Elementary generation and canonicity for varieties of Boolean algebras with operators
Every Boolean algebra \({\mathfrak A}\) with (finitely additive) operators is a (certain special kind of) subalgebra of a complete atomic Boolean algebra with operators called the canonical embedding algebra or the perfect extension of \({\mathfrak A}\). A variety \(V\) of Boolean algebras with operators is said to be canonical if \(V\) contains the perfect extensions of its members. Jónsson and Tarski proved that a variety \(V\) of Boolean algebras with operators is canonical if \(V\) has an axiomatization consisting exclusively of equations in which no variable appears in the scope of an odd number of complementations. The canonical embedding algebra can be obtained by first constructing a relational structure, called the canonical structure of \({\mathfrak A}\), by defining from each \(n\)-ary operator a corresponding \(n + 1\)-ary relation on the ultrafilters of \({\mathfrak A}\), and then taking the complex algebra of that structure. The complex algebra of a relational structure \({\mathfrak S}\) is the complete atomic Boolean algebra of subsets of the universe of \({\mathfrak S}\) augmented with an \(n\)-ary operator for every \(n + 1\)-ary relation in \({\mathfrak S}\). For example, the unary operation on sets that corresponds to a binary relation \(R\) in \({\mathfrak S}\) is the operation of forming the image of a set under \(R\). Operators obtained in this way distribute over arbitrary unions. It frequently happens that a variety \(V\) of Boolean algebras with operators can be characterized as the class of algebras isomorphic to subalgebras of complex algebras of structures in some interesting elementary class \(K\). The construction of complete atomic Boolean algebras with operators from relational structures induces a duality between algebraic and model-theoretic notions. For example, the algebraic notions of subalgebra, homomorphism, and direct product correspond to bounded morphisms, inner substructures, and disjoint unions. This duality is the guiding theme of the paper. The first section summarizes many of the relationships that hold among the model-theoretic closure operators for inner substructures, bounded epimorphisms, bounded unions, disjoint unions, canonical extensions, ultraproducts, and ultraroots. One key fact is that the closure operators for bounded unions and ultraproducts commute. The next section applies these tools to show that, for a fixed dimension, the class of representable cylindric algebras is a canonical variety and the class of cylindric-relativized set algebras is also a canonical variety (that the latter variety is canonical is new). A variety \(V\) of Boolean algebras with operators is said to be generated by a class \(K\) of relational structures if every algebra in \(V\) is a homomorphic image of a subalgebra of a direct product of complex algebras of structures in \(K\). However, if \(K\) is elementary, then the variety it generates is canonical and has a simpler form, namely, there is a class \(M\) of relational structures such that \(V\) is the class of algebras isomorphic to subalgebras of complex algebras of structures in \(M\). It is shown that this is true when \(M\) is the closure of \(K\) under the formation of inner substructures, bounded unions, and bounded epimorphic images, or when \(M\) is the class of models of the ``pseudo-equational'' first-order sentences that are true in \(K\). Although it is still not known whether every canonical variety \(V\) is generated by an elementary class \(K\), it is shown that if \(V\) is generated by an elementary class, then it is generated by the class of models of the first-order theory of the canonical structure of the free algebra in \(V\) on countably many generators. If \(W\) is a class of atomic Boolean algebras with operators that is closed under ``atom-preserving'' atomic subalgebras, is closed under ultraproducts, and contains all the complex algebras of atom structures of atomic algebras in \(W\), then the class of atom structures of atomic