All Regular-Solid Varieties of Idempotent Semirings

Abstract The lattice of all regular-solid varieties of semirings splits in two complete sublattices: the sublattice of all idempotent regular-solid varieties of semirings and the sublattice of all normal regular-solid varieties of semirings. In this paper, we discuss the idempotent part.


Introduction
Varieties of semirings are varieties of algebras of type (2,2), where both binary operations are associative and satisfy the two usual distributive laws. Single semirings as well as classes of semirings form important structures in Automata and Formal Languages Theories [5]. To get more insight into the complete lattice of all varieties of semirings, all solid and all pre-solid varieties of semirings were determined [1,2]. Now, we are interested in the complete lattice of all regular-solid varieties of semirings by characterizing all regular-solid varieties of idempotent semirings. To achieve our aim, we recall some basic concepts.
Let F and G be the both binary operation symbols and let W (2,2) (X 2 ) be the set of all binary terms of type (2,2) built up by variables from the alphabet X 2 = {x, y}. Hypersubstitutions of type (2, 2) are mappings σ : {F, G} → W (2,2) (X 2 ).

H. Hounnon
The set of all hypersubstitutions of type (2, 2) will be denoted by Hyp. A hypersubstitution σ ∈ Hyp can be extended on the set W (2,2) (X) of all terms of type (2,2), where X is an arbitrary countably infinite alphabet of variables, by the following steps: where σ(f ) can be interpreted as the term operation σ(f ) F (2,2) (X) induced by the term σ(f ) on the free algebra F (2,2) (X) := (W (2,2) (X); (F , G)) with f : It is easy to prove that the algebra (Hyp; • h , σ id ), is a monoid with • h (where σ 1 • h σ 2 :=σ 1 • σ 2 and • is the usual mapping composition) as binary operation and σ id , defined by σ id (f ) := f (x, y) for all f ∈ {F, G}, as an identity element. Hypersubstitutions can be applied to algebras as follows: given an algebra A = (A; (F A , G A )) of type (2, 2) and a hypersubstitution σ ∈ Hyp, one defines the algebra σ(A) := (A; (σ(F) A , σ(G) A )). This algebra of type (2, 2) is called the derived algebra by A and σ.
The hypersubstitution σ ∈ Hyp such that σ(F ) = t and σ(G) = s will be denoted by σ t,s . For all variables u and v, the term F (u, v) and G(u, v) will be denoted by u + v and uv, respectively.
A hypersubstitution σ ∈ Hyp is called a regular hypersubstitution if σ maps both F and G to binary terms containing both variables x and y. It is easy to verify that the set Reg of all regular hypersubstitutions of type (2, 2) forms a submonoid of the monoid Hyp. An identity s ≈ t in a variety V of semirings is called a regular hyperidentity if for every σ ∈ Reg, the equationσ[s] ≈σ[t] belongs to the set IdV of all identities satisfied in V . A variety V of semirings is called regular-solid if all identities in V are satisfied as regular hyperidentities. For more information about hypersubstitutions and varieties of algebras see in [3,7].
In the next section, we will provide some necessary conditions for a variety of semirings to be a regular-solid one. This leads to a description of the lattice of all regular-solid varieties of semirings. The last section will be devoted to the determination of the lattice of all regular-solid varieties of idempotent semirings. An equation s ≈ t is called normal if either both terms s and t are equal to the same variable or none of them is a variable, that is, if s = t or the complexity (number of occurrences of operation symbols) of both terms s and t is greater or equal to 1. A variety in which all identities are normal is called a normal variety. Now, we can derive some necessary conditions for varieties of semirings to be regular-solid. Proposition 1. Let V be a regular-solid variety of semirings. The following properties are:

