L -subvarieties of the variety of idempotent semirings

Algebra univers. 46 (2001) 75 – 96 0002–5240/01/020075 – 22 $ 1.50 + 0.20/0 © Birkh¨auser Verlag, Basel, 2001

L-subvarieties of the variety of idempotent semirings Zhao Xianzhong, K. P. Shum∗ and Y. Q. Guo† In memory of the late Professor Viktor Aleksandrovich Gorbunov

Abstract. We show that certain varieties of idempotent semirings are determined by some properties of Green0 s relations, provide equational bases for them and give conditions guaranteeing that some Green0 s relations are congruences.

1. Introduction and preliminaries A semirings (S, +, ·) is an algebra with two binary operations + and · such that both the additive reduct (S, +) and the multiplicative reduct (S, ·) are semigroups which satisfies the identities: x(y + z) ≈ xy + xz,

(x + y)z ≈ xz + yz.

The set of all natural numbers under usual addition and multiplication is a natural example of a semiring. One can also easily find many other examples of semirings in almost all branches of mathematics. Semirings can be regarded as a common generalization of both rings and distributive lattices. By a band, we mean a semigroup in which every element is an idempotent. We call a semiring (S, +, ·) an idempotent semiring if both (S, +) and (S, ·) are bands. Given a class of idempotent semirings, one can regard it as a class of type (2, 2) algebra satisfying two additional identities x · x ≈ x and x + x ≈ x. Thus, the class of all idempotent semirings is an equational class, or equivalently a variety. We denote this variety of all idempotent semirings by I . A special subvariety of the variety I is the subvariety of ID of all distributive semirings, first studied by Pastijn and Romanowska in [14] and Romanowska in [19], [20]. In fact, ID is a special variety of idempotent semirings satisfying the additional dual distributive identities, namely, x + yz ≈ (x + y)(x + z), yz + x ≈ (y + x)(z + x). Presented by Professors Kira Adaricheva and Wieslaw Dziobiak. Received November 2, 1999; accepted in final form May 2, 2000. 2000 Mathematics Subject Classification: 16A78, 08B05, 20M07. Key words: Idempotent semirings, Green0 s relations, congruences, variety, identities. ∗ The research of K. P. Shum is partially supported by a UGC (HK) grant #2160126 (1999/2001). † The research of Y. Q. Guo is supported by a grant of NSF, China #19761004 and a grant of Yunnan Provincial Applied Fundamental Research Foundation #96a001z. 75

76

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

It is noted that the bisemilattices, investigated by Romanowska [18], [21], [22] and by McKenzie and Romanowska [11], form a variety of idempotent semirings which satisfy the additional identities xy ≈ yx, x + y ≈ y + x. There are a series of papers in the literature considering variety of idempotent semirings and similar topics, for example, see [2]–[7], [13]–[14] and [20]–[24]. The variety I has recently been investigated by using Green0 s relations by Guo, Pastijn and Sen [8], Pastijn and Guo [13], Sen, Guo and Shum [23] and Zhao and Guo [24]. The D-relation for the multiplicative reduct (S, ·) of an idempotent semiring (S, +, ·) has been investigated in detail by Pastijn and Zhao [15]. In [20], Zhao and Guo have given some descriptions for the so-called D-subvarieties of the variety I of idempotent semirings by listing down their defining sets of identities and their Mal´cev product decompositions. For a semirings (S, +, ·), the Green0 s relations D on its multiplicative reduct (S, ·) and its additive reduct (S, +) will be denoted by •D and +D, respectively. Thus, a D-subvariety of I is a subvariety of I with D-relations +D and •D defined on (S, +) and (S, ·), respectively. If we let 1 be the identity relation, 5 the universal relation and Con(S) the set of semiring congruences on an idempotent semiring S, then we can list the related D-subvarieties of the variety I by the following additional conditions: Table 1

{S {S {S {S

∈I ∈I ∈I ∈I

: •D = 1}, : +D = 1}, : •D ∨ +D = 1}, : •D ∧ +D = 1},

{S {S {S {S

∈I ∈I ∈I ∈I

: •D = ∇}, : +D = ∇}, : •D ∨ +D = ∇}, : •D ∧ +D = ∇},

{S {S {S {S

∈I ∈I ∈I ∈I

: •D ∈ Con(S)}, : +D ∈ Con(S)}, : •D ∨ +D ∈ Con(S)}, : •D ∧ +D ∈ Con(S)},

In the above table, the lattice meet C1 ∧ C2 means C1 ∩ C2 and the lattice join C1 ∨ C2 means the smallest congruence generated by C1 and C2 . In this connection, we consider the following classes of idempotent semirings related to the Green0 s L-relation +L and/or •L of an idempotent semiring, where •L and +L are the Green0 s relations L on the multiplicative reduct (S, ·) and the additive reduct (S, +) of the semiring (S, +, ·), respectively. Table 2

{S {S {S {S {S {S {S {S {S

∈I ∈I ∈I ∈I ∈I ∈I ∈I ∈I ∈I

: •L = 1}, : •L ∧ +D = 1}, : •L ∧ +L = 1}, : •L ∧ +R = 1}, : •L ∨ +L = 1}, : •L ∨ +R = 1}, : +L = 1}, : +L ∧ •D = 1}, : +L ∨ +D = 1},

{S {S {S {S {S {S {S {S {S

∈ I : •L = 1}, ∈ I : •L ∧ +D = 1}, ∈ I : •L ∧ +L = 1}, ∈ I : •L ∧ +R = 1}, ∈ I : •L ∨ +L = 1}, ∈ I : •L ∨ +R = 1}, ∈ I : +L = 1}, ∈ I : +L ∧ •D = 1}, ∈ I : +L ∨ •D = 1},

{S {S {S {S {S {S {S {S {S

∈I ∈I ∈I ∈I ∈I ∈I ∈I ∈I ∈I

: •L ∈ Con(S)} : •L ∧ +D ∈ Con(S)} : •L ∧ +L ∈ Con(S)} : •L ∧ +R ∈ Con(S)} : •L ∨ Lp ∈ Con(S)} : •L ∨ Rp ∈ Con(S)} : +L ∈ Con(S)} : +L ∧ •D ∈ Con(S)} : •L ∨ •D ∈ Con(S)}

