The Lattice of Subvarieties of Semilattice Ordered Algebras

This paper is devoted to the semilattice ordered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{V}$\end{document}-algebras of the form (A, Ω, + ), where + is a join-semilattice operation and (A, Ω) is an algebra from some given variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{V}$\end{document}. We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{V}$\end{document}-algebras in relation to the lattice of subvarieties of the variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{V}$\end{document}.


Introduction
Ordered algebraic structures, particularly ordered fields, ordered vector spaces, ordered groups and semigroups, have a well-established tradition in mathematics. It is due to their internal interest and due to their applications in other areas.
In 1979 McKenzie and Romanowska [10] showed that there are exactly five varieties of dissemilattices-semilattice ordered semilattices. Apart from the variety of all dissemilattices and the trivial variety, there are the variety of distributive lattices, the variety of "stammered" semilattices (where both basic semilattice operations are equal) and the variety of distributive dissemilattices.
In 2005 Ghosh et al. [5] described the lattice of all subvarieties of the variety generated by all ordered bands (semirings whose multiplicative reduct is an idempotent semigroup and additive reduct is a chain). They showed that the lattice is distributive and contains precisely 78 varieties. Each of them is finitely based and generated by a finite number of finite ordered bands.
In the same year, Kuřil and Polák [9] introduced the certain closure operators on relatively free semigroup reducts and applied them to describe the lattice of subvarieties of the variety of all semilattice ordered semigroups.
But, in general, very little is known about varieties of modals. In 1995 Kearnes [8] proved that to each variety V of entropic modals one can associate a commutative semiring R(V), whose structure determines many of the properties of the variety. In particular,

Theorem 1.2 ([8]) The lattice of subvarieties of the variety V of entropic modals is dually isomorphic to the congruence lattice ConR(V) of the semiring R(V).
Using Theorem 1.2,Ślusarska [17] described the lattice of subvarieties of entropic modals (D, ·, +) whose groupoid reducts (D, ·) satisfy the additional identity: Just recently (see [13]) we described some family of fully invariant congruences on the free semilattice ordered idempotent and entropic algebras which can partially result in describing subvariety lattices of modals.
The main aim of this paper is to describe a structure of the subvariety lattice of all semilattice ordered -algebras of a given type and study how it is related to the subvariety lattice of all -algebras. Kuřil and Polák introduced in [9] the notion of an admissible closure operator and applied it to describe the subvariety lattice of the variety of semilattice ordered semigroups. Description proposed by them is complicated and strictly depends on properties of semigroups. Applying the techniques from power algebras theory we simplify their description. Moreover, we generalize their results and obtain another type of description of the subvariety lattice of all semilattice ordered algebras.
We start with the following easy observation.
There are two other basic properties worth mentioning.

Lemma 1.4
Let ( A, , +) be a semilattice ordered -algebra, and x ij ∈ A for 1 ≤ i ≤ n, 1 ≤ j ≤ r. Then for each 0 = n-ary operation ω ∈ we have Lemma 1.5 Let ( A, , +) be a semilattice ordered -algebra and let ω ∈ be an 0 = n-ary operation. The algebra ( A, , +) satisf ies for any x ∈ A the condition

4)
if and only if for any x 1 , . . . , x n ∈ A the following holds In particular, if a semilattice ordered -algebra ( A, , +) is idempotent then Eq. 1.5 holds.
It is easy to see that in general both Eqs. 1.2 and 1.3 hold also for term operations. The proofs go just by induction on the complexity of terms. Also, in such a case, for any term operation t, we obtain the inequality (1.6) that generalizes the distributive law Eq. 1.1. The paper is organized as follows. In Section 2 examples of semilattice ordered V-algebras are provided, for some given varieties V. Among the others, we mention semilattice ordered n-semigroups, extended power algebras and modals. In Section 3 we describe the free semilattice ordered algebras (Theorem 3.4) and next we apply the result to describe the identities which are satisfied in varieties of semilattice ordered algebras (Corollary 3.10). Section 4 is devoted to present the "main knots" in the subvariety lattice of semilattice ordered algebras (Theorem 4.7). In Section 5 we characterize so called V-preserved subvarieties of the variety of all semilattice ordered algebras. In Theorem 5.9 we show that there is a correspondence between the set of all V-preserved subvarieties and the set of some fully invariant congruence relations on the extended power algebra of the V-free algebra. Theorem 5.12 contains the main result of this paper, i.e. the full description of the lattice of all subvarieties of the variety of all semilattice ordered algebras. The last Section 6 summarizes all results in the convenient form of the algorithm.
The notation t(x 1 , . . . , x n ) means the term t of the language of a variety V contains no other variables than x 1 , . . . , x n (but not necessarily all of them). All operations considered in the paper are supposed to be finitary. We are interested here only in varieties of algebras, so the notation W ⊆ V means that W is a subvariety of a variety V.