Some Properties
(1) V is medial, distributive and satisfies the identities: (2) V is either idempotent or normal.
Proof. (1) It is clear that the usual distributive laws are satisfied in V . The application of the regualar hypersubstitutions σ xy,x+y to them gives the other distributive laws since V is a regular-solid variety of semirings. Moreover, applying the regular hypersubstitutions σ xy,xy and σ yx,yx to the distributive law of V , we get the identities xyz ≈ xyxz and zyx ≈ zxyx, respectively, in V.
It is folklore that the identities xyz ≈ xyxz ≈ xzyz imply the medial law xyzu ≈ xzyu and the identities xyz ≈ x 2 yz ≈ xy 2 z ≈ xyz 2 . The application of the regular hypersubstitution σ xy,x+y to these identities gives the remaining identities.
(2) Suppose that t ≈ x is an identity in V which is not normal. This provides x k ≈ x ∈ IdV for some k ≥ 2 (by using the regular hypersubstitution σ xy,xy and identifying all variables with x). From the identity x 2 yz ≈ xyz ∈ IdV , we get x 4 ≈ x 3 ∈ IdV and together with x k ≈ x ∈ IdV , we obtain the idempotent law x 2 ≈ x ∈ IdV . Therefore, V is idempotent by using the regular hypersubstitution σ xy,x+y . Proposition 1 (2), leads to a description of the complete lattice Reg(Sr) of all regular-solid varieties of semirings. Denoting by L(2, 2) the lattice of all varieties of type (2, 2), we have: Proof. The lattice L N (2, 2) of all normal varieties of type (2, 2) and the lattice L Idem (2, 2) of all idempotent varieties of type (2,2) are complete sublattices of L(2, 2) (see [4,7]). Therefore, since Reg N (Sr) = Reg(Sr) ∩ L N (2, 2) (the intersection of two complete sublattices) and since Reg Idem (Sr) = Reg(Sr) ∩ L Idem (2, 2) (the intersection of two complete sublattices), it arises that both lattices Reg Idem (Sr) and Reg N (Sr) are complete sublattices. By Proposition 1 (2) the lattices Reg Idem (Sr) and Reg N (Sr) are disjoint and their union is Reg(Sr).

All Regular-Solid Varieties of Idempotent Semirings
In this section, the lattice of all regular-solid varieties of idempotent semirings will be determined. An equation s ≈ t is outermost if the terms s and t start with the same variable (we write lef tmost(s) = lef tmost(t)) and end also with the same variable (we write rightmost(s) = rightmost(t)). A variety V is called outermost if all equations in IdV are outermost. A variety V of semirings is commutative if x + y ≈ y + x ∈ IdV and xy ≈ yx ∈ IdV . The following result gives a description of idempotent regular-solid varieties of semirings.
Proposition 3. Each idempotent regular-solid variety of semirings is either outermost or commutative.
Proof. Let V be an idempotent regular-solid variety of semirings. Assume that V is not outermost. We will show that V is commutative. Since V is not outermost, without loss of generality, we can assume that there exists an equation s ≈ t in IdV such that lef tmost(s) = x = y = lef tmost(t). Applying the regular hypersubstitution σ xy,xy to the identity s ≈ t ∈ IdV , we get the following identity s 1 ≈ t 1 in V (where lef tmost(s 1 ) = x = y = lef tmost(t 1 )). Let us consider the function h : X → W (2,2) (X), w → x if w = x y otherwise. It is well known that this function can be uniquely extended to an endomorphism h on F (∈,∈) (X ). Then, h(s 1 ) ≈ h(t 1 ) ∈ IdV and h(s 1 )yx ≈ h(t 1 )yx ∈ IdV , so xyx ≈ yx ∈ IdV because of the idempotent law. Applying the regular hypersubstitution σ yx,yx to the latter identity, the following equations xy ≈ xyx ≈ yx hold in V as identities. The application of σ xy,x+y to xy ≈ yx shows that V is commutative. Now, we determine the commutative part of Reg Idem (Sr). Proposition 1 (1) shows that every regular-solid variety of idempotent semirings is a subvariety of the variety V M ID of all medial idempotent and distributive semirings. But the subvariety lattice of V M ID is fully described by Pastjin in [6] as follows: Let us consider the two-element algebras (using the same notations as in [6]): The algebra J generates the variety DL of all distributive lattices and L generates the variety SL of bi-semilattices. Then we have Lemma 4 [6]. The subvariety lattice of the variety V M ID of all medial idempotent and distributive semirings is a Boolean lattice with 10 atoms and 10 dual atoms, i.e., with 2 10 elements. The atoms are exactly the varieties Proof. Let V be a regular-solid variety of commutative and idempotent semirings. By Proposition 1 (1), the variety V is a commutative subvariety of V M ID . So V is either trivial or a join of some commutative atoms listed in Lemma 4. This means that either V is trivial or V ∈ {SL, DL, SL ∨ DL}. But the varieties DL and SL ∨ DL are not regular-solid. Indeed, the application of σ x+xy,x+xy to the commutative identity xy ≈ yx gives the identity x+xy ≈ y +yx which cannot be satisfied in DL because of the absorption laws. IdSL is the set of all regular identities of type (2,2). It is clear that applying regular hypersubstitution to any regular identity, one gets a regular identity. So SL is regular-solid.
We are now interested in the outermost part of Reg Idem (Sr). Some definitions and facts will be referred.