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

77

In this paper, we shall give a complete description for the above classes of idempotent semirings by using some sets of defining identities so that the classes form subvarieties of I . We shall call them L-subvarieties. In this context, the concept of Mal0 cev product is needed. By the Mal0 cev product of two classes V and W of the semirings, denoted by V ◦ W , we mean the class of all semirings S in which there exists a congruence ρ on S such that S/ρ ∈ W and the ρ-classes are all in V . Thus, in this way, some new classes of semirings can be obtained via the Mal0 cev products of some known semirings. Another aim of this paper is to consider the decomposition of Mal0 cev product of some L-subvarieties into other L-subvarieties. Finally, we show that •L is a congruence on an idempotent semiring (S, +, ·) if and only if (•L) ∧ (+D) and (•L) ∨ (+D) are both congruences on S. Throughout this paper, unless otherwise stated, S is always an idempotent semiring. We use the symbol V to denoted the band variety. Thus, +V (•V ) is a subvariety of I consisting of the semirings whose additive (multiplicative) reducts belong to V . For example, •R is the subvariety of I satisfying the identity xyz ≈ x, while +R is the subvariety of I satisfying the identity x + y + x ≈ x. The following notations will be used throughout our paper. Lz Rz R S` Ln Rn N Re Lr Rr

left zero bands right zero bands rectangular bands Semilattices left normal bands right normal bands normal bands regular bands left regular bands right regular bands

xy ≈ x xy ≈ y xyx ≈ x xy ≈ yx xyz ≈ xzy xyz ≈ yxz xyzx ≈ xyxzx xyzx ≈ xyxzx xy ≈ xyx yx ≈ xyx

For the sake of completeness, we note that +L and •L are defined on (S, +) and (S, ·) by (∀ a, b ∈ S) a(+L)b ⇔ a + b = a, b + a = b;

(1)

(∀ a, b ∈ S) a(•L)b ⇔ a + b = a, ba = b,

(2)

It is noted in [8] and [19] that if (S, ·) is a band, then we may extend (S, ·) to an algebra (S, +, ·) by stipulating that x +y ≈ x for all x, y ∈ S. The resulting (S, +, ·) then becomes an idempotent semiring which belongs to +Lz and it is obvious that every member of +Lz can be so obtained. Similar observation holds for the members of +Rz , •Lz and •Rz respectively. The following result on idempotent semirings obtained in [15] are particularly useful in the sequel. LEMMA 1.1. [15, Theorem 1.5] Let S = (S, +, ·) be an idempotent semiring. Then the Green0 s relation +D on the additive reduct (S, +) of S is a congruence on the idempotent

78

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

semiring (S, +, ·) and, consequently, +L and +R are congruences on the multiplicative reduct (S, ·). For notations and terminologies not given in this paper, the reader is referred to Howie [9] and Petrich [16] for a background on semigroup theory and to McKenzie, McNulty and Taylor [10] for information concerning universal algebra. We shall assume that the reader is familiar with the basic results in this area. 2. L-subvarieties related to the meets of L-relations with some other Green0 s relations In this section, we are going to describe those of idempotent semirings related to the lattice meets of the L-relation with some other Green0 s relations. We will show that these classes are varieties. We first establish the following lemma. LEMMA 2.1. Let a, b be any elements of an idempotent semiring S. Then, we have (i)

a(•L ∧ +D)b if and only if (∃u, v ∈ S)a = uuu + vu, b = vu + uvu;

(ii)

a(•L ∧ +L)b if and only if (∃u, v ∈ S)a = vu + uvu + uv, b = vu + uvu;

(iii)

(4)

a(•L ∧ +R)b if and only if (∃u, v ∈ S)a = vu + uvu + uv, b = uvu + vu;

(iv)

(3)

(5)

a(+L ∧ •D)b if and only if (∃u, v ∈ S)a = uv + u + v + u, b = vu + u + v + u;

Proof. (i) If a (•L ∧ +D)b, then by letting u = a + b and v = b + a, we have uvu + vu = (a + b)(b + a)(a + b) + (b + a)(a + b) = (a + b) + (b + a) = a, and uv + uvu = (b + a)(a + b) + (a + b)(b + a)(a + b) = (b + a) + (a + b) = b+a+b = b.

(6)

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

79

Conversely, let u, v be elements of S and suppose that a = uvu + vu and b = vu + uvu. Then it is obvious that a(+D)b and uvu(•L)vu. As the congruence class is a subsemiring of the idempotent semiring S, we have a(•L)b. This implies that a(•L ∧ +D)b and hence (3) holds on S. (ii) If a(•L ∧ +L)b, then by letting u = b and v = a, we have vu + uvu + vu = ab + bab + ab = a + b + a = a, and uvu + vu = bab + ab = b + a = b. For the converse part, we let a = vu + uvu + vu, b = uvu + vu + vu, for u, v ∈ S. Then it is trivial to see that a + b = a, b + a = a and uvu(•L)uv. Thus, we have a(•L)b and by (1), we also have a(+L)b. This implies that a(•L ∧ +L)b and so (4) holds on S. This part is similar to (ii) and we omit the details. (iii) Let a(+L ∧ •D)b. Then by letting u = ab, v = ba and by Lemma 1.1, we have uv + u + v + u = (u + v + u)(v + u) = (ab + ba + ab)(ba + ab) = ab · ba = a and vu + u + v + u = (ba + ab)(ab + ba + ab) = ba · ab = b. Conversely, if u, v ∈ S, a = (u + v + u)(v + u) and b = (v + u)(u + v + u), then it is obvious that a•Db and (u + v + u)(+L)(v + u). So by Lemma 1.1 again, we have a(+L)b. This implies a(+L ∧ •D)b and hence (6) holds in S. ¨ LEMMA 2.2. The following statements hold for an idempotent semiring S = (S, +, ·). (i) •L∧+D is a congruence on S if and only if S satisfies the following set of identities: (yx + xyx + z)(xyx + yx + z) ≈ yx + xyx + z,

(7)

(z + yx + xyx)(z + xyx + yx) ≈ z + yx + xyx,

(8)

z(yx + xyx)z(xyx + yx) ≈ z(yx + xyx).

(9)

80

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

(ii) •L∧+L is a congruence on S if and only if S satisfies the following set of identities: (z + xyx + yx) + (z + yx + xyx + yx) ≈ z + xyx + yx,

(10)

(z + xyx + yx)(z + yx + xyx + yx) ≈ z + xyx + yx,

(11)

(xyx + yx + z)(yx + xyx + yx + z) ≈ xyx + yx + z,

(12)

z(xyx + yx)z(yx + xyx + yx) ≈ z(xyx + yx).

