Varieties of idempotent semirings with commutative addition

Algebra univers. 54 (2005) 301–321 0002-5240/05/030301 – 21 DOI 10.1007/s00012-005-1947-8 c Birkh¨ auser Verlag, Basel, 2005

Algebra Universalis

Varieties of idempotent semirings with commutative addition Francis Pastijn and Xianzhong Zhao Abstract. The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13 distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings free in these varieties are given.

1. Introduction A semiring is an algebra (S, +, ·) with two binary operations + and · such that both the reducts (S, +) and (S, ·) are semigroups and such that the distributive laws x(y+z) = xy+xz and (x+y)z = xz +yz hold. A band is a semigroup in which every element is an idempotent. A semiring is said to be an idempotent semiring if the two reducts are bands. For any variety V of bands, V+ (resp. V ) denotes the variety of idempotent semirings whose additive (resp. multiplicative) reduct belongs to V. The variety of all idempotent semirings with commutive addition is accordingly denoted by S+ . We shall be interested in the lattice L(S+ )of subvarieties of S+ . There are plenty of examples, natural or artificially fabricated, of idempotent semirings with commutative addition. In the category of the natural examples we mention the ordered bands investigated by T. Saitˆ o in [14] and [16]. His results extend earlier work done by M. L. Dubreil-Jacotin and T. Merlier for special cases. An ordered band (resp. ordered semilattice) is a band (resp. semilattice) endowed with a total order which is compatible with the band (resp. semilattice) operation. Ordered semilattices were also considered in [15]. The varieties contemplated by R. N. McKenzie and A. Romanowska in [4] are each generated by finite ordered semilattices. In Section 2 of this paper we slightly modify a construction introduced in [17] and obtain for every band F a member P f (F ) of S+ . We hope that the reader Presented by B. M. Schein. Received April 22, 2004; accepted in final form June 3, 2005. 2000 Mathematics Subject Classification: 16Y60, 08B05, 20M07. Key words and phrases: Idempotent semiring, band, order, variety, identity. The research of X. Z. Zhao is supported by a grant of NSF, China number 10471112 and a grant of Shaanxi Provincial Natural Science Foundation number 2003A10. 301

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will hesitate to merely lump the class of examples thus obtained into the category of the artificially fabricated ones, because every member of S+ is a homomorphic image of a semiring thus constructed and having a closely related structure, and moreover, as we shall find out in Section 3, semirings free in various subvarieties of S+ , including S+ itself, are of this form. Since an exploration into the lattice L(S+ )of subvarieties of S+ is the topic of this paper, a basic understanding of the structure of such semirings P f (F ) a prerequisite. Such details are provided in Section 3. The lattice L(B) of subvarieties of the variety B of all bands is well understood. We refer to [1] for a comprehensive treatment and for references to the original papers. For a first approach to the investigation of L(S+ ), it therefore makes sense to consider the complete ∩-homomorphism L(B) → L(S+ ),

V → V ∩ S+ .

(1.1)

We cannot expect that this mapping is a ∨-homomorphism, and as we shall see in Section 4, it is not. Neither can we expect that the image under this mapping generates L(S+ ). Again, as we shall see in Section 4, it does not. Various known subvarieties of S+ do not show up in this sublattice of L(S+ ), the most notorious one being the variety of distributive lattices. Other varieties not showing up may be found in [4], [5], [6], [7] and [11]. But at least one should hope that the lattice generated by the image of the mapping (1.1) yields a basic skeleton for L(S+ ). The main result of Section 4 gives a description of this sublattice of L(S+ ). Although copies of the countably infinite lattice L(B) can be found in several places in the lattice of idempotent semiring varieties (see [6]), the mapping (1.1) is not even injective; its image instead consists of only 13 varieties. In Section 3 we find that the multiplicative reducts of the members of S+ are regular bands. Therefore, if RB denotes the variety of regular bands, then S+ = (RB) ∩ S+ . For a good understanding of this paper we therefore don’t need the full force of [1] but can confine ourselves to the information concerning the lower part of L(B) as given in the relevant chapters of [2] and [9]. We shall assume that the reader is familiar with these particulars. For general background information we urge the reader to consult [2] and [3]. That the multiplicative reducts of the members of S+ are regular bands is in a sense not so surprising. The orderable bands considered by Saitˆ o in [14] and [16] are bands in which each D-class is either an L-class or an R-class, and so they are regular bands. Further motivation for exploring L(S+ )stems from the fact that it provides a strategy for finding examples of quasivarieties of regular bands. After all, for any W ∈ L(S+ ), the class of all bands embeddable in the multiplicative reduct of a member of W is a quasivariety of regular bands.

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In addition to the standard notation, which can be found in [2], [3] and [9], we continue to use the particular notation for idempotent semirings which was introduced in [6] and [8]. 2. The least regular band congruence We recall a construction given by Zhao in [17] and present a refinement of some of the results obtained there. Let F be any band. A subband C of F is said to be closed if for all a, b ∈ F and s, t ∈ F 1 we have that sat, sbt ∈ C ⇒ sabt ∈ C. For any nonempty subset A of F , We denote by A the smallest closed subband of F containing A and we say that A is the closed subband generated by A. A closed subband of F is said to be finitely generated if it is of the form A for some finite nonempty subset A of F . The closed subband generated by a ∈ F will be denoted by a. The set of finitely generated closed subbands of F will be denoted by P f (F ). We let P (F ) be the set of subsets of F . For A, B ∈ P (F ) we define A ◦ B = {ab | a ∈ A, b ∈ B}. Then (P (F ), ∪, ◦) is a semiring which is called the power semiring of F . Pf (F ) = {A | ∅ = A ⊆ F, A finite} constitutes a subsemiring of P (F ). By Lemma 2.2 of [17], we have A ∪ B = A ∪ B = A ∪ B,

A◦B =A◦B = A◦B

for all A, B ∈ Pf (F ). Therefore, if we introduce an addition + and a multiplication · on P f (F ) by putting for all A, B ∈ Pf (F ) A + B = A ∪ B,

A · B = A ◦ B,

then (P f (F ), +, ·) is an idempotent semiring with commutative addition and Pf (F ) → P f (F ),

A → A.

is a semiring homomorphism. Then, since F → Pf (F ),

a → {a}

is an embedding, ϕ : F → P f (F ),

a→a

(2.1)

is a multiplicative homomorphism. Later in this section we shall prove that the congruence relation induced on F by ϕ is the least regular band congruence. It should be clear that the semiring P f (F ) is generated by F ϕ. Indeed, for every A ∈ Pf (F ), A = a∈A a.

