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Properties of Lattices of Subquasivarieties

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Properties of Lattices of Subquasivarieties APPENDIX A Properties of Lattices of Subquasivarieties A.1. Representations of Quasivariety Lattices The span of nearly two decades since the publication of Viktor Gor- bunov’s book on quasivarieties [77] has made it desirable to have an updated review of what is known about lattices of quasivarieties. This appendix pro- vides a summary of their basic properties, with references to the literature. A.2. Representations of Quasivariety Lattices Representations of a lattice can reveal structural properties, so let us be- gin there. For a quasivariety K,letLq(K) denote the lattice of quasivarieties contained in K, ordered by inclusion. No assumption is made yet on the type of the structures in K, but the type will generally have both operations and relations. Of course Lq(K) is a complete lattice. The theory of a quasivariety K is the set of all quasi-equations holding in K. The theory of a subquasivariety Q ≤ K contains the theory of K. (More quasi-equations means fewer models.) The lattice QTh(K) of all quasi- equational theories extending the theory of K, ordered by containment, is dually isomorphic to the lattice of subquasivarieties Lq(K). In Adaricheva and Nation [19], it is shown that QTh(K)canberepre- sented as the congruence lattice of a semilattice with operators. That is, ∼ there is an algebra S such that QTh(K) = Con S,whereS =(S, ∨, 0,F)isa join semilattice with 0 endowed with a set of operations f : S → S such that f(x ∨ y)=f(x) ∨ f(y)andf(0) = 0 for all f ∈ F . In this representation, S is the semilattice of compact K-congruences of the free structure FK(ω), and the operators are derived from endomorphisms of the free structure. In turn, the congruence lattice of a semilattice with operators is du- ally isomorphic to the lattice Sp(L,H)ofallH-closed algebraic subsets of an algebraic lattice L,whereH is a monoid of operators preserving arbi- trary meets and nonempty directed joins (Hyndman, Nation, and Nishida [87]). An algebraic subset of a complete lattice is one closed under arbi- trary meets and nonempty directed joins. So in the case when L is finite, © Springer International Publishing AG, part of Springer Nature 2018 135 J. Hyndman, J. B. Nation, The Lattice of Subquasivarieties of a Locally Finite Quasivariety, CMS Books in Mathematics, https://doi.org/10.1007/978-3-319-78235-5 136 A. PROPERTIES OF LATTICES OF SUBQUASIVARIETIES Sp(L,H) is just the subalgebra lattice Sub(L, ∧, 1,H). In the correspon- ∼d dence Con(S, ∨, 0,F) = Sp(L,H), we can take L to be the filter lattice of S, ordered by set containment, with the operators of H induced by those in F . ∼d Thus, we have two pairs of dual isomorphisms: Lq(K) = QTh(K)and ∼d Con(S, ∨, 0,F) = Sp(L,H). Starting with a quasivariety K, we can find a ∼ ∼ semilattice with operators such that QTh(K) = Con S, and hence Lq(K) = Sp(L,H). Most (but not all) properties of subquasivariety lattices can be derived from this latter representation. The reverse inclusion is not true: there is a lattice Sp(L,H)thatisnot isomorphic to any subquasivariety lattice Lq(K); see the last paragraph of Section A.3. Under certain additional restrictions on the pair (L,H), for example if 1L is compact and H is a group, one can show that Sp(L,H) can be represented as Lq(K) for some quasivariety K [19]. Moreover, if S ∼ is a lattice such that S = Sp(L,H) for some algebraic lattice L and monoid of operators H, then the linear sum 1 + S is isomorphic to Lq(K)fora quasivariety K [17]. Alternatively, if we expand the notion of quasivariety to include impli- cational classes in languages that need not contain equality, with the corre- sponding adjustments, then every lattice Sp(L,H)withL algebraic and H a set of operators is isomorphic to Lq(I) for some implicational class I [132]. Gorbunov and Tumanov [79] represent a quasivariety lattice Lq(K)asthe lattice Sp(L,R) of all algebraic subsets of an algebraic lattice L that are closed with respect to a distributive binary relation R,whereR corresponds to either isomorphism or embedding of quotients of the lattice of K-congruences of the free structure FK(ω). The representation using operators is perhaps more convenient for describing the structure of Lq(K). A.3. Basic Consequences of the Representations The congruence lattice of a semilattice is meet semidistributive and al- gebraic [142]. So the fact that a