On relative universality and \(Q\)-universality of finitely generated varieties of Heyting algebras. (Q2842082)

A \textit{quasivariety} is a class \(\mathbb Q\) of similar algebras which is closed under the formation of subalgebras, Cartesian products and ultraproducts, or what amounts to the same thing, a class defined as all the models of a set of first order sentences each of which has the form NEWLINE\[NEWLINE\forall x,\;\sigma_1(x)=\tau_1(x)\wedge\ldots\wedge \sigma_n(x)=\tau_n(x)\to\sigma(x)=\tau(x)NEWLINE\]NEWLINE where the \(\sigma_i\), \(\tau_i\), \(\sigma\) and \(\tau\) are terms.NEWLINENEWLINE When \(\mathbb Q\) is a quasivariety, the lattice whose members are all the quasivarieties \(\mathbb Q'\subseteq\mathbb Q\), and whose ordering is set-inclusion, is denoted \(\mathrm{Lat}(\mathbb Q)\) and called the lattice of subquasivarieties of \(\mathbb Q\).NEWLINENEWLINE A lattice \(L\) is called a factor of a lattice \(L'\) if \(L\) is a homomorphic image of a sublattice of \(L'\). A quasivariety \(\mathbb Q\) is called \(Q\)-universal if every lattice \(\mathrm{Lat}(\mathbb V)\), where \(\mathbb V\) is a quasivariety of algebras of finite type, is a factor of \(\mathrm{Lat}(\mathbb Q)\) [\textit{M. V. Sapir}, Algebra Univers. 21, 172-180 (1985; Zbl 0599.08014)].NEWLINENEWLINE The principal aim of this paper is to prove: Theorem 1.3. For each \(i=0,1,\ldots,10\) the variety of Heyting algebras generated by \(DQ_i\) is var-relatively alg-universal modulo \(C_2\) or \(C_2^0\) and contains an \(A\)-\(D\) family, thus it is \(Q\)-universal. The posets \(Q_i\) are shown in Figure 1. For \(i=0,1,2\) the variety of Heyting algebras generated by \(\{DF_i,DG_i\}\) is var-relatively ff-alg-universal and contains an \(A\)-\(D\) family, so that it is \(Q\)-universal. The posets \(F_i\) and \(G_i\) with \(i=0,1,2\) are shown in Figure 2.NEWLINENEWLINE By \(DQ_i\), for each finite poset \(Q_i\), \(i=0,1,\ldots,10\), the authors mean the Heyting algebra whose Priestley dual is \(Q_i\), and similarly for the finite posets \(F_i\) and \(G_i\), \(i=0,1,2\), all illustrated on page 66. The two element cyclic group is denoted as \(C_2\), which makes \(C_2^0\) the trivial group. The notions of var-relatively alg-universal modulo a monoid, var-relatively ff-alg-universal, as well as the notion of \(A\)-\(D\) family, are explained at the introduction of the paper. References and explanations on why these conditions are sufficient to get \(Q\)-universality are also given at the introduction.NEWLINENEWLINE The authors use Priestley duality and the category of functors from a poset into compact totally disconnected spaces. The Priestley duality for Heyting algebras is recalled in section 2. The third section is devoted to a construction of relatively full embeddings from a suitable subcategory of \(\mathbb C^P\) (the functor category from a poset \(P\) into the category \(\mathbb C\) of compact totally disconnected spaces and continuous maps between them) to categorical duals of varieties of Heyting algebras. The subsequent sections apply these results to finitely generated varieties of Heyting algebras. The last section summarizes the results in detail.