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Theorem 2.1. Let F be a band, S ∈ S+ and ψ : F → S a multiplicative homomorphism. Then with ϕ as in (2.1), there exists a unique semiring homomorphism ψ : P f (F ) → S which makes the diagram

ϕ

F

P f (F )

ψ

ψ

S commutative. Proof. Let A ∈ Pf (F ) and b ∈ A. By Theorem 2.1 of [17] there exists a positive integer n and subbands A1 , . . . , An of F such that (i) A1 is the subband of F generated by A; (ii) for 1 ≤ i < n, Ai+1 is the subband of F generated by the set {scdt | sct, sdt ∈ Ai , c, d ∈ F, s, t ∈ F 1 }; (iii) b ∈ An . We shall prove by induction that for every 1 ≤ i ≤ n , if q ∈ Ai , then qψ + a∈A aψ = a∈A aψ. Let q ∈ A1 . Thus there exists k and a1 , . . . , ak ∈ A such that q = a1 · · · ak . Then qψ = a1 ψ · · · ak ψ, and thus a1 ψ + · · · + ak ψ + qψ = (a1 ψ + · · · + ak ψ)k + qψ = (a1 ψ + · · · + ak ψ)k = a1 ψ + · · · + ak ψ

since qψ is one of the terms in the expansion of (a1 ψ + · · · + ak ψ)k . Therefore in particular, qψ + a∈A aψ = a∈A aψ. We now assume that 1 ≤ i < n and that for every element p ∈ Ai we have pψ +

a∈A

aψ =

a∈A

aψ.

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Let sct, sdt ∈ Ai for some c, d ∈ F, s, t ∈ F 1 . If 1 is the identity of F 1 we let 1ψ be the identity of the multiplicative semigroup S 1 . Then by the induction hypothesis, aψ = (sct)ψ + (sdt)ψ + aψ a∈A

a∈A

= sψ(cψ + dψ)tψ +

aψ

a∈A

= sψ(cψ + dψ)2 tψ +

aψ

a∈A

= (sct)ψ + (scdt)ψ + (sdct)ψ + (sdt)ψ +

aψ

a∈A

and therefore (scdt)ψ + a∈A aψ = a∈A aψ. Using the same method as in the basis of our proof by induction we can now show that for every q ∈ Ai+1 , qψ + a∈A aψ = a∈A aψ. This completes our proof by induction, and in particular we have aψ = aψ. bψ + a∈A

a∈A

Let A, B ∈ Pf (F ) such that A = B. Since B ⊆ A, we have from the foregoing that b∈B bψ + a∈A aψ = a∈A aψ, and since A ⊆ B we have aψ + bψ = bψ, whence

b∈B

bψ =

a∈A

b∈B

b∈B

a∈A aψ.

Therefore the mapping ψ : P f (F ) → S, A → aψ, A ∈ Pf (F )

(2.2)

a∈A

is well-defined. This mapping ψ is a semiring homomorphism since for all A, B ∈ Pf (F ), aψ + bψ = A ψ + B ψ, (A + B)ψ = A ∪ B ψ = a∈A

(A · B) ψ = A ◦ B ψ =

b∈B

(ab)ψ = (

a∈A, b∈B

a∈A

aψ)(

bψ) = (A ψ)(B ψ).

b∈B

Obviously ϕψ = ψ and ψ is the unique homomorphism which makes the diagram commutative since P f (F ) is generated by F ϕ. In view of Theorem 2.1 we can call P f (F ) the semiring of S+ which is freely generated by the band F .

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Corollary 2.2. [17] Let X be any set and (F BX , ı) the free band on X. Let ϕ : F BX → P f (F BX ), a → a. Then (P f (F BX ), ıϕ) is free in S+ on X. Proof. F BX

ı X

ϕ ψ

θ

P f (F BX )

ψ

S

(2.3)

Let S ∈ S+ and θ : X → S a mapping. Then there exists a unique band homomorphism ψ : F BX → S such that ıψ = θ. By Theorem 2.1, there exists a unique semiring homomorphism ψ : P f (F BX ) → S such that ψ = ϕψ. Then of course, θ = ıϕψ. Theorem 2.3. The multiplicative reduct of every member of S+ is a regular band. Proof. We use the same notation as in Corollary 2.2, and we let x, y, z ∈ X. In F BX we put c = xyx, d = zxy, s = xy, t = xyzxy and find that in F BX sct = xy · xyx · xyzxy = xyzxy = xy · zxy · xyzxy = sdt, scdt = xy · xyx · zxy · xyzxy = xyxzxy, and therefore xyxzxy ∈ xyzxy.

(2.4)

We now put c = xy, d = zxy, s = xyx, t = xyxzxy and find sct = xyx · xy · xyxzxy = xyxzxy = xyx · zxy · xyxzxy = sdt, scdt = xyx · xy · zxy · xyxzxy = xyzxy and therefore xyzxy ∈ xyxzxy.

(2.5)

Comparing (2.4) and (2.5) we find that xyzxy = xyxzxy. Thus in P f (F BX ) the equality xyzxy = xyxzxy is true. Therefore each member of S+ satisfies the identity xyzxy ≈ xyxzxy. It should be noted that this is equivalent . to the following structural property: for every S ∈ S+ , R is a congruence relation on the multiplicative reduct of S.

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.