Examples
In this section we discuss some basic and natural examples of semilattice ordered algebras.
Example 2.1 Semilattice ordered semigroups. An algebra ( A, ·, +), where ( A, ·) is a semigroup, ( A, +) is a semilattice and for any a, b , c ∈ A, a · (b + c) = a · b + a · c and (a + b ) · c = a · c + b · c is a semilattice ordered semigroup. In particular, semirings with an idempotent additive reduct [11,18], distributive bisemilattices [10] and distributive lattices are semilattice ordered SG-algebras, where SG denotes the variety of all semigroups.
An algebra ( A, f, +), where ( A, f ) is an n-semigroup, ( A, +) is a semilattice and the Eq. 1.1 are satisfied for the operation f , is a semilattice ordered n-semigroup.
Let SG n be the variety of all n-semigroups, ( A, +) be a semilattice and End( A, +) be the set of all endomorphisms of ( A, +). For each 2 ≤ n ∈ N, we define n-ary composition Then the algebra (End( A, +), is a semilattice ordered n-semigroup (semilattice ordered SG n -algebra). The algebra (End( A, +), {ω n : n ∈ N}, ∨) is also an example of a semilattice ordered algebra.
Another example of a semilattice ordered n-semigroup can be obtained as follows. Let A be a set and Rel( A) be the set of all binary relations on A. For each 2 ≤ n ∈ N, we define n-ary relational product Then the algebra (Rel( A), ω n , ∪), where ∪ is a set union, is a semilattice ordered SG n -algebra and the algebra (Rel( A), {ω n : n ∈ N}, ∪) is a semilattice ordered algebra.

Example 2.3 Extended power algebras.
For a given set A denote by P >0 A the family of all non-empty subsets of A. For any n-ary operation ω : A n → A we define the complex operation ω : (P >0 A) n → P >0 A in the following way: where ∅ = A 1 , . . . , A n ⊆ A. The set P >0 A also carries a join semilattice structure under the set-theoretical union ∪. Jónsson and Tarski proved in [7] that complex operations distribute over the union ∪. Hence, for any algebra ( A, ) ∈ , the extended power algebra (P >0 A, , ∪) is a semilattice ordered -algebra. The algebra (P <ω >0 A, , ∪) of all finite non-empty subsets of A is a subalgebra of (P >0 A, , ∪).

Example 2.4
Modals. An idempotent (in the sense that each singleton is a subalgebra) and entropic algebra (any two of its operations commute) is called a mode.
Each modal is in fact a semilattice ordered M-algebra. Modes and modals were introduced and investigated in detail by Romanowska and Smith [15,16].
If a modal (M, , +) is entropic, then (M, , +) is a mode and it is an example of a semilattice mode. Semilattice modes were described by Kearnes in [8].

Free Semilattice Ordered Algebras and Identities
Results of Kuřil and Polák [9] show that the problem of the characterization of semilattice ordered algebras requires the knowledge of the structure of the power algebra of a given algebra. In this section we will describe the free semilattice ordered algebras using the concept of extended power algebras. Next we will apply the result to describe the identities which are satisfied in varieties of semilattice ordered algebras.
Let be the variety of all algebras of finitary type τ : → N and let V be a subvariety of . Let (F V (X), ) be the free algebra over a set X in the variety V and let S V denote the variety of all semilattice ordered V-algebras.

