A Refinement of the Equaclosure Operator

In Part I of [3], Adaricheva and Nation introduced a new property that holds (dually) for the natural equaclosure operator on congruence lattices of semilattices with operators, and hence holds in lattices of quasivarieties Lq(K). The new property implied some previously known conditions as well. Using the duality for congruences of semilattices with operators from [7], we will refine that to a version that is both stronger and simpler. Wieslaw Dziobiak [5] observed that there is a natural closure operator Γ on the lattice of subquasivarieties Lq(K) of a quasivariety K, viz., for Q ≤ K let Γ(Q) = K ∩ HSP(Q). Moreover, there is a least subquasivariety L = T(Q) with Γ(L) = Γ(Q), which is the quasivariety generated by the Q-free algebra FQ(ω). The map Γ is called the natural equaclosure operator on Lq(K). Every subquasivariety lattice Lq(K) is can be represented as the lattice Sp(S, H) of all H-closed algebraic subsets of an algebraic lattice S, where H is a monoid of operators that preserve arbitrary meets and nonempty directed joins (combining [3] and [7]). The corresponding natural operator for a lattice of algebraic subsets Sp(S, H) can be described easily. Each H-closed algebraic subset A has a least element a0 ∈ S. Then Γ(A) =↑a0 is the filter generated by a0, and T(A) is the H-closed algebraic subset generated by a0. Adaricheva and Gorbunov [2] defined an equaclosure operator abstractly to have those properties that we know to hold for the natural equaclosure operator on the lattice of subquasivarieties of a quasivariety. Not every lattice supports an equaclosure operator (e.g., M3), and its existence has structural consequences for those that do. The modern version reads as follows. An equaclosure operator on a dually algebraic lattice L is a map γ : L→ L satisfying the following properties. (I1) x ≤ γ(x). (I2) x ≤ y implies γ(x) ≤ γ(y). (I3) γ(x) = γ(x).