By symmetry, for every S ∈ S+ , both L and R are congruences on the multiplicative reduct of S. Therefore this multiplicative reduct is a regular band. Theorem 2.4. If F is a regular band, then the multiplicative homomorphism ϕ : F → P f (F ),

a→a

is injective. Proof. We shall prove that for every a ∈ F, a = {a}. Let a = sct = sdt for some c, d ∈ F and s, t ∈ F 1 . If s = 1, then a = ct = dt, whence a = ca = cdt = scdt. Similarly, if t = 1, then a = scdt. We now assume that s, t ∈ F . We have that a = sca = adt and thus a = scsdtsctdt = scdt since the identity xyxzx ≈ xyzx holds in RB. Theorem 2.5. For any band F the multiplicative homomorphism ϕ : F → P f (F ),

a→a

induces the least regular band congruence θ on F and P f (F ) ∼ = P f (F/θ). Proof. By Theorem 2.3 the multiplicative reduct of P f (F ) is a regular band and so the congruence ϕϕ−1 induced by ϕ on F is a regular band congruence, that is, θ ⊆ ϕϕ−1 . We let θ : F → F/θ be the canonical homomorphism of F onto F/θ and ϕ : F/θ → P f (F/θ), aθ → aθ. By Theorem 2.1 there exists a unique semiring homomorphism ϕ : P f (F ) → P f (F/θ) which makes the diagram F

ϕ

ϕ

θ

F/θ

P f (F )

ϕ

P f (F/θ)

(2.6)

commutative. Thus ϕϕ−1 ⊆ (ϕϕ)(ϕϕ)−1 = (θ ϕ )(θ ϕ )−1 . By Theorem 2.4, ϕ is an injection and so (θ ϕ )(θ ϕ )−1 = θ. We conclude that ϕϕ−1 = (ϕϕ)(ϕϕ)−1 = θ. From this it follows that ϕϕ−1 is the equality relation on P f (F ), whence ϕ is injective. But certainly ϕ is surjective and so P f (F ) ∼ = P f (F/θ). We can now improve on Corollary 2.2.

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Porism 2.6. Let X be any set and (F RBX , ε) the free regular band on X. Let ξ : F RBX → P f (F RBX ), a → a. Then (P f (F RBX ), εξ) is free in S+ on X. Proof. If in the proof of Theorem 2.5 we put F = F BX and we identify F BX /θ with F RBX and ϕ with ξ. We obtained the commutative diagram F BX

ϕ

ϕ

θ

F RBX

P f (F BX )

ξ

P f (F RBX )

(2.7)

where ϕ is a semiring isomorphism. If ı is as in Corollary 2.2, then ıϕϕ = εξ, and so from Corollary 2.2 it follows that (P f (F RBX ), εξ) is free in S+ on X. 3. On the structure of the semirings P f (F ) In the following we shall assume that F is a regular band. For A ∈ Pf (F ) we put (3.1) RA = {Ra | a ∈ A} and RA = {Rb | b ∈ A},

(3.2)

where for every c ∈ F, Rc denotes the R-class of c in F . For A, B ∈ Pf (F ) we then define A δ B ⇔ RA = RB (3.3) Recall that F/R is a left regular band. Clearly RA ∈ Pf (F/R) for every A ∈ Pf (F ). Lemma 3.1. Let F be a regular band. For A ∈ Pf (F ), RA = RA ∈ P f (F/R). Proof. We first show that RA is a closed subband of F/R. Therefore, let Rs Rc Rt , Rs Rd Rt ∈ RA for some c, d ∈ F and s, t ∈ F 1 . Since R is a congruence on F 1 this implies that Rsct , Rsdt ∈ RA , and so there exist a, b ∈ A such that a R sct and b R sdt in F . Again, since R is a congruence on F , we have in F that abaRsctaba and so aba = sctaba = sctba ∈ A. Similarly, we also have sdtba = ba ∈ A. Since sctba, sdtba ∈ A and A is closed, so scdtba ∈ A. Since in F , sct ∈ Da , sdt ∈ Db , so scdt ∈ Dba , whence scdt R scdtba ∈ A, that is, Rs Rc Rd Rt ∈ RA . Thus, RA is closed. We let Rb ∈ RA with b ∈ A. We shall show that Rb ∈ RA . We let the subbands A1 , . . . , An of F be as in the proof of Theorem 2.1. We shall show by induction that

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for every 1 ≤ i ≤ n, if q ∈ Ai , then Rq ∈ RA . If q ∈ A1 , then q = a1 · · · ak for some a1 , . . . , ak ∈ A and thus Rq = Ra1 · · · Rak ∈ RA . Now assume that 1 ≤ i < n and that Rp ∈ RA for every p ∈ Ai . Let sct, sdt ∈ Ai for some c, d ∈ F and s, t ∈ F 1 . Thus, by the induction hypothesis, Rs Rc Rt = Rsct , Rs Rd Rt = Rsdt ∈ RA , and so Rscdt = Rs Rc Rd Rt ∈ RA , since RA is closed. Using the same method as in the basis of our induction, we then find that Rq ∈ RA for every q ∈ Ai+1 . So in conclusion, Rb ∈ RA , as required. From the first part of the proof it follows that RA ⊆ RA , where the second part yields the reverse inclusion. Theorem 3.2. For a regular band F , ϕ : P f (F ) → P f (F/R),

A → RA

(3.4)

is a surjective homomorphism of semirings. Proof. For A, B ∈ Pf (F ), by Lemma 3.1, and since R is a congruence on F , we have (A · B)ϕ =(A ◦ B)ϕ = RA◦B

= RA◦B = RA ◦ RB

=RA RB = RA RB = AϕBϕ, (A + B)ϕ =(A ∪ B)ϕ = RA∪B

= RA∪B = RA ∪ RB

=RA +RB = RA +RB = Aϕ + Bϕ. Every element of Pf (F/R) is of the form RA for some A ∈ Pf (F ). Therefore every element of P f (F/R) is of the form RA for some A ∈ Pf (F ). By Lemma 3.1, RA = RA = Aϕ, and so ϕ is surjective. The relation δ defined in (3.3) is obviously the congruence relation induced on P f (F ) by the homomorphism (3.4). For A ∈ Pf (F ) we can define LA and LA in a way dual to the definitions (3.1) and (3.2) and we obtain the duals of Lemma 3.1 and Theorem 3.2. The congruence relation induced by the surjective homomorphism of semirings P f (F ) → P f (F/L), A → LA (3.5) will be denoted by σ. Lemma 3.3. Let F be a regular band. Then σ ∩ δ is the equality on P f (F ). Proof. Assume that A( σ ∩ δ )B for some A, B ∈ Pf (F ), and let a ∈ A. Then Ra ∈ RA = RB

and La ∈ LA = LB

and so there exist b, c ∈ B such that b R a L c in F . Then a = bc ∈ B. By symmetry, A = B.

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Theorem 3.4. Let F be a regular band. Then P f (F ) → P f (F/L) × P f (F/R),

A → (LA , RA ),

A ∈ Pf (F ),

is an injective homomorphism of semirings. .