Theorem 3.1 (Universality Property for Semilattice Ordered Algebras) Let X be an arbitrary set and (
So any mapping h : X → A may be uniquely extended to an -homomorphism h : Further, -homomorphism h can be extended to a unique { , ∪}-homomorphism where T is a non-empty finite subset of F V (X).
To show that h is an -homomorphism, consider an n-ary operation ω ∈ and non-empty finite subsets T 1 , . . . , T n ⊆ F V (X). Then Moreover, h is a semilattice homomorphism because The uniqueness of h is obvious.
By Theorem 3.1, for an arbitrary variety V ⊆ , the algebra (P <ω >0 F V (X), , ∪) has the universality property for semilattice ordered algebras in S V , but in general, the algebra itself doesn't have to belong to the variety S V .
Example 3.2 Let V be a variety of idempotent groupoids satisfying the identities: Consider the free groupoid (F V (X), ·) over a set X in the variety V and its two generators x, y ∈ X. One can easily see that This shows that the algebra (P <ω >0 F V (X), ·, ∪) does not belong to the variety S V .
Note that the semilattice ordered algebra (P <ω then it is, up to isomorphism, the unique algebra in S V generated by a set X, with the universal mapping property.
With each subvariety S ⊆ S of semilattice ordered -algebras we can associate a least subvariety V of with the property S ⊆ S V . But, for two different subvarieties V and W of , the varieties S V and S W can be equal.
Example 3.5 A dif ferential groupoid is a mode groupoid (D, ·) satisfying the additional identity: Each proper non-trivial subvariety of the variety D of differential groupoids (see [14]) is relatively based by a unique identity of the form for some i ∈ N and positive integer j. Denote such a variety by D i,i+ j . Obviously, the variety D 0,1 is exactly the variety LZ of left-zero semigroups (groupoids ( A, ·) such that a · b = a for all a, b ∈ A). Let S D denote the variety of all dif ferential modals (modals whose mode reduct is a differential groupoid) and let S D0, j ⊆ S D be the variety of semilattice ordered D 0, jgroupoids. We showed in [12] that for each positive integer j, one has S D0, j = S LZ .
Example above arises the question, for which different subvarieties V 1 = V 2 ⊆ , the varieties S V1 and S V2 are different, too.
To answer this, for an arbitrary binary relation on the set P <ω Example 3.6 It is easy to see that for the least equivalence relation id P <ω >0 FV (X) and the greatest equivalence relation T on the set P <ω Clearly, if is an equivalence relation, then is an equivalence relation, too. Additionally, if we consider the algebra (F V (X), ) and is a congruence on (P <ω >0 F V (X), , ∪), then also is a congruence relation on (F V (X), ).

Then the relation is a fully invariant congruence on
Obviously, α + is also a homomorphism with respect to ∪. If α is an endomorphism of the algebra (F V (X), ), then α + is an endomorphism of the algebra Since, by assumption, is a fully invariant congruence on (P <ω Let Con f i (F V (X)) be the set of all fully invariant congruences of the algebra (F V (X), ) and denote the set of all fully invariant congruences of the algebra . From now through all over the paper, we assume X is an infinite set. By Lemma 3.7, each fully invariant congruence relation on (P <ω >0 F (X), , ∪) determines a subvariety of : Of course, it may happen that for 1 = 2 , the varieties 1 and 2 are equal.
Example 3.8 By results of McKenzie and Romanowska [10] we have that there are at least five subvarieties of the variety of semilattice ordered groupoids which are also semilattice ordered semilattices. But the variety of semilattices has only two subvarieties.
We will show in Theorem 3.14 that for 1 , 2 ∈ Con f i (P <ω >0 F (X)), different congruences 1 = 2 ∈ Con f i (F (X)) always determine different subvarieties of the variety S . But first we prove some auxiliary results.

"Main Knots" in the Lattice of Subvarieties
In previous section we have shown that for different congruences 1 = 2 ∈ Con f i (F (X)), with 1 , 2 ∈ Con f i (P <ω >0 F (X)), the subvarieties S 1 and S 2 are always different. To describe all fully invariant congruences of the algebra (P <ω >0 F (X), , ∪) which uniquely determine subvarieties of the form S V , for some V ⊆ , we have to introduce the next binary relation.
Let us define a binary relation on the set Con f i (P <ω >0 F (X)) in the following way: for 1 , 2 ∈ Con f i (P <ω >0 F (X)) Obviously, is an equivalence relation.