(13)

(iii) •L∧+R is a congruence on S if and only if S satisfies the following set of identities: (yx + xyx + z) + (yx + xyx + yx + z) ≈ yx + xyx + z,

(14)

(yx + xyx + z)(yx + xyx + yx + z) ≈ yx + xyx + z,

(15)

(z + yx + xyx)(z + yx + xyx + yx) ≈ z + yx + xyx,

(16)

z(yx + xyx)z(yx + xyx + yx) ≈ z(yx + xyx).

(17)

(iv) +L∧•D is a congruence on S if and only if S satisfies the following set of identities: (z + xy + x + y + x) + (z + yx + x + y + x) ≈ z + xy + x + y + x,

(18)

(z + xy + x + y + x)(z + yx + x + y + x)(z + xy + x + y + x) ≈ z + xy + x + y + x, (xy + x + y + x + z)(yx + x + y + x + z) ≈ xy + x + y + x + z.

(19) (20)

Proof. We only prove (ii), as by similar arguments, we can also prove (i), (iii) and (iv). First, we let •L ∧ +L be a congruence on the idempotent semiring (S, +, ·). Then, by (iv) in Lemma 2.1, for any u, v ∈ S we have (vu + uvu + uv)(•L ∧ +L)(vu + uvu). Consequently, for w ∈ S, we have (w + vu + uvu + vu)(•L ∧ +L)(vu + uvu + vu + w) (•L ∧ +L)(uvu + vu + w)w(vu + uvu + vu + w). These imply that (w + vu + uvu + vu)(•L)(w + uvu + vu), (w + vu + uvu + vu)(•L)(w + uvu + vu), (vu + uvu + vu + w)(•L)(uvu + vu + w), w(vu + uvu + vu)(•L)w(uvu + vu). Thus, by (1) and (2), we obtain (w + uvu + vu) + (w + vu + uvu + vu) ≈ w + uvu + vu, (w + uvu + vu)(w + vu + uvu + vu) ≈ w + uvu + vu, (uvu + vu + w)(vu + uvu + vu + w) ≈ uvu + vu + w, w(uvu + vu)w(vu + uvu + vu) ≈ w(uvu + vu).

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

81

Thereby, we can easily see that S satisfies (10), (11), (12) and (13). Conversely, we suppose that S is an idempotent semiring satisfying (10), (11), (12) and (13), and also suppose that a(•L ∧ +L)b for some a, b ∈ S. Then by Lemma 2.1, there exist u, v ∈ S such that a = vu + uvu + vu and b = vu + uvu. By replacing c by a, x by u, also y by v in (10) and (11), we have (c + b) + (c + a) = c + b,

(c + b)(c + a) = c + b.

By interchanging a and b above, we can similarly obtain that (c + a) + (c + b) = c + a,

(c + a)(c + b) = c + a.

By invoking (1) and (2), we can show that (c + a)(•L ∧ +L)(c + a). Consequently, •L ∧ +L is a left congruence on the additive reduct (S, +) of (S, +, ·). Next, we claim that •L ∧ +L is also a right congruence on (S, +). Since a(+L)b and +L is a right congruence on (S, +), we have (a + c)(+L)(b + c). By replacing z by c, x by u, and y by v in (12), we have (b + c)(a + c) = b + c. By interchanging a and b again, we can similarly show that (a + c)(b + c) = a + c. Thus, our claim is established. Now, by the above two formulas, we get (a + c)(•L)(b + c) and therefore (a + c)(•L ∧ +L)(b + c). This shows that •L ∧ +L is a right congruence (S, +). For the multiplicative reduct (S, ·), we know, by Lemma 1.1, that +L is a congruence on (S, ·) and •L is left congruence on (S, ·) so •L ∧ +L is a left congruence on (S, ·). Finally we prove that •L ∧ +L is also a right congruence on (S, ·). Since •L is a left congruence on (S, ·) we have cb(+L)ca for any c ∈ S. Replacing z by c, x by u, and y by v in (13), we get cbca = cb, and by interchanging a and b, we can similarly obtain cacb = ca. In conclusion, we have cb(•L)ca, and so ca(•L ∧ +L)cb. Thus, •L ∧ +L is a right congruence on (S, ·). As we have shown that •L ∧ +L is not only a congruence on (S, +) and is also a congruence on (S, ·), (•L ∧ +L) is a semiring congruence on (S, +, ·). Our proof is now completed. ¨ By the above theorem, we can easily observe that the following classes of idempotent semirings {S {S {S {S

∈I ∈I ∈I ∈I

: •L ∧ +D ∈ Con(S)} : •L ∧ +L ∈ Con(S)} : •L ∈ +R ∈ Con(S)} : +L ∧ •D ∈ Con(S)}

and

are L-subvarieties of the variety I of idempotent semirings. The above four L-subvarieties of I defined by the congruences (•L ∧ +D), (•L ∧ +L), (•L ∧ +R) and (+L ∧ •D) on S are denoted by Ld , L` , Lr and Ld∗ , respectively. For other classes of idempotent semirings in Table 2, we can also give the descriptions using some defining identities. We first state the following lemmas which are consequences of Lemma 2.1 and so we omit the proofs.

82

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

LEMMA 2.3. Let S be an idempotent semiring. Then (1) S satisfies •L ∧ +D = 1 if and only if S satisfies the identity yx + xyx ≈ xyx + yx.

(21)

(2) S satisfies •L ∧ +L = 1 if and only if S satisfies the identity yx + xyx + yx ≈ xyx + yx.

(22)

(3) S satisfies •L ∧ +R = 1 if and only if S satisfies the identity yx + xyx + yx ≈ yx + xyx.

(23)

(4) S satisfies •L ∧ +D = 1 if and only if S satisfies the identity yx + x + y + x ≈ yx + x + y + x.

(24)

LEMMA 2.4. Let S be an idempotent semiring. Then we have (i) S satisfies •L ∧ +L = ∇ if and only if S satisfies the identities xy ≈ x,

(25)

x + y + x ≈ x.

(26)

Thus S ∈ •Lz ∧ +R. (ii) S satisfies •L ∧ +L = ∇ if and only if S satisfies the identities (25) and x + y ≈ x.

(27)

Thus S ∈ •Lz ∧ +Lz . (iii) S satisfies •L ∧ +R = ∇ if and only if S satisfies the identities (25) and x + y ≈ x.

(28)

Thus S ∈ •Lz ∧ +Rz . (iv) S satisfies +L ∧ •D = ∇ if and only if S satisfies the identities (27) and xyx ≈ x.

