p,j),(j,p)) of S. Put T i = j,l=0 H (i,j),(l,i) (i = 0,1,,p 1), T p+j = H(p,j),(j,p) (j = 0,1,,q 1) and let α i,j be the linking homomorphisms as in the proof of Lemma 2.2(iii). By the proof of Lemma 2.2(i), T i (i = 0,1,,p 1) is a type A ω-semigroup in which D = D. Evidently, T p+j (j = 0,1,,q 1) isacancellativemonoid. Define α p+q 1,p+q : T p+q 1 K byxα p+q 1,p+q = xe p,q where e p,q is the identity of K. Since xe p,q,e p,q x H(p,q),(q,p), then xe p,q = e p,q xe p,q = e p,q x. Consequently, (xy)α p+q 1,p+q = xye p,q = x(e p,q ye p,q ) = (xe p,q )(ye p,q ) = xα p+q 1,p+q yα p+q 1,p+q, showing that α p+q 1,p+q is a homomorphism. Furthermore, H (p,q),(q,p) contains the identity e p,q of K. So defining θ as in the statement of the theorem, we see that for x A, y K, xy = x(e p,q y) = (xe p,q )y = (xe p,q 1 e p,q )y = (xθ)y, and in a similar manner yx = y(xθ). This completes the proof. We conclude this section with a criterion for isomorphisms of two type A ω 2 -semigroups with proper -kernel and D = D. Let S = T K, S = T K be type A ω 2 -semigroups containing the -kernel K S (resp. K S ) and D = D on S (resp. S ) and T = p+q 1 i=0 T i (resp. T = p +q 1 i=0 T i ) be a chain of pairwise-disjoint semigroups T 0,T 1,,T p+q 1 (resp. T 0,T 1,,T p +q 1 ) where T 0,T 1,,T p 1 (resp. T 0,T 1,,T p 1 ) are the sets of type A ω-semigroups in which D = D for some p > 0 (resp. p > 0) and T p,t p+1,,t p+q 1 (resp. T p,t p +1,,T p +q 1 ) are the sets of cancellative monoids for some q > 0 (resp. q > 0) with linking homomorphisms α i,j (0 i < j p+q 1) (resp. α i,j (0 i < j p + q 1)), where image Imα i,j H(j,0),(0,j) (resp. Imα i,j H (j,0),(0,j) ) for i < j p (resp. i < j p ) and K (resp. K ) is a -simple type A ω 2 -semigroup with D = D, and α p+q 1,p+q : T p+q 1 K (resp. α p +q 1,p +q : T p +q 1 K ) is a homomorphism with image Imα p+q 1,p+q H1 (resp. Im α p +q 1,p +q H 1 ) where H 1 (resp. H 1 ) is the H -class which contains the identity element of K (resp. K ). We denote by T p+q (resp. T p +q ) the H -class containing the identity of K (resp. K ). Theorem 2.3 The semigroups S and S are isomorphic if and only if the following conditions are satisfied: (i) (p,q) = (p,q ); (ii) There exists an isomorphism ψ of K onto K ; (iii) For 0 i < j p + q 1, there exists an isomorphism θ i of T i onto T i such that α i,j θ j = θ i α i,j ;

10 10 (iv) θψ = θ i θ for 0 i p+q 1. Proof First suppose that there exists an isomorphism φ of S onto S. Let E(S) = {e m,n : m,n N 0 } and E(S ) = {e m,n : m,n N0 } where e 0,0 > e 0,1 > > e 1,0 > e 1,1 > > e 2,0 > e 2,1 > and e 0,0 > e 0,1 > > e 1,0 > e 1,1 > > e 2,0 > e 2,1 >. Now φ preserves clear order of these idempotents and so Then we have that R m,n φ is an R -class of S and e m,n φ = e m,n (m,n N0 ). (1) R m,n(s)φ = R m,n(s ), L p,q(s)φ = L p,q(s ). (2) In particular, H (m,n),(q,p) (S)φ = H (m,n),(q,p) (S ). (3) By Lemma 2.4, (p,q) (resp. (p,q )) is the maximal pair (m,n) (resp. (m,n )) such that Rm,n (S) i=0 H (m,n),(i,m) (S) (resp. R m,n (S ) i =0 H (m,n ),(i,m ) (S )). Hence, (p,q) = (p,q ) and so p = p and q = q. By (2) and (3), we can define an isomorphism θ i of T i onto T i by the rule that θ i = φ Ti (i = 0,1,,p+q 1). It is clear that φ maps the -kernel K of