We next compare the relation R on P f (F ) with the δ-relation. Lemma 3.5. Let F be a regular band. Then for A, B ∈ Pf (F ), ˙ A R˙ B in P f (F ) ⇐⇒ A δ B and LA RL B in P f (F/L). If this is the case, then B = A ◦ B and A = B ◦ A. ˙ in P f (F ). Assume that for some s, t ∈ F 1 and c, d ∈ F we have Proof. Let ARB that sct, sdt ∈ A ◦ B, and let p = scdt. In any case p ∈ A ◦ B = A B = B. Further, sct = ab for some a ∈ A and b ∈ B. Since D is the least semilattice congruence on F and sct = asct, we have in F ap D asctsdt = sctsdt D p and thus p R pa ∈ B ◦ A ⊆ B ◦ A = B A = A. Therefore p = pap ∈ A ◦ B. We proved that A ◦ B is a closed subband of F . Therefore A ◦ B = A ◦ B = A B = B. By symmetry, B ◦ A = A. ˙ in P f (F ) and b ∈ B. By the foregoing, b = ab1 for some a ∈ A Again, let ARB and b1 ∈ B. Then b R ab1 a in F and ab1 a = ba ∈ B ◦ A = A. Thus, Rb ∈ RA . ˙ in P f (F ) and since (3.5) is a By symmetry RA = RB , that is, A δ B. From ARB ˙ homomorphism, we have LA RLB in P f (F/L). ˙ To prove the converse, assume that A δ B and LA RL B in P f (F/L). By the dual of Lemma 3.1, LA = LA and LB = LB . By the above, LB = LA ◦ LB . Let a ∈ A and b ∈ B, thus ab ∈ A ◦ B. Since A δ B there exists b1 ∈ B such that Ra = Rb1 , and then ab R b1 b ∈ B since R is a congruence on F . Since L is a congruence on F , Lab = La Lb ∈ LA ◦ LB = LB and so ab L b2 ∈ B in F . Thus ab = b1 bb2 ∈ B. We proved that A ◦ B ⊆ B. Let b ∈ B. Since A δ B there exists a ∈ A such that a R b and so b = ab ∈ A ◦ B. Hence B ⊆ A ◦ B and the equality A ◦ B = B prevails. Therefore A B = B, and by ˙ symmetry also B A = A. Hence ARB. Theorem 3.6. For a regular band F the following statements are equivalent: (i) F ∈ LRB. (ii) P f (F ) ∈ (LRB) ∩ S+ . (iii) δ is the equality on P f (F ). Proof. If δ is the equality on P f (F ), then so is R˙ because R˙ ⊆ δ by Lemma 3.5. Therefore the multiplicative reduct of P f (F ) is a left regular band. Therefore also F is a left regular band by Theorem 2.4. We have proved (iii)⇒(ii)⇒(i).

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Let F be a left regular band and A δ B for some A, B ∈ Pf (F ). For a ∈ A there exists b ∈ B such that Ra = Rb in F . Since R is the equality on F , a = b. Therefore A ⊆ B, and by symmetry, A = B. We proved (i)⇒(iii). Corollary 3.7. Let F be a regular band. Then P f (F ) is a subdirect product of P f (F/R) ∈ (LRB) ∩ S+ and P f (F/L) ∈ (RRB) ∩ S+ . Proof. The proof follows immediately from Theorems 3.2, 3.4 and 3.6.

Theorem 3.8. Let F be a regular band. Then δ is the least (LRB) -congruence ˙ on P f (F ), and δ is the congruence relation generated by R. Proof. By Theorems 3.2 and 3.6, P f (F )/δ ∼ = P f (F/R) ∈ (LRB) , where δ is a (LRB) -congruence. ˙ For A, B ∈ Let R˙ ∗ denote the congruence relation on P f (F ) generated by R. Pf (F ), let A δ B in P f (F ). For a ∈ A choose a1 ∈ B such that Ra = Ra1 in F, and for any b ∈ B, choose b1 ∈ A such that Rb = Rb1 in F . If we put A1 = A∪{b1 |b ∈ B} and B1 = B ∪ {a1 | a ∈ A}, then A1 , B1 ∈ Pf (F ) and A = A1 , B = B 1 . In P f (F ) we have a R˙ a1 , a ∈ A and b R˙ b1 , b ∈ B, and thus A = A1 = a+ b1 R˙ ∗ a+ b = B 1 = B. a∈A

b∈B

a∈A

b∈B

Therefore δ ⊆ R˙ ∗ , and this entails that δ = R˙ ∗ since δ is a congruence which ˙ contains R. ˙ in P f (F ) for some A, B ∈ Let ξ be any (LRB) -congruence on P f (F ). If ARB ˙ Pf (F ), then (Aξ)R(Bξ) in P f (F )/ξ, and since P f (F )/ξ ∈ (LRB) , so Aξ = Bξ. Therefore R˙ ⊆ ξ and thus also δ = R˙ ∗ ⊆ ξ. Thus, δ is the least (LRB) -congruence. We now introduce notation in analogy with (3.1) and (3.2). For A ∈ Pf (F ), with F a regular band, we put DA = {Da | a ∈ A} and DA = {Db | b ∈ A} Theorem 3.9. Let F be a regular band. Then δ ∨ σ = δ ◦ σ = σ ◦ δ is the least ˙ For Bi-congruence on P f (F ), and δ ∨ σ is the congruence relation generated by D. A, B ∈ Pf (F ), A (δ ∨ σ) B ⇔ DA = DB . (3.6) Proof. From Theorem 3.8 and its dual, δ ∨ σ is the least ((LRB) ∩ (RRB) )congruence on P f (F ). Since (LRB) ∩ (RRB) = S , and since P f (F ) ∈ S+ in

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any case, so δ ∨ σ is the least (S ∩ S+ )-congruence, that is, δ ∨ σ is the least Bi-congruence. Assume that for A, B ∈ Pf (F ), A (δ ∨ σ) B. Then there exists a positive integer n and A1 , . . . , An ∈ Pf (F ) such that A = A1 δ A2 σ A3 δ · · · δ An−1 σ An = B. If for 1 ≤ i < n, Ai δ Ai+1 , then RAi = RAi+1 and thus also DAi = DAi+1 . Dually, if Ai σAi+1 , then again DAi = DAi+1 . Therefore A (δ ∨ σ) B entails DA = DB . Assume that for some A, B ∈ Pf (F ), DA = DB . For any a ∈ A, choose a1 ∈ B such that Da = Da1 , and for any b ∈ B, choose b1 ∈ A such that Db = Db1 . Then put A1 = A ∪ {b1 | b ∈ B},