Lemma 4.3 Let I be a set and for each i ∈ I, let i be an equivalence relation on
Let ∈ Con f i (P <ω >0 F (X)), I be a set and for each i ∈ I, let i ∈ / . Then by Lemma 4.3, also i∈I i ∈ / . This shows that in each class of the relation there is the least element with respect to the set inclusion. Let be the set of all such the least congruences in each -class. Directly by Theorem 4.1 we obtain Corollary 4.4 Let ∈ Con f i (P <ω >0 F (X)). Then S = HSP((P <ω >0 F (X)/ , , ∪)).
We can say that subvarieties of the form S , ∈ Con f i (P <ω >0 F (X)), are some kinds of "main knots" in the lattice of subvarieties of semilattice ordered algebras.

Lemma 4.6
The ordered set (Con f i (P <ω >0 F (X)), ⊆) is a complete lattice. Let I be a set and for each i ∈ I, let i ∈ Con f i (P <ω >0 F (X)). Then, the relation is the least upper bound of { i } i∈I , and the congruence Let us introduce the following notation: Clearly, the set L ( ) is partially ordered by the set inclusion. Moreover, for Directly from Lemma 4.6 we obtain the following has not to be equal to the variety 1 ∩ 2 . By Theorem 3.14 we immediately obtain is their greatest lower bound (see Fig. 1). for ∈ Con f i (P <ω >0 F (X)).

Lemma 5.1
Let ∈ Con f i (P <ω >0 F (X)), I be a set and for each i ∈ I, i ∈ / . Then ( / , ⊆) is a complete meet-semilattice with the relation i∈I i as the greatest lower bound of { i } i∈I .
Let ∈ / . Note that by Corollary 3.10 the algebras (P <ω >0 F (X)/ , ) and (F (X)/ , ) satisfy exactly the same identities. This shows that the variety is a subvariety of S and is not included in S W for any proper subvariety W ⊂ . This justifies the introduction of the following definition. Proof By Lemma 3.12, for each subvariety S = HSP((P <ω >0 F (X)/ , , ∪)) ⊆ S , one has ⊆ . Now let S be -preserved subvariety of S . By definition, S is not included in any variety S W for a proper subvariety W ⊂ . This means that the algebra (P <ω >0 F (X)/ , ) does not belong to any proper subvariety of . Then, by Corollary 3.10, = .
Let ∈ Con f i (P <ω >0 F (X)). By Lemma 5.3 with each congruence ∈ / one can associate the -preserved subvariety HSP((P <ω >0 F (X)/ , , ∪)) of S . This mapping is the restriction of the dual isomorphism we have between lattice (Con f i (P <ω >0 F (X)), ⊆) and the lattice of all subvarieties of S , to the set / . By Lemmas 4.3, 5.1 and 5.3 we immediately have Theorem 5.4 Let ∈ Con f i (P <ω >0 F (X)). The semilattice of all -preserved subvarieties of S is dually isomorphic to the complete semilattice ( / , ⊆). For any -preserved subvarieties {S i } i∈I of S , the variety i∈I S i is their least upper bound.
In Theorem 5.9 we will show that there is also a correspondence between the set of -preserved subvarieties of S and the set of some fully invariant congruence relations on the algebra (P <ω >0 (F (X)/ ), , ∪). First we will prove some auxiliary technical results. We introduce the following notation. For a set Q ∈ P <ω >0 F (X) and a congruence ∈ Con f i (P <ω >0 F (X)), Now, we define a relation δ ⊆ P <ω >0 (F (X)/ ) × P <ω >0 (F (X)/ ) in the following way:
By Corollary 5.8 and Theorem 5.4 we obtain Theorem 5.9 Let ∈ Con f i (P <ω >0 F (X)). The complete semilattice of all preserved subvarieties of S is dually isomorphic to the complete semilattice (Con id f i (P <ω >0 F (X)), ⊆).
Before we formulate Theorem 5.12, the main result of the paper, let us summarize what we have already known. For each ∈ Con f i (P <ω >0 F (X)) such that = , the subvariety S = HSP((P <ω >0 F (X)/ , , ∪)) of S is an -preserved subvariety of S ⊆ S according to Lemma 5.3. Therefore, to find any subvariety S of S we should proceed as follows: first find the proper "main knot" and then choose one of its subvarieties.
By Theorem 3.14, Lemma 5.3, and Theorem 5.9 one can uniquely associate with the variety S two congruence relations: ∈ Con f i (P <ω >0 F (X)) such that = (for the "main knot" S ), and then δ ∈ Con id f i (P <ω >0 F (X)) (for the chosen preserved subvariety).