Definition.
A variety V of semirings is s-outermost if for any identity s ≈ t ∈ IdV , the equations s ≈ t as well asσ x+y,yx [s] ≈σ x+y,yx [t] are outermost.
This definition coincides with that one given in [1] and it is clear that every outermost regular-solid variety of semirings is s-outermost since the hypersubstitution σ x+y,yx is regular.
A variety V of semirings is said to be a solid variety if for all s ≈ t ∈ IdV and for all σ ∈ Hyp, we getσ[s] ≈σ[t] ∈ IdV . It is well known that the variety RA (2,2) generated by all projection algebras of type (2, 2) is a variety of semirings and it is defined by RA (2,2) [1]. It is already proved: Lemma 6 [1]. The lattice of all solid varieties of semirings is the four-element chain represented by T ⊂ RA (2,2)

Now, we can prove:
Lemma 8. Let V be an outermost regular-solid variety of idempotent semirings. If V is different from RA (2,2) then V is regular i.e all equations in IdV are regular.
Proof. We will prove that if V is not regular then V = RA (2,2) . Since V is outermost regular-solid variety of semirings, V is s-outermost and we have RA (2,2) ⊆ V (Lemma 7). It left to prove that V ⊆ RA (2,2) i.e Id(RA (2,2) ) ⊆ IdV . Since V is not regular, there exists an identity s ≈ t in IdV such that, without loss of generality, a variable x i occurs in s but not in t. Applying σ xy,xy to s ≈ t and identifying all variables different from x i with x, we get xx i x ≈ x ∈ IdV because V is outermost and idempotent. Therefore, xyz ≈ xz ∈ IdV . The application of σ xy,x+y to this identity gives x + y + z ≈ x + z ∈ IdV . Moreover, using the previous identity, the distributivity and the idempotency, the basis identities of RA (2,2) are also identities in V . This finishes the proof of Id(RA (2,2) ) ⊆ IdV . Now, we have all tools to prove our main result: Theorem 9. The lattice of all regular-solid varieties of idempotent semirings is the lattice Proof. Let V be a regular-solid variety of idempotent semirings. Then V is either commutative or outermost (Proposition 3). If V is commutative then V ∈ {T , SL} (Theorem 5). Otherwise, V is outermost. Then V = RA (2,2) or V is regular (Lemma 8). Therefore, V = RA (2,2) or SL ⊆ V since Id(SL) is the set of all regular identities of type (2, 2). Moreover, V is s-outermost and thus RA (2,2) ⊆ V (Lemma 7). Altogether, we have V = RA (2,2) or RA (2,2) ∨ SL ⊆ V .