B1 = B ∪ {a1 | a ∈ A}

and thus A1 , B1 ∈ Pf (F ) with A = A1 and B = B 1 . We next put C = {cd | c ∈ A1 , d ∈ B1 , Dc = Dd }. Clearly RC ⊆ RA1 . If a ∈ A, then aa1 ∈ C with Ra = Raa1 . If b1 ∈ A1 for some b ∈ B, then b1 b ∈ C with Rb1 = Rb1 b . Therefore RC = RA1 and by Lemma 3.1, RC = RC = RA1 = RA1 = RA . Dually, LC = LB1 and LC = LB1 = LB . Hence A δ C σ B and in particular A (δ ∨ σ) B. We conclude that (3.6) holds true. We also showed that A (δ ∨ σ) B for some A, B ∈ Pf (F ), then A (δ ◦ σ) B, and therefore δ ∨ σ = δ ◦ σ. By symmetry, also δ ∨ σ = σ ◦ δ holds. By Lemma 3.5 and its dual, R˙ ⊆ δ and L˙ ⊆ σ, whence D˙ = L˙ ∨ R˙ ⊆ δ ∨ σ. Since δ ∨ σ is a congruence relation, so δ ∨ σ contains the congruence generated by D˙ . Since R˙ ⊆ D˙ , so by Theorem 3.8, δ is contained in the congruence generated by D˙ . By symmetry, δ ∨ σ is contained in the congruence generated by D˙ . Therefore δ ∨ σ is the congruence generated by D˙ . We are now in the position to give analogues of Lemma 3.1 and Theorem 3.2. For a regular band F , F/D is a semilattice. Clearly DA ∈ Pf (F/D) for every A ∈ Pf (F ). Lemma 3.10. Let F be a regular band. For A ∈ Pf (F ), DA = DA ∈ P f (F/D). Proof. We first show that DA is closed. Let Ds Dc Dt , Ds Dd Dt ∈ DA for some c, d ∈ F and s, t ∈ F 1 . Then sct D a and sdt D b for some a, b ∈ A, and thus scdt D ab ∈ A. Therefore Ds Dc Dd Dt ∈ DA , and so DA is indeed closed. For Db ∈ DA , with b ∈ A we can show that Db ∈ DA : the proof is analogous to the second part of the proof of Lemma 3.1. Thus DA = DA , as required. Theorem 3.11. For a regular band F , ϕ : P f (F ) → P f (F/D),

A → DA

(3.7)

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is a surjective homomorphism of semirings. The homomorphism ϕ induces the congruence δ ∨ σ on P f (F ). P f (F/D) is the greatest bisemilattice homomorphic image of P f (F ). Proof. That ϕ is a homomorphism of semirings can be proved in a way analogous to the first part of the proof for Theorem 3.2: Lemma 3.10 is used here. Every element of P f (F/D) is of the form DA = DA for some A ∈ Pf (F ), and so the homomorphism given by (3.7) is surjective. It follows from Theorem 3.9 that ϕ induces δ ∨ σ and that P f (F/D) is the greatest bisemilattice homomorphic image of P f (F ). From Theorem 3.8 it follows that, for a regular band F , R˙ is a congruence on ˙ We investigate under which conditions this will happen. P f (F ) if and only if δ = R. We first have a lemma of a more general nature. Lemma 3.12. Let S ∈ S+ with R˙ a congruence on S. Then S ∈ (LQNB) ∩ S+ . ˙ and p = ep = pe, q = eq = qe for Proof. Assume that in S we have that pRq ˙ some p, q, e ∈ S. Then pRp + q, and since R˙ is a congruence we have also that ˙ + p + q, whence e + pRe e + p = (e + p + q)(e + p) = e + p + q since pq = q and qp = p. Therefore p = p(e + p) = p(e + p + q) = p + q and similarly, q = p + q. Thus p = q. Therefore the multiplicative reduct of S is a left quasinormal band. Theorem 3.13. For a regular band F the following are equivalent: (i) F ∈ LQNB. (ii) P f (F ) ∈ (LQNB) ∩ S+ . (iii) δ = R˙ on P f (F ). ˙ then R˙ is a congruence on P f (F ) by Theorem 3.8, and so (ii) Proof. If δ = R, holds true by Lemma 3.12. The proof for (ii)⇒(i) follows from Theorem 2.4. It remains to show that (i) implies (iii). Let F ∈ LQNB. Assume that A δ B in P f (F ) for some A, B ∈ Pf (F ). For b ∈ B there exists b1 ∈ A such that Rb1 = Rb in F , whence b = b1 b ∈ A ◦ B and so certainly B ⊆ A ◦ B. Now let a ∈ A and consider ab ∈ A ◦ B. There exists a1 ∈ B such that Ra = Ra1 in F . Since R is a congruence on F , so ab R a1 b and thus also bab R ba1 b. Since F ∈ LQNB it follows that bab = ba1 b and therefore that ab = a1 b ∈ B. We conclude that A ◦ B = B and so A B = B in P f (F ). By

314

F. Pastijn and X. Zhao ·

Algebra univers.

·

symmetry, ARB in P f (F ). Since R ⊆ δ by Lemma 3.5 we can now conclude that ·

δ =R on P f (F ).

From Theorem 3.9 we know that for a regular band F , D˙ is a congruence on P f (F ) if and only if δ ∨ σ = D˙ . We now characterize the regular band F for which this is the case. We have the following analogue of Lemma 3.12 : its proof follows the same lines as the proof for Lemma 3.12. The content of this lemma forms part of Lemma 4.4 of [18]. +