Consider the set
Con id f i (P <ω >0 F (X)).
To stress the fact that the congruence α depends on we denote it by α . Now, define on the set Con id f i ( ) a binary relation in the following way: for α ∈ Con id f i (P <ω >0 F (X)) and β ∈ Con id f i (P <ω >0 F (X)), with , ∈ Con f i (P <ω >0 F (X)) α β ⇔ ⊆ and ∀(a 1 , . . . , a k , b 1 , . . . , b m ∈ F (X)) Clearly, the relation is a partial order.

Theorem 5.11
The ordered set (Con id f i ( ), ) is a complete lattice. Let I be a set and for each i ∈ I, α i ∈ Con id f i (P <ω >0 F i (X)). The binary relations are, respectively, the greatest lower bound and the least upper bound of {α i } i∈I with respect to .
Proof Let , ∈ Con f i (P <ω >0 F (X)). By Lemma 5.6, for any α ∈ Con id f i ( ) we have α = . Then by Lemma 4.3, one obtains i∈I Hence, by Lemmas 5.5 and 5.6, the congruences δ i∈I α i and δ i∈I α i belong to the set Con id f i ( ).
Obviously, for each i ∈ I, i∈I i ⊆ i . Moreover, for t 1 , . . . , t k , u 1 , . . . , u m ∈ F (X) This implies that for each i ∈ I, Now let γ ∈ Con id f i (P <ω >0 F (X)) and for each i ∈ I, γ α i . Then, for each i ∈ I, ⊆ i , and consequently, ⊆ i∈I i . Further, for any t 1 , . . . , t k , u 1 , . . . , u m ∈ F (X) This shows the relation δ i∈I α i is the greatest lower bound of {α i } i∈I with respect to . Now note that for each i ∈ I, and for t 1 , . . . , t k , u 1 , . . . , u m ∈ F (X) we have Thus, for each i ∈ I, α i δ i∈I α i .
Finally, let γ ∈ Con id f i (P <ω >0 F (X)) and for each i ∈ I, α i γ . Then, for each i ∈ I, i ⊆ and for t 1 , . . . , t k , u 1 , . . . , u m ∈ F (X) Then, for each i ∈ I, α i ⊆ γ . Hence Moreover, by Lemma 5.7 we have which means the relation δ i∈I α i is the least upper bound of {α i } i∈I , and completes the proof. By Theorem 5.11 we obtain a full description of the lattice of all subvarieties of the variety S .
Let L(S ) denote the set of all subvarieties of the variety S . As we have already noticed each subvariety S of S may be uniquely described by two congruences: ∈ Con f i (P <ω >0 F (X)) and α ∈ Con id f i (P <ω >0 F (X)). Hence, we can denote each subvariety in the set L(S ) by S α . Thus, one has L(S ) = S α | ∈ Con f i (P <ω >0 F (X)), α ∈ Con id f i (P <ω >0 F (X)) .
Theorem 5.12 allows to describe the subvariety lattice L(S ) without knowledge of the set Con f i (P <ω >0 F (X)). But, if we know the latter, each congruence α ∈ Con id f i (P <ω >0 F (X)) may be replaced by the congruence δ , for ∈ Con f i (P <ω >0 F (X)) such that = . Then, Theorem 5.12 may be considerably simplified.

Corollary 5.13
For any S δ , S δ ∈ L(S ) we have:

The Lattice LºS »-Practical Computations
Now, for a subvariety V ⊆ , we will present how to find the lattice (L(S V ), ⊆) of all subvarieties of the variety S V , knowing the lattice (L(V), ⊆). Of course, any lattice (L(S V ), ⊆) is a sublattice of the lattice (L(S ), ⊆).
On the other hand, Example 3.5 shows that not for each subvariety V ⊆ , the variety S V is uniquely defined. Let us consider two sets By Lemma 5.6, if Con V = ∅ then S V is not equal to S W for any proper subvariety W ⊂ V. Therefore, to find the lattice (L(S V ), ⊆), Theorems 5.9, 5.11 and 5.12 lead us to the procedure described on Fig. 3.
This confirmed previous results of McKenzie and Romanowska (see [10]) that there are exactly five subvarieties of the variety S SL .