(29)

Thus S ∈ +Lz ∧ •Rz . For the sake of convenience, we denote the above classes of idempotent semirings by Ld0 = {S ∈ I L`0 = {S ∈ I Lr0 = {S ∈ I Ld∗0 = {S ∈ I

: •L ∧ +D = 1}, : •L ∧ +L = 1}, : •L ∧ +R = 1}, : •L ∧ +D = 1},

Ld1 = {S ∈ I L`1 = {S ∈ I Lr1 = {S ∈ I Ld∗1 = {S ∈ I

: •L ∧ +D = ∇} : •L ∧ +L = ∇} : •L ∧ +R = ∇} : •L ∧ +D = ∇}

In closing this section, we are now going to study the Mal0 cev product of the above subvarieties so that the relations between Ld , L` , Lr , Ld∗ of the variety I can be found.

L-subvarieties of the variety of idempotent semirings

Vol. 46, 2001

83

THEOREM 2.5. (The First Mal0 cev Product Theorem) (i) (ii) (iii) (iv)

Ld L` Lr Ld∗

= = = =

Ld1 ◦ Ld0 . L`1 ◦ L`0 . Lr1 ◦ Lr0 . Ld∗1 ◦ Ld∗0 .

Proof. (i) If S ∈ Ld , then •L ∧ +D ∈ Con(S). Now, the (•L ∧ +D)-classes are in Ld1 and so S/(•L ∧ +D) is clearly in Ld0 . This shows that S ∈ Ld1 ◦ Ld0 . In other words, we have Ld ⊆ Ld1 ◦ Ld0 . Conversely, let S ∈ Ld1 ◦ Ld0 and ρ ∈ Con(S). Suppose that the ρ-classes ρu ∈ Ld1 , for all S/ρ ∈ Ld0 . Then by S/ρ ∈ Ld0 and by Lemma 2.3(i), we can see that for any a, b ∈ S, we have ρba+aba = ρaba+ba . Thus, for any c ∈ S, we have ρba+aba+c = ρaba+ba+c , ρc+ba+aba = ρc+aba+ba , ρc(ba+aba) = ρc(aba+ba) . However, since ρu ∈ Ld1 for all u ∈ S and by Lemma 2.4(i), we can see that ρu satisfies the identity (25) for all u ∈ S. By using the above formulas of ρu , we immediately obtain the following equalities: (ba + aba + c)(aba + ba + c) = ba + aba + c, (c + ba + aba)(c + aba + ba) = c + ba + aba, c(ba + aba)c(aba + ba) = c(ba + aba). Hence, we have shown that S satisfies (7), (8) and (9). By Lemma 2.2(1), we have S ∈ Ld . ¨ This shows that Ld1 ◦ Ld0 ⊆ Ld , as required. The equalities (2), (3), (4) can be proved similarly and their proofs are omitted. 3. L-subvarieties related to the joins of L-relations with some other Green0 s relations In this section, we study the L-subvarieties of I related to the joins of the L-relations with some other Green0 s relations satisfying certain conditions on idempotents semirings.

84

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

LEMMA 3.1. The following equalities hold for any idempotent semiring S. (i) •L ∨ +D = (+D)(•L)(+D),

(30)

(ii) •L ∨ +L = (+L)(•L)(+L),

(31)

(iii) •L ∨ +R = (+R)(•L)(+R),

(32)

(iv) +L ∨ •D = (+L)(•D)(+L).

(33)

Proof. Let S be an idempotent semiring. Then, by a direct application of [9, Proposition 1.5.11], we know that each left side of (30)–(33) is contained in the right side. Thus, we only need to prove the inverse inclusion. Here we prove (30) and the other proofs are similar. To show that •L ∧ +D ⊆ (+D)(•L)(+D), we need to show that (•L)(+D)(•L) ⊆ (+D)(•L)(+L). Suppose a(•L)(+D)(•L)b for a, b ∈ S. Then there exist u, v ∈ S such that a(•L)u(+D)u(•L)b. Because +D is a congruence on S and a(•L)u(+D)v, we have a = au(+D)av. Also, by u(+D)v(•L)b, we have b = bv(+D)bu. Hence a = au = au·u(+D)avu by a(+D)av. Furthermore, because •L is also a right congruence on (S, ·), we have auv(•L)uvu by a(•D)u, and bu(•L)vu by b(•D)v. Thus, we have a(+D)avu(•L)uvu(•L)uv(•L)bu(+D)b and thereby a(+D)(•L)(+D)b. This shows that (•L)(+D)(•L) ⊆ (+D)(•L)(+D) as required. ¨ LEMMA 3.2. Let S be an idempotent semiring. Then for any a, b ∈ S, we have (1) (2) (3) (4)

a(•L ∨ +D)b if and only if +Dab = +Da and +Dba = +Db . a(•L ∨ +L)b if and only if +Lab = +La and +Lba = +Lb . a(•L ∨ +R)b if and only if +Rab = +Ra and +Rba = +Rb . a(+L ∨ •D)b if and only if +Laba = +La and +Lbab = +La .

Proof. Let S be an idempotent semiring. We will prove (i) only as the proof of the other cases are similar. If a, b ∈ S such that a(•L ∨ +D)b, then by (30), there exist u, v ∈ S such that a(+D)u(•L)v(+D)b. Since +D is a congruence on (S, ·), by a(•D)u, we have ab(+D)ub, also u = uv(+D)ub by u(•L)v and v(+D)b. So a(+D)u = uv(+D)uv(+D)ab. This proves that +Dab = +Da . Similarly, we can prove that +Dba = +Db . Conversely, if +Dab = +Da and +Dba = +Db , for a, b ∈ S, then we obtain a(+D)ab(+D)aba(•L) ba(+D)b. Thus, by (30), we can easily see that a(•L ∨ +D)b, as required. ¨ By summing up Lemmas 3.1 and 3.2, we obtain the following descriptive results.

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

85

THEOREM 3.3. Let S be an idempotent semiring and x, y ∈ S. Then (1) S satisfies •L ∨ +D = ∇ if and only if +Dab = +Da for a, b ∈ S, that is, S satisfies the following identities x + xy + x ≈ x

(34)

xy + x + xy ≈ xy.

(35)

(2) S satisfies •L ∨ +L = ∇ if and only if +Lab = +La for a, b ∈ S, that is, S satisfies the following identities x + xy ≈ x

(36)

xy + x ≈ xy.

(37)

(3) S satisfies •L ∨ +R = ∇ if and only if +Rab = +Ra for a, b ∈ S, that is S satisfies the following identities x + xy ≈ xy

(38)

yx + x ≈ x.

(39)

(4) S satisfies +D ∨ •D = ∇ if and only if +Laba = +La for a, b ∈ S, that is S satisfies the following identities x + xyx ≈ x

(40)

xyx + x ≈ xy.