Lemma 3.14. Let S ∈ S such that D˙ is a congruence on S. Then S ∈ (NB) ∩S+ . Theorem 3.15. For a regular band F the following statements are equivalent: (i) F ∈ NB. (ii) P f (F ) ∈ (NB) ∩ S+ . (iii) δ ∨ σ = D˙ on P f (F ). Proof. The proof for (iii)⇒(ii)⇒(i) follows from Theorems 2.4, 3.9 and Lemma 3.14. If F ∈ NB then in particular F ∈ LQNB and so by Theorem 3.13, δ = R˙ on P f (F ). Using the dual of Theorem 3.13 we also have that σ = L˙ on P f (F ). Therefore δ ∨ σ = R˙ ∨ L˙ = D˙ on P f (F ). We proved that (i)⇒(iii). Remark. If F is a normal band, then P f (F ) consists of the finite subbands of F (see also [12] and [13]) Theorem 3.16. Let V is a subvariety of the variety NB. For a normal band F the following statements are equivalent: (i) F ∈ V. (ii) P f (F ) ∈ V ∩ S+ . Proof. (ii)⇒(i) follows from Theorem 2.4. (i)⇒(ii) follows from Theorem 3.15 and Theorem 3.6 and its dual if V is one of the varieties S, LNB, RNB or NB. If V = ReB, then one verifies that for all A, B ∈ P f (F ) we have that ABA = A ◦ B ◦ A = A, and thus P f (F ) ∈ ReB ∩ S+ . The situation is similar if V is one of the varieties Lz or Rz. Of course, if V = T, then P f (F ) is the one-element semiring. Therefore (i)⇒(ii) holds true in all cases. We are now ready for the main theorem of this section. For each subvariety V of RB and every nonempty set X we give a model for the free semiring in V ∩ S+ on X. Theorem 3.17. Let X be any set and V a subvariety of RB. Let (F VX , ε) be free in V on X. Let ξ : F VX → P f (F VX ), a → a. Then (P f (F VX ), εξ) is free in V ∩ S+ on X.

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Proof. The proof is based on Theorem 2.1 and proceeds along the lines of the proof for Corollary 2.2. In addition we need to show that for every subvariety V of RB, P f (F VX ) belongs to V ∩S+ . For V = RB this follows from Theorem 2.3 and this +

˙ ∩ S gave rise to Porism 2.6. If V is a proper subvariety of RB, then P f (F VX ) ∈ V follows from Theorems 3.6, 3.13, 3.15 and their duals. Corollary 3.18. The mapping θ : L(RB) → L(S+ ),

V → V ∩ S+

is an injective ∩-homomorphism. Corollary 3.19. Let S ∈ S+ . Then R˙ is a congruence on S if and only if S ∈ (LQNB) ∩ S+ . Corollary 3.20. [Lemma 4.4 in 19] For S ∈ S+ the following are equivalent: (i) (ii) (iii)

S ∈ (NB) ∩ S+ , L˙ and R˙ are congruence relations on S, D˙ is a congruence relation on S.

4. A sublattice of L(S+ ) In this section we calculate the sublattice of L(S+ ) generated by the 13 varieties in the image of the mapping θ of Corollary 3.18. We shall find that this lattice consists of exactly 20 varieties. Lemma 4.1. Each of {V ∩ S+ | S ⊆ V ⊆ RB}

(4.1)

{T, (Lz) ∩ S , (Rz) ∩ S , ReB ∩ S } +

+

+

(4.2)

forms a sublattice of L(S+ ). Proof. We first look at the diagram on the left of Figure 1, which represents the ∩-semilattice consisting of the varieties mentioned in (4.1). We need to verify that the joins are as suggested by this diagram. It suffices to show that for any S ⊆ V ⊆ RB in L(RB) we have that V ∩ S+ = (V ∩ (LRB) ∩ S+ ) ∨ (V ∩ (RRB) ∩ S+ ). This equality however follows from the fact that for every nonempty set X, the free semiring on X in V ∩ S+ is a subdirect product of the free semirings on X in V ∩ (LRB) ∩ S+ and V ∩ (RRB) ∩ S+ , respectively: we apply Theorems 3.2, 3.4, 3.8 and 3.17. The diagram on the right can be handled in the same way. This simple situation occurs already in [6].

316

Algebra univers.


(ReB) ∩S+ S+ =(RB) ∩S+ (LQNB) ∩(RB) ∩S+

(RB) ∩(RQNB) ∩S+

+

=(RQNB) ∩S+

=(LQNB) ∩S

(Lz) ∩S+

(Rz) ∩S+

(NB) ∩S+

+

(RRB) ∩S+

(LRB) ∩S

T (LNB) ∩S+

(RNB) ∩S+

Bi

Figure 1

The two lattices we found in Figure 1 must be assembled into a bigger one. Before we do so, we shall calculate the interval [Bi, (NB) ∩ S+ ] of L(S+ ). We first investigate the interval [Bi, (LNB) ∩ S+ ]. The variety Bi ∨ ((Lz) ∩ S+ ) obviously belongs to this interval. In the following X will be a countably infinite set of variables. Let V ∈ L(RB). In view of Theorem 3.17 we can, in the presence of determining identities for V ∩S+ , write any identity in the form u ≈ v where u = u1 +· · ·+uk , v = v1 +· · ·+vl , and where the terms ui and vj belong to F VX . Then u ≈ v is satisfied in V ∩ S+ if and only if the finite sets {u1 , . . . , uk } and {v1 , . . . , vl } generate the same closed subband of F VX . If V ∈ L(NB), then by the remark made before Theorem 3.16, u ≈ v is satisfied in V ∩ S+ if and only if these finite sets generate the same subband of F VX . In any case, if no particular V ∈ L(RB) is specified, and since S+ = RB ∩ S+ , we can, in the presence of the determining identities for S+ , write every identity in the form u ≈ v as given above, where the terms ui and vj belong to F RBX . In this case we shall call u ≈ v a S+ -identity, in the former case we call u ≈ v a (V ∩ S+ )-identity. The ∪-semilattice of finite nonempty subsets of X is a model of the free semilattice on the set X. If we denote this semilattice by F SX and put ε : X → F SX , x → {x}, then (F SX , ε) is free on X in S. Accordingly, with the notation of Theorem 3.17, (P f (F SX ), εξ) is free in Bi on X. Here P f (F SX ) consists of the finite subsemilattices of F SX . For u ∈ F RBX , let c(u) ⊆ X be the set of variables which occur in u. If u = u1 + · · · + uk with ui ∈ F RBX , let C(u) = {c(ui ) | 1 ≤ i ≤ k} ⊆ F SX . Thus, with u = u1 + · · · + uk , v = v1 + · · · + vl ,