S onto the -kernel K of S. Thus we can define an isomorphism ψ of K onto K by the rule that ψ = φ K. Write T p+q = He p,q (S) (resp. T p+q = He (S )) where e p,q p,q (resp. e p,q) is the identity of K (resp. K ). For x T i (0 i p+q 1), e j,0 T j (0 j p+q 1) and i < j, (xe j,0 )φ = (xα i,j e j,0 )φ = (xα i,j e j,0 )θ j = xα i,j θ j e j,0 = xα i,jθ j. But (xe j,0 )φ = xφe j,0 φ = xθ i e j,0 θ j = xθ i α i,j e j,0 = xθ iα i,j. Hence, we have xα i,j θ j = xθ i α i,j and so α i,jθ j = θ i α i,j. Suppose that x T i (0 i p+q 1) and y T p+q. Then (xy)φ = (xθy)ψ = xθψyψ. But (xy)φ = xφyφ = xθ i yφ = xθ i θ yψ. Thus, since yψ T p,q, we have xθψ = xθ i θ and so θψ = θ i θ. Conversely, let (p,q) = (p,q ). Suppose also that there exists an isomorphism ψ of K onto K and there exist isomorphisms θ i : T i T i (0 i p + q 1) so that α i,jθ j = θ i α i,j for 0 i < j p+q 1 and θψ = θ i θ for 0 i p+q 1. Since the sets T i (0 i p+q 1), K are disjoint, we can use the maps θ i (0 i p+q 1) and ψ to piece together a map φ : S S : { xθ i, if x T i, 0 i p+q 1, xφ = xψ, if x K. Clearly, φ is a bijection. We show below that φ is also a homomorphism. It is convenient to consider separately the following three cases.

11 , «: Type A ω 2 -semigroups 11 Case I Let x T i,y T j and 0 i j p+q 1. Then (xy)φ = (xα i,j y)φ = (xα i,j y)θ j = xθ i α i,jyθ j = xθ i yθ j = xφyφ. Case II Suppose that x T i (0 i p+q 1) and y K. Then (xy)φ = (xθy)φ = (xθy)ψ = xθψyψ = xθ i θ yψ = xφyφ. Case III x K and y K. In this case, it is clear that (xy)φ = xφyφ. This completes the proof. References [1] Asibong-Ibe, U., -bisimple type A ω-semigroups - I, Semigroup Forum, 1985, 31: [2] Asibong-Ibe, U., -simple type A ω-semigroups, Semigroup Forum, 1993, 47: [3] Chen, H., Huang, H. and Guo, X.J., On q -bisimple IC semigroups of type E, Semigroup Forum, 2009, 78(1): [4] Fountain, J.B., Abundant semigroups, Proc. London Math. Soc., 1982, 44(1): [5] Fountain, J,B., Adequate semigroups, Proc. Edinburgh Math. Soc. (Ser. 2), 1979, 22(2): [6] Guo, X.J. and Shum, K.P., On translational hulls of type-a semigroups, J. Algebra, 2003, 269(1): [7] Howie, J.M., Fundamentals of Semigroup Theory, Oxford: Clarendon Press, [8] Lawson, M.V., The natural partial order on an abundant semigroup, Proc. Edinburgh Math. Soc. (Ser. 2), 1987, 30(2): [9] Munn, W.D., Regular ω-semigroups, Glasgow Math. J., 1968, 9(1): [10] Reilly, N.R., Bisimple ω-semigroups, Proc. Glasgow Math. Assoc., 1966, 7(3): [11] Shang, Y. and Wang, L.M., -bisimpletype A ω 2 -semigroups as generalized Bruck-Reilly -extensions, Southeast Asian Bull. Math., 2008, 32(2): [12] Shum, K.P., Du, L. and Guo, Y.Q., Green s relations and their generalizations on semigroups, Discuss. Math. Gen. Algebra Appl., 2010, 30(1): [13] Shum, K.P. and Guo, Y.Q., Regular semigroups and their generalizations, In: Rings, Groups, and Algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 181, New York: Marcel Dekker, 1996, [14] Wang, L.M. and Shang, Y., Regular bisimple ω 2 -semigroups, Advances in Mathematics(China), 2008, 37(1): (in Chinese). [15] Warne, R.J., Bisimple inverse semigroups mod groups, Duke Math. J., 1967, 34(4): A ω 2-1, µ²³ 2 (1. Æ, Æ,, ; 2. Å,,, ) ± - ½º D = D º A ω 2 -, ¾¹ º ¼». ¾¹ º ¼». A ; ω 2 - ; ¼