(41)

By using Theorem 3.3, it can be easily seen that the following classes of semirings in Table 2 {S {S {S {S

∈I ∈I ∈I ∈I

: •L ∨ +D = ∇}, : •L ∨ +L = ∇}, : •L ∨ +R = ∇} and : +L ∨ •L = ∇}

can be described by a set of identities and so they form the L-subvarieties of the variety I . L`1 , e Lr1 and e Ld∗1 respectively. We also consider We denote the above subvarieties by e L d1 , e the dual of these four subvarieties of the variety of the idempotent semiring I in Table 2. Denote these subvarieties by e Ld0 = {S ∈ I : •L ∨ +D = 1}, e L`0 = {S ∈: •L ∨ +L = 1}, e Lr0 = {S ∈ I : •L ∨ +R = 1} and e Ld∗0 = {S ∈ I : +L ∨ •D = 1}, respectively. The following theorem now follows directly from definitions.

86

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

THEOREM 3.4. Let (S, +, ·) be an idempotent semiring. Then (i) S satisfies •L ∨ +D = 5 if and only if the multiplicative reduct (S, ·) of the semiring (S, +, ·) is a right regular band and the additive reduct (S, +) of (S, +, ·) is a semilattice, that is, S ∈ •Rr ∧ •S` such that S satisfies the following identities yx ≈ xyx

(42)

x + y ≈ y + x.

(43)

(ii) S satisfies •L ∨ +L = 5 if and only if (S, ·) are right regular bands, that is, S ∈ •Rr ∧ +Rr such that S satisfies the identity (42) and the identity y + x ≈ x + y + x.

(44)

(iii) S satisfies •L ∨ +R = 5 if and only if (S, ·) is a right regular band and (S, +) is a left regular band, that is, S ∈ •Rr ∧ +Lr such that S satisfies the identity (42) and the identity x + y ≈ x + y + x.

(45)

(iv) S satisfies +L ∨ •L = 5 if and only if (S, ·) is a semilattice and (S, +) is a right regular band, that is, S ∈ •L ∧ +Rr such that S satisfies (44) and the identity xy ≈ xyx.

(46)

L`0 , e Lr0 and e Ld∗0 can be described by a set of identities and Thus, by Theorem 3.4, e Ld0 , e they form some L-subvarieties of I . There remains four classes of idempotent semirings in Table 2 that have not yet been discussed. We now denote these four classes of idempotent semirings by e Ld = {S ∈ I e L` = {S ∈ I e Lr = {S ∈ I e L∗ = {S ∈ I d

: •L ∨ +D ∈ Con(S)}, : •L ∨ +L ∈ Con(S)}, : •L ∨ +R ∈ Con(S)}, : +L ∨ •D ∈ Con(S)}, respectively.

L` , e Lr and e Ld∗ are indeed subvarieties of I . We now verify that e Ld , e THEOREM 3.5. (The Second Mal0 cev Product Theorem)

(1) (2) (3) (4)

e Ld1 ◦ e Ld0 ; Ld = e e e L` = L`1 ◦ e L`0 ; e Lr1 ◦ e Lr0 ; Lr = e ∗ e L =e L∗ ◦ e L∗ . d

d1

d0

Proof. We only prove (i). The proofs of the other equalities are similar and we omit them.

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

87

If S ∈ e Ld then ρ = •L ∨ +D ∈ Con(S). Also it is obvious that ρu ∈ e Ld , for all u ∈ S. By •L ⊆ ρ and •Lba = •Laba , for any a, b ∈ S, we have ρba = ρaba , i.e., ρb ρa = ρa ρb ρa . This implies that S/ρ satisfies the identity (42). By +D ⊆ ρ and +Da+b = +Db+a , for any a, b ∈ S, we have ρa+b = ρb+a . This is equivalent to say that ρa + ρb = ρb + ρa . Ld0 . Hence S/ρ satisfies the identity (43). Thus, by Theorem 3.4(i), S ∈ e Ld1 ◦ e 0 Conversely, if S ∈ e Ld1 ◦ e Ld0 , then by the definition of Mal cev products, there exists Ld1 for all u ∈ S and S/ρ ∈ e Ld0 . By ρu ∈ e Ld for all u ∈ S we ρ ∈ Con(S) such that ρu ∈ e e have ρ ⊆ •L ∨ +D. Since S/ρ ∈ Ld0 , we know that S/ρ satisfies the identities (42) and (43). By identity (42), it follows that •L ⊆ ρ, and by identity (43) the condition +D ⊆ ρ also holds. Therefore, we have (•L) ∨ (+D) ⊆ ρ. It is now clear that ρ = (•L) ∨ (+D). ¨ This proves that S ∈ e Ld , as required. We now show that the above subvarieties of I can be determined by certain identities. THEOREM 3.6. e Ld is a subvariety of I determined by the following identities (yx + z) + (yx + z)(xyx + z) + (yx + z) ≈ yx + z,

(47)

(yx + z)(xyx + z) + (yx = z) + (yx + z(xyx + z) ≈ (yx + z)(xyx + z),

(48)

zyx + zyxzxyx + zyx ≈ zyx,

(49)

zyxzxyx + zyx + zyxzxyx ≈ zyxzxyx.

(50)

Ld1 for all u ∈ S and Proof. If S ∈ e Ld then ρ = •L ∨ +D ∈ Con(S), where ρu ∈ e e e S/ρ ∈ Ld0 . Since S/ρ ∈ Ld0 , by Theorem 3.4(1), we have ρba = ρaba , for any a, b ∈ S. It follows that ρba+c = ρaba+c and ρcba = ρcaba , for any c ∈ S. By Theorem 3.3(i), we have (ba + c) + (ba + c)(aba + c) + (ba + c) = ba + c, (ba + c)(aba + c) + (ba + c) + (ba + c)(aba + c) = (ba + c)(aba + c), cba + cbacaba + cba = cba, cbacaba + cba + cbacaba = cbacaba. This shows that S satisfies (45), (46), (47), and (48).

¨

Conversely, let S be an idempotent semiring satisfying (47), (48), (49) and (50), and suppose that a(•L ∨ +D)b, for a, b ∈ S. Then by formula (30) there exist c, d ∈ S such that a(+D)c(•L)d(+D)b. First we claim that (•L ∨ +D) is a congruence on (S, +). That is, we need to prove (a + w)(•L ∨ +D)(b + w) and (w + a)(•L ∨ +D)(w + b) hold for any w ∈ S. In view of (47) and (48), we have (cd + w)(+D)(cd + w)(dcd + w). Moreover,

88

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

since c(•L)d, we also have (c + w)(+D)(c + w)(d + w). By interchanging c and d, we find similarly that (d + w)(+D)(d + w)(c + w). As (+D) is a congruence on S, we have (a + w)(+D)(c + w)(+D)(c + w)(d + w)(c + w) (•L)(d + w)(c + w)(+D)(d + w)(+D)(b + w). So, it follows that (w + a)(+D)(w + c)(+D)(w + c)(w + d)(w + c) (•L)(w + d)(w + c)(+D)(w + d)(+D)(w + b). Thus, by (30), we have (a + w)(•L) ∨ (+D)(b + w) and (w + a)(•L ∨ +D)(w + b). Our claim is now established. Next, we prove that (•L ∨ +D) is a left congruence on (S, ·). In fact, by (49) and (50), we have wcd(+D)wcdwdcd. Applying the condition that c(•L)d, we have wc(+D)wcwd. By interchanging c and d, we prove that wd(+D)wdwc. Since (+D) is a congruence on S, we have wa(+D)wx(+D)wcwd(+D)wcwdwc(•L)wdwc(+D)wd(+D)wb. Thus, by (30), we have wa((•L) ∨ (+D))wb and so (•L ∨ +D) is a left congruence on (S, ·). Finally, since (•L) is a right congruence on (S, ·) and (+D) is a congruence on S, by (30) again, (•L ∨ +D) is a right congruence on (S, ·). Hence (•L ∨ +D) is, indeed, a congruence on S. THEOREM 3.7. e L` is a subvariety of I determined by the additional identities (yx + z) + (yx + z)(xyx + z) ≈ yx + z,

(51)

(yx + z)(xyx + z) + (yx + z) ≈ (yx + z)(xyx + z),

(52)

(z + yx) + (z + yx)(z + xyx) ≈ z + yx,

(53)

(z + yx)(z + xyx) + (z + yx) ≈ (z + yx)(z + xyx),

(54)

(z + y + x) + (z + y + x)(z + x + y + x) ≈ z + y + x,

(55)

(z + y + x)(z + x + y + x) + (z + y + x) ≈ (z + y + x)(z + x + y + x),

(56)

zyx + zyxzxyx ≈ zyx,

(57)

zyxzxyx + zyx ≈ zyxzxyx.

(58)

L`1 for all u ∈ S, Proof. If S ∈ e L` , then ρ = (•L ∨ +D) ∈ Con(S), where ρu ∈ e and S/ρ ∈ e L`0 . By Theorem 3.4(2), we see that S/ρ satisfies (42) and (44), i.e., for any a, b ∈ S, we have ρba = ρaba

and ρb+a = ρa+b+a .

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

89

Furthermore, since ρ ∈ Con(S) we have, ρba+w = ρaba+w , ρw+ba = ρw+aba , ρw+b+a = ρw+a+b+a , ρwba = ρwaba , L`1 for all u ∈ S, and by Theorem 3.3(2), we deduce that for any w ∈ S. However, as ρu ∈ e (ba + w) + (ba + w)(aba + w) = ba + w, (ba + w)(aba + w) + (ba + w) = (ba + w)(aba + w), (w + ba) + (w + ba)(w + aba) = (w + ba)(w + aba), (w + b + a) + (w + b + a)(w + a + b + a) = w + b + a, (w + b + a)(w + a + b + a) + (w + b + a) = (w + b + a)(w + a + b + a), wba + wbawaba = wba, wbawaba + wba = wbawbab. This shows that S satisfies (51) to (58).

¨

Conversely, suppose that an idempotent semiring S satisfies the identities (51) to (58), and a, b ∈ S are such that a(•L ∨ +Db). Then by (31), there exist c, d ∈ S such that a(+L)c(•L)d(+L)b. Also, by (51) and (52) and by c(•L)d, we have, for any w ∈ S, (c + w)(+L)(c + w)(d + w) and (d + w)(+L)(d + w)(c + w). Since (+L) is a right congruence on (S, +) and it is a congruence on (S, ·) we have (a + w)(+L)(c + w)(+L)(c + w)(d + w)(+L)(c + w)(d + w)(c + w) (•L)(d + w)(c + w)(+L)(d + w)(+L)(b + w). By (31), we obtain (a + w)((•L) ∨ (+L))(b + w). This shows that (•L) ∨ (+L) is a right congruence on (S, +). Next, we prove that (•L ∨ +L) is a left congruence on (S, +). That is, we need to prove (w + a)(•L ∨ +L)(w + b). However, by (53), (54) and c(•L)d, we have (w + c)(+L)(w + c)(w + d),

(w + d)(+L)(w + d)(w + c).

Again by (55) and (56), and noting that a(+L)c and d(+L)b, we have (w + a)(+L)(w + a)(w + c),

(w + c)(w + a)(+L)w + c,

(w + d)(+L)(w + d)(w + b),

(w + b)(w + d)(+L)w + b.

90

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

Now, from the last six formulas and from the fact that (+L) is a congruence on (S, ·), we derive that (w + a)(+L)(w + a)(w + c)(+L)(w + a)(w + c)(w + a) (•L)(w + c)(w + a) + L(w + c), (w + c) + L(w + c)(w + d)(+L)(w + c)(w + d)(w + c) (•L)(w + d)(w + c)(+L)(w + d), (w + d)(+L)(w + d)(w + b)(+L)(w + d)(w + b)(w + d) (•L)(w + b)(w + d)(+L)(w + b). This leads to (w + a)(•L ∨ +L)(w + b). It remains to prove that (•L ∨ +L) is a left congruence on (S, ·). Indeed, by (57), (58) and c(•L)d, we have wcwd(+L)wc,

wdwc(+L)wd.

Furthermore, since (+L) is a congruence on (S, ·), we have wa(+L)wc(+L)wcwd(+L)wcwdwc(•L)wdwc(+L)wd(+L)wb. Thus, we have wa(•L ∨ +L)wb. In other words, (•L ∨ +L) is a left congruence on (S, ·). Finally, we notice that (+L) is a congruence on (S, ·) and •L is a right congruence on (S, ·). This shows that (•L ∨ +L) is a right congruence on (S, ·). Thus, (•L ∨ +L) ∈ Con(S), as required. Similarly, we can deduce the following results. Their proofs are omitted. THEOREM 3.8. e Lr is a subvariety of I determined by the following set of additional identities (z + yx) + (z + yx)(z + xyx) ≈ (z + yx)(z + xyx),

(59)

(z + yx)(z + xyx) + (z + xy) ≈ z + yx,

(60)

(yx + z) + (yx + z)(xyx + z) ≈ (yx + z)(xyx + z),

(61)

(yx + z)(xyx + z) + (yx + z) ≈ yx + z,

(62)

(x + y + z) + (x + y + z)(x + y + x + z) ≈ (x + y + z)(x + y + x + z),

(63)

(x + y + z)(x + y + x + z) + (x + y + z) ≈ x + y + z,

(64)

zyx + zyxzxyx ≈ zyxzxyx,

(65)

zyxzxyx + zyx ≈ zyx.

(66)

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

91

f∗ is a subvariety of I determined by the following set of additional THEOREM 3.9. L d identities (yx + z) + (xy + z)(yx + z)(xy + z) ≈ yx + z,

(67)

(xy + z)(yx + z)(xy + z) + (xy + z) ≈ (xy + z)(yx + z)(xy + z),

(68)

(z + xy) + (z + xy)(z + yx)(z + xy) ≈ z + xy,

(69)

(z + xyxz + yx)(z + xy) + (z + xy) ≈ (z + xy)(z + yx)(z + xy),

(70)

(z + y + x) + (z + y + x)(z + x + y + x)(z + y + x) ≈ z + y + x,

(71)

(z + y + x)(z + x + y + x)(z + yx) + (z + y + x) ≈ (z + y + x)(z + x + y + x)(z + y + x).

(72)

4. Other kinds of L-subvarieties In this section, we discuss the following classses of idempotent semirings: L0 = {S ∈ I : •L = 1}, L1 = {S ∈ I : •L = ∇}, L = {S ∈ I : •L ∈ Con(S)}, +L0 = {S ∈ I : +L = 1}, +L1 = {S ∈ I : +L = ∇}, +L = {S ∈ I : +L ∈ Con(S)}. To start with, we state the following results. The proofs are easy and we omit them. THEOREM 4.1. (1) L0 is a subvariety of I determined by the identity xy ≈ yxy. Thus L0 = •Rr . (2) L1 is a subvariety of I determined by the identity xy ≈ x. Thus L1 = •Lz . (3) L is a subvariety of I determined by the following set of additional identities (xy + z)(yxy + z) ≈ xy + z,

(73)

(z + xy)(z + yxy) ≈ z + xy,

(74)

zxyzyxy ≈ zxy.

(75)

(4) L = L1 ◦ L0 .

92

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

THEOREM 4.2. (i) (+L0 ) is a subvariety of I determined by the identity x + y ≈ y + x + y. Thus (+L0 ) = (+Rr ). (ii) (+L1 ) is a subvariety of I determined by the identity x + y ≈ x. Thus L1 = Lz . (iii) (+L) is a subvariety of I determined by the identity z + x + y + z + y + x + y ≈ z + x + y. (iv) (+L) = (+L1 ◦ +L0 ). The following theorem describes the property of the relation (•L) on the idempotent semiring S = (S, +, ·). THEOREM 4.3. Let S = (S, +, ·) be an idempotent semiring such that its additive reduct (S, +) is a semilattice. If (•L) is a congruence on (S, +) then S satisfies the identity zxy ≈ zyxy.

(76)

Proof. Since S is an idempotent semiring with (S, +) being a semilattice, we have, for all x, y ∈ S, x + y ≈ y + x. If (•L) is a congruence on (S, +) then, since aba(•L)ba, for any a, b, c ∈ S, we have (ab + c) = (ab + c)(bab + c)

(by •L ∈ Con(S))

= ab + abc + cbab + c = ab + c + abc + cbab

(by the additive commutative law)

= (ab + c) + abc + cbab 2

= (ab + abc + cab + c) + abc + cbab = ab + (abc + abc) + cab + c + cbab = (ab + abc + cab + c) + cbab = ab + c + cbab. Thus, we obtain ab + c ≈ ab + c + cbab.

(by the additive commutative law)

L-subvarieties of the variety of idempotent semirings

Vol. 46, 2001

93

By interchanging ab and bab we can similarly find (bab + c) = bab + c + cab. Now, by multiplying c on the left side of the last two formulas, we have cab + c = cab + c + cbab

(by the additive commutative law)

= cbab + c + cab = cbab + c. Similarly, by multiplying ab on the right side of the last formula, we get cbab + cab = cab. Interchanging ab and bab, we obtain cab + cbab = cbab. By using the last two formulas, we further obtain cab = cbab + cab = cab + cbab

(by the additive commutative law)

= cbab. Thus, we finally get cab ≈ cbab and hence S satisfies (76) as required.

¨

The following corollaries are consequence of Theorem 4.3. COROLLARY 4.4. Let (S, +, ·) be an idempotent semiring whose additive reduct (S, +) is a semilattice. Then the following statements are equivalent: (1) •L is a congruence on semiring (S, +, ·); (2) •L is a congruence on the additive reduct (S, +) of (S, +, ·); (3) the semiring (S, +, ·) satisfies the additional identity (73) (xy + z)(yxy + z) ≈ xy + z, for all x, y, z ∈ (S, +, ·). COROLLARY 4.5. Let S be an idempotent semiring. Then the following statements are equivalent: (1) S ∈ e Ld , that is, •L ∨ +D ∈ Con(S/+D); (2) •LS/+D ∈ Con(S/+D);

94

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

(3) S satisfies the identity (47), that is, (xy + z) + (xy + z)(yxy + z) + (xy + x) ≈ xy + z, (xy + z)(yxy + z)(xy + z) + (xy + z) + (xy + z) (yxy + z) ≈ (xy + z)(yxy + z). Proof. The equivalence of (1) and (2) follows directly from Theorem 3.6. Also, it is easy to see that S satisfies (47) if and only if S/+D satisfies (73). So by Corollary 4.4, (2) is equivalent to (3). ¨ PROPOSITION 4.6. Let S be an idempotent semiring in Ld . Then S ∈ L if and only if S/(•L ∧ +D) ∈ L. In other words, •L ∈ Con(S) if and only if •LS/(•L∧+D) ∈ Con(S)/(•L ∧ +D). Proof. Suppose that S is an idempotent semiring in Ld , that is, (•L ∧ +D) ∈ Con(S). If S ∈ L then since L is a variety, we have S/(•L ∧ +D) ∈ L. Conversely, if S/(•L ∧ +D) ∈ L, that is, •LS/(•L∧+D) ∈ Con(S/(•L ∧ +D), then by Theorem 4.1(3), S/(•L ∧ +D) satisfies (73), (74) and (75). Hence, for any a, b, c ∈ S, we have (ab + c)(bab + c)(•L ∧ +D)(ab + c), (c + ab)(c + bab)(•L ∧ +D)(c + ba), cabcbab(•L ∧ +D)cab. This implies that (ab + c)(bab + c)(•L)(ab + c), (c + ab)(c + bab)(•L)(c + ab), cbacbab(•L)cab, and thereby, we obtain ab + c = (ab + c) · (ab + c)(bab + c) = (ab + c)(bab + c), c + ab = (c + ab) · (c + ab)(c + bab) = (c + ab)(c + bab), cab = cab · cabcbab = cabcbab. Thus, S satisfies (73), (74) and (75) and so by Theorem 4.1, we have S ∈ L, as required. ¨

Vol. 46, 2001

L-subvarieties of the variety of idempotent semirings

95

We now arrive at our main theorem. THEOREM 4.7. Let (S, +, ·) be an idempotent semiring. Then (•L) is a congruence on (S, +, ·) if and only if both (•L ∧ +D) and (•L ∨ +D) are congruences on (S, +, ·). In other words, we have Ld . L = Ld ∩ e Proof. The necessary part is obvious. We prove the sufficiency. Let S be an idempotent semiring such that both (•L ∧ +D) and (•L ∨ +D) are congruences on S. We show that (•L) is a congruence on S. Let a, b ∈ S such that a(•L)b. This implies that (a + c)(•L ∨ +D)(b + c) ⇒ (a + c)(+D)(a + c)(b + c) (by Lemma 3.2 (i)) = (a + cb + ac + c)(+D)(a + c + ac + cb) (since (x + y)(+D)(y + x) for all x, y ∈ S) = ((a + c)2 + ac + cb)(+D)(a + c + cd). Similarly (b + c)(+D)(b + c + ca). These imply (ca + c)(+D)(ca + c + cb) and (cd + c)(+D)(cd + c + ca). Thus (ca + c)(+D)(cb + c) which implies ca(+D)(cb + ca)(+D)cb. Then ca + cb + ca = ca and cb + ca + cb = cb. Also, since a(•L)b, we have ab = a, ba = b. Then (a + b)(b + a) = a + b, (b + a)(a + b) = b + a ⇒ (a + b)(•L ∧ +D)(b+a). Thus ca = (ca +cb+ca)(•L∧+D)(ca +cb) (•L∧+D)(cb+ca +cb) = cb. Now (a + c)(b + c) = (a + cb + ac + c)(•L ∧ +D)(a + ca + ac + c) = a + c. Similarly, (b + c)(a + c)(•L ∧ +D)(b + c). Therefore (a + c)(b + c)(•L)(a + c) ⇒ (a + c)(b + c) = (a + c). Similarly, (b + c)(a + c) = b + c. Thus (a + c)(•L)(b + c). Proceeding in the same way, we can show that (c + a)(•L)(c + b). Also in the process of the above proof, we have already shown that ca(•L)cb. Thus it follows that (•L) is a congruence on the semiring S as it is a right congruence on the semigroups (S, ·). ¨ Acknowledgment The authors are particularly grateful to Professors F. Pastijn and H. J. Shyr for their helpful discussions contributed to this paper. The authors also thank the referees for their valuable comments and suggestions which lead to a substantial improvement of this paper. REFERENCES [1] [2] [3] [4]

Bandelt, H. J. and Petrich, M., Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. 25 (1982), 144–171. Evans, T., The lattice of semigroup varieties, Semigroup Forum 2 (1) (1971), 1–43. Fennemore, C. F., All varieties of bands, I, Math. Nachr. 48 (1971), 237–251. Fennemore, C. F., All varieties of bands, II, Math. Nachr. 48 (1971), 253–262.

96 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

zhao xianzhong, k. p. shum and guo yuqi

algebra univers.

Ghosh, S. M., Lattice congruences on an idempotent commutatives semirings, Soochow J. Math. 22 (1996) no. 3, 357–362. Gierz, G. and Romanowska, A., Duality for distributive bisemilattices, J. Austral. Math. Soc. 51 (1991), 247–275. Golan, J. S., The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Pitman Monographs and Surveys in Pure and Appl. Maths. 54, Longman, New York, 1992. Guo, Y. Q., Pastijn, F. and Sen, M. K., Semirings which are union of rings, (to appear in Arch. Math.). Howie, J. M. Fundamentals of Semigroups Theory, Oxford Science Publications, Oxford, 1995. McKenzie, R. N., McNulty, G. F. and Taylor, W. F., Algebras, Lattices, Varieties, Vol. 1, Wadsworth and Brooks, Monterey 1987. McKenzie, R. N. and Romanowska, A., Varieties of distributive bisemilattices, Contributions to General Algebra (Proc. Klagenfurt Conf., Klagenfurt 1978), 213–218. Pastijn, F., Idempotent distributive semirings II, Semigroup Forum 26 (1983), 151–166. Pastijn, F. and Guo, Y. Q., The lattice of idempotent distributive semiring varieties, Science in China, Series A 29 (1999), 391–407. Pastijn, F. and Romanowska, A., Idempotent distributive semirings I, Acta Sci. Math. (Szeged) 44 (1982), 239–253. Pastijn, F. and Zhao, X. Z., Green0 s D-relation for the multiplicative reduct of an idempotent semirings, (to appear). Petrich, M., Lectures in Semigroups, Wiley, London, 1997. Plonka, J., On distributive quasi-lattices, Fund. Math. 60 (1967), 191–200. Romanowska, A., On bisemilattices with one distributive law, Algebra Univers. 10 (1980), 36–47. Romanowska, A., Idempotent distributive semirings with a semilattice reduct, Math. Japonica. 27 (1982), 483–493. Romanowska, A., Free idempotent distributive semigroups with a semilattice reduct, Math. Japonica. 27 (1982), 467–481. Romanowska, A., Subdirectly irreducible meet-distributive bisemilattices, II, Disc. Math. Algebra and Stoch. Meth. 15 (1995), 145–161. Romanowska, A., From bissemilattices to snack algebra, Fund. Inf. 31 (1997), 65–77. Sen, M. K., Guo, Y. Q. and Shum, K.P., A class of idempotent semirings, Semigroup Forum 60 (2000), 351–367. Zhao, X. Z. and Guo, Y. Q., On D-subvarieties of the variety of idempotent semirings, (to appear).

Z. Xianzhong Department of Mathematics Northwest University Xian, 710069 China K. P. Shum Department of Mathematics The Chinese University of Hong Kong Shatin, Hong Kong China (SAR) Y. Q. Guo Institute of Mathematics Yunnan University Kunming, 650091 China