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we have that the S+ -identity u ≈ v is satisfied in Bi if and only if C(u) and C(v) generate the same subsemilattice of F SX . We call c(u) the content of u ∈ F RBX and C(u) the content of u = u1 +· · ·+uk , ui ∈ F RBX . If V ∈ L(RB) with S ⊆ V, then we can define c(u), u ∈ F VX , and C(u), u = u1 + · · · + uk , ui ∈ F VX , in the same way, and again we have that the (V ∩ S+ )-identity u ≈ v is satisfied in Bi if and only if C(u) and C(v) generate the same subsemilattice of F SX . When investigating the interval [Bi, (LNB) ∩ S+ ] of L(S+ ) we shall be concerned only with ((LNB) ∩ S+ )-identities which are satisfied in Bi. Let V ∈ L(RB) and Lz ⊆ V. For u ∈ F VX we can define h(u) ∈ X to be the first letter (from the left) which occurs in u. If u = u1 + · · · + uk , ui ∈ F VX , we let H(u) = {h(ui ) | 1 ≤ i ≤ k} ⊆ X. We call h(u) the head of u ∈ F VX and H(u) the head of u = u1 + · · · + uk , ui ∈ F VX . It is easy to see that the (V ∩ S+ )-identity u ≈ v is satisfied in (Lz) ∩ S+ if and only if H(u) = H(v). Lemma 4.2. Bi ∨ ((Lz) ∩ S+ ) is the subvariety of (LNB) ∩ S+ determined by the identity xy + xyz + yxz ≈ yx + xyz + yxz. (4.3) Proof. The identity (4.3) is satisfied in Bi ∨ ((Lz) ∩ S+ ) since the content of both sides of (4.3) is {{x, y}, {x, y, z}} and the head of both sides of (4.3) is {x, y}. We now show that every ((LNB) ∩S+ )-identity which holds in Bi∨((Lz) ∩S+ ) is a consequence of (4.3). We let u ≈ v be such an identity. We can as well assume that u = u1 + · · · + uk and v = v1 + · · · + v where {u1 , . . . , uk } and {v1 , . . . , v } each constitute subbands of F LNBX of orders k and , respectively. Since this identity is satisfied in both Bi and ((Lz) ∩ S+ ) we have that C(u) = C(v) and H(u) = H(v). Here C(u) = C(v) constitutes a subsemilattice of F SX . Let A be the largest set in C(u) = C(v), and let n = |H(u)| = |H(v)|. By appropriately ordering the terms which occur in u and in v we have that uk−n+1 , . . . , uk and v−n+1 , . . . , v are precisely the n terms in each expression, each with content A, and then {uk−n+1 , . . . , uk } = {v−n+1 , . . . , v }, {h(uk−n+1 ), . . . , h(uk )} = H(u) = H(v) = {h(v−n+1 ), . . . , h(v )}. Assume that for some i, ui = vj for all j. We choose vi such that c(ui ) = c(vi ). Then, since ui = vi , in F LNBX , x = h(ui ) = h(vi ) = y. Clearly i ≤ k − n, and thus also c(ui ) = c(vi ) is a proper subset of A. Whence i ≤ l − n. Let uk , uk−1 , v , vl−1 be the terms with content A where {h(uk ), h(uk−1 )} = {x, y} = {h(vl ), h(vl−1 )}. Then ui + uk−1 + uk ≈ vi + vl−1 + v

318


Algebra univers.

is a consequence of (4.3). By symmetry it follows that u ≈ v is a consequence of (4.3). Lemma 4.3. Bi ∨ ((Lz) ∩ S+ ) is a proper subvariety of ((LNB) ∩ S+ ) which covers Bi. Proof. Bi∨((Lz) ∩S+ ) = ((LNB) ∩S+ ) since (4.3) is not satisfied in ((LNB) ∩ S+ ): {xy, xyz, yxz} and {yx, xyz, yxz} constitute different subbands of F LNBX . Let W be a variety such that Bi ⊆ W ⊆ Bi ∨ ((Lz) ∩ S+ ) and Bi = W. Then there exists S ∈ W such that S has a nontrivial L˙ -class. Such a nontrivial L˙ -class is subsemiring of S which generates ((Lz) ∩ S+ ). Therefore W = Bi ∨ ((Lz) ∩ S+ ). Lemma 4.4. The interval [Bi, ((LNB) ∩ S+ )] of L(S+ ) consists of the three varieties Bi, Bi ∨ ((Lz) ∩ S+ ) and ((LNB) ∩ S+ ). Proof. Let W be a variety in the interval such that W = ((LNB) ∩S+ ). Let u ≈ v be a ((LNB) ∩ S+ )-identity which is satisfied in W, but not in ((LNB) ∩ S+ ). We can assume that u = u1 + · · · + uk , v = v1 + · · · + v where {u1 , . . . , uk } and {v1 , . . . , v } constitute (different) subbands of F LNBX . Then, since u ≈ v is satisfied in Bi, C(u) = C(v) is a subsemilattice of F SX . If H(u) = H(v), then u ≈ v is not satisfied in ((Lz) ∩ S+ ) and so W = Bi by the argument used in the proof of Lemma 4.3. We shall henceforth assume that H(u) = H(v). Therefore u ≈ v is satisfied in Bi ∨ ((Lz) ∩ S+ ) and has the same features as the identity u ≈ v discussed in the proof of Lemma 4.2. By symmetry we may assume that for some i, ui = vj in F LNBX for all j. We shall use the same notation concerning u ≈ v as in the proof of Lemma 4.2. In the identity uui ≈ vui we substitute x by x, all other variables of c(ui ) by y, and the all the remaining variables by xyz. From uui ≈ vui we then derive the consequence xy + yx + xyz + yxz ≈ yx + xyz + yxz.

From this ((LNB) ∩ S+ )-identity we have (4.3) as a consequence. Therefore W = Bi ∨ ((Lz) ∩ S+ ) by Lemma 4.2. We shall need the following well-known result concerning congruence relations on normal bands. Lemma 4.5. Let F be a normal band, ı the equality on F , and consider the intervals I = [ı, D], I1 = [L, D], I2 = [R, D] of the congruence lattice on F . Then the mappings I → I1 × I2 , ρ → (ρ ∨ L, ρ ∨ R) I1 × I2 → I, are pairwise inverse lattice isomorphisms.

(ξ, ζ) → ξ ∩ ζ

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319

(NB) ∩S+

((LNB) ∩S+ )∨((Rz) ∩S+ )

(LNB) ∩S+

((Lz) ∩S+ )∨((RNB) ∩S+ )

Bi∨((ReB) ∩S+ )

Bi∨((Lz) ∩S+ )

(RNB) ∩S+

Bi∨((Rz) ∩S+ )

Bi

Figure 2 Theorem 4.6. The interval [Bi, ((NB) ∩ S+ )] of L(S+ ) is given by Figure 2. Proof. We need to find the fully invariant congruences on the free semiring in (NB) ∩ S+ on a countably infinite set X which are contained in the least bisemilattice congruence. By Theorem 3.17 this free object is P f (F NBX ). The relations L˙ and R˙ are fully invariant congruences on P f (F NBX ); by Theorems 3.8 and 3.13 and their duals L˙ is the least ((RNB) ∩ S+ )-congruence and R˙ is the least ((LNB) ∩ S+ )-congruence. So by Theorems 3.9 and 3.15, D˙ = L˙ ∨ R˙ is the least Bi-congruence. Therefore, by Lemma 4.5, every fully invariant congruence contained in the least bisemilattice congruence can be uniquely written as the intersection of a fully invariant congruence of the interval [L˙ , D˙ ] and a fully ˙ D˙ ]. Rephrased in terms of varieties, every invariant congruence of the interval [R, + variety in [Bi, (NB) ∩ S ] can be written uniquely as the join of a variety of [Bi, (LNB) ∩ S+ ] and a variety of [Bi, (RNB) ∩ S+ ]. By Lemma 4.4 and its dual the desired result follows. We now arrive at the main theorem of this section. Theorem 4.7. Let θ be the mapping of Corollary 3.18. Then the sublattice of L(S+ ) generated by L(RB)θ is given by Figure 3. Proof. When collating the lattices which appear in Lemma 4.1 and Theorem 4.6, two new varieties appear : ((Lz) ∩ S+ ) ∨ ((RRB) ∩ S+ ) and its dual. Clearly (RRB) ∩ S+ ⊆ ((Lz) ∩ S+ ) ∨ ((RRB) ∩ S+ ) ⊆ (RQNB) ∩ S+

320

Algebra univers.


S+ =(RB) ∩S+

(LQNB) ∩S+

(RQNB) ∩S+ (NB) ∩S+

((LRB) ∩S+ )∨((Rz) ∩S+ )

((Lz) ∩S+ )∨((RRB) ∩S+ )

(LRB) ∩S+

(RRB) ∩S+ Bi∨((ReB) ∩S+ )

(LNB) ∩S+

(RNB) ∩S+

Bi∨((Lz) ∩S+ )

Bi∨((Rz) ∩S+ )

Bi

(ReB) ∩S+ (Lz) ∩S+

(Rz) ∩S+

T

Figure 3 and the first inclusion is strict since (RRB) ∩ S+ does not contain any nontrivial member of (Lz) ∩ S+ . To see that the second inclusion is strict, it suffices to verify that the identity xy + xyzxy + yxzxy ≈ yxy + xyzxy + yxzxy is satisfied in ((Lz) ∩ S+ ) ∨ ((RRB) ∩ S+ ) but not in (RQNB) ∩ S+ . It is now clear from Lemma 4.1 and Theorem 4.6 that the varieties mentioned in Figure 3 form a ∨-semilattice. To verify that these varieties also form a ∩-semilattice,

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it helps to see that result of the intersection of ((Lz) ∩ S+ ) ∨ ((RRB) ∩ S+ ) with any variety of Figure 3 which is incomparable with it necessarily belongs to the interval [Bi, (NB) ∩ S+ ] and is thus one of the varieties listed in Theorem 4.6. References [1] J. A. Gerhard and M. Petrich, Varieties of bands revisited, Proc. London Math. Soc. 3 58 (1989), 323–350. [2] J. M. Howie, Fundamentals of Semigroup Theory, Oxford Science Publications, Oxford, 1995. [3] R. N. McKenzie, G. F. McNulty and W. F. Taylor, Algebras, Lattices, Varieties, Vol.1, Wadsworth and Brooks/Cole, Monterey, 1987. [4] R. N. McKenzie and A. Romanowska, Varieties of ·-distributive bisemilattices, in: Contributions to General Algebra (Proc. Klagenfurt Conf., Klagenfurt, 1978), Heyn, Klagenfurt, 1979, 213–218. [5] F. Pastijn, Idempotent distributive semirings II, Semigroup Forum, 26 (1983), 153–166. [6] F. Pastijn and Y. Q. Guo, The lattice of idempotent distributive semiring varieties, Science in China, (Ser. A) 48(8) (1999), 785–804. [7] F. Pastijn and A. Romanowska, Idempotent distributive semirings, Acta Sci. Math. (Szeged), 44 (1982), 239–253. [8] F. Pastijn and X. Z. Zhao, Green’s D-relation for the multiplicative reduct of an idempotent semiring, Arch. Math. (Brno), 36 (2000), 77–93. [9] M. Petrich, Lectures in Semigroups, Wiley, London, 1977. [10] A. Romanowska, Free idempotent distributive semirings with a semilattice reduct, Math. Japonica, 27 (1982), 467–481. [11] A. Romanowska, Idempotent distributive semirings with a semilattice reduct, Math. Japonica, 27 (1982), 483–493. [12] A. Romanowska and J. D. H. Smith, Modal Theory, Heldermann Verlag, Berlin, 1985. [13] A. Romanowsk, and J. D. H. Smith, Modes , World Scientific, Singapore,2002. [14] T. Saitˆ o, Ordered idempotent semigroups, J. Math. Soc. Japan, 14(2) (1962), 150–169. [15] T. Saitˆ o, Ordered inverse semigroups, Trans. Amer. Math. Soc., 153 (1971), 99–138. [16] T. Saitˆ o, The orderability of idempotent semigroups, Semigroup Forum, 7 (1974), 264–285. [17] X. Z. Zhao, Idempotent semirings with a commutative additive reduct, Semigroup Forum, 64 (2002), 289–296. [18] X. Z. Zhao, Y. Q. Guo, and K. P. Shum, D-subvariety of idempotent semirings, Algebra Colloquium, 9 (2002), 15–28. [19] X. Z. Zhao, K. P. Shum, and Y. Q. Guo, L-subvariety of idempotent semirings, Algebra Universalis, 46 (2001), 75–96. Francis Pastijn Department of Mathematics, Statistics, and Computer Science, Marquette University, P.O. Box 1881, Milwaukee, Wisconsin 53201-1881, USA e-mail : [email protected] Xianzhong Zhao Department of Mathematics, Northwest University, Shaanxi Xi’an 710069, The People’s Republic of China e-mail : [email protected]