subquasivariety lattice Lq(K) is dually isomorphic to the congruence lattice of a semilattice with operators imme- diately implies that Lq(K) is dually algebraic and join semidistributive, i.e., satisfies (SD∨) x ∨ y ≈ x ∨ z → x ∨ y ≈ x ∨ (y ∧ z). Indeed, since being dually algebraic implies lower continuity, Lq(K)satisfies the more general join semidistributive law u ≈ x ∨ zi for all i ∈ I implies u ≈ x ∨ zi. i∈I A complete lattice K has the J´onsson-Kiefer Property if every element a ∈ K is the join of elements that are (finitely) join prime in the ideal ↓ a. This property holds in all finite join semidistributive lattices [93]. An easy argument shows that in any complete lattice, the J´onsson-Kiefer Property implies join semidistributivity. On the other hand, there is a dually A.3. BASIC CONSEQUENCES OF THE REPRESENTATIONS 137 algebraic, join semidistributive lattice that has no join prime elements [18]. Thus the J´onsson-Kiefer Property is strictly stronger than join semidistribu- tivity. Gorbunov showed that the property holds in lattices of subquasivarieties [76]. Corollary A.1. For any quasivariety Q, the lattice Lq(Q) has the J´onsson-Kiefer Property. This is a consequence of a slightly more general result. Theorem A.2. If L is an algebraic lattice and H a monoid of operators on L,thenSp(L,H) has the J´onsson-Kiefer Property. We can sketch the proof of Theorem A.2 as follows. Let Sp(L)denotethe lattice of all algebraic subsets of L, with no operators. We use Gorbunov’s lemma that the join of finitely many algebraic subsets in the lattice Sp(L)is given by X1 ∨···∨Xn = {x1 ∧···∧xn : xj ∈ Xj for 1 ≤ j ≤ n}, along with the observation that the join of H-closed algebraic sets is H- closed. Hence if q ∈ K is meet irreducible, then q ∈ X1 ∨···∨Xn implies q ∈ Xj for some j. Therefore the H-closed algebraic subset generated by a meet irreducible element q is join prime in Sp(L,H). But every element of an algebraic lattice is a meet of completely meet irreducible elements. Thus the join of all the join prime algebraic sets in Sp(L,H)isL itself. Finally, an algebraic subset of an algebraic lattice is itself an algebraic lattice. Relativizing the argument to a principal ideal ↓ A in Sp(L,H) gives the statement in the property. For a more complete discussion of the J´onsson-Kiefer property, see [18]. Recall that a lattice is atomic if it has a least element 0 and for every x>0 there exists an atom a such that x ≥ a 0. Theorem A.3. For any quasivariety Q, the lattice Lq(Q) is atomic. Gorbunov’s argument that Lq(K) is atomic lattice [77] uses the special quasivariety x ≈ y.LetT denote the least subquasivariety of K,andlet T < Q ≤ K. If it happens that Q ∩ x ≈ y = T (including the case when T = x ≈ y), then since x ≈ y is finitely based and hence dually compact, there is a quasivariety H such that T ≺ H ≤ Q. On the other hand, if T < Q ∩ x ≈ y,thenQ contains a 1-element structure S in which not all the relations of the language of K hold. Letting T0 denote the 1-element structure in which all the relations hold, we see that {S, T0} is a subquasivariety of Q (it is closed under S, P, U) which is of course an atom of Lq(K). We note that a lattice Sp(L,H)ofH-closed algebraic sets need not be atomic. Here is an example from [19]. Let Ω be ω+1 with the single operator ∼ d h such that h(k)=k+1 for k<ωand h(ω)=ω.ThenSp(Ω, {h}) = (ω+1) , which is not atomic. 138 A. PROPERTIES OF LATTICES OF SUBQUASIVARIETIES A.4. Equaclosure Operators Wieslaw Dziobiak [55] observed that there is a natural closure operator Γ on the lattice of subquasivarieties Lq(K), viz.,forQ ≤ K let Γ(Q)= K ∩ HSP(Q). In the terminology of Section 4.3, Γ(Q) is the least equational subquasivariety of K containing Q. Moreover, there is a least quasivariety L =T(Q)withΓ(L)=Γ(Q), which is T(Q)= FQ(ω). The map Γ is called the natural equaclosure operator on Lq(K), and it has important structural consequences. We now define an equaclosure operator abstractly to have those proper- ties that we know to hold for the natural equaclosure operator on the lattice of subquasivarieties of a quasivariety. These properties are from Adaricheva and Gorbunov [13], with refinements from Adaricheva and Nation [19], Na- tion and Nishida [133], and Adaricheva, Hyndman, Nation, and Nishida [17]. An equaclosure operator on a dually algebraic lattice L is a map γ : L → L satisfying the following properties. (For the map τ of property (I9), determined by γ,seebelow.) (I1) x ≤ γ(x). (I2) x ≤ y implies γ(x) ≤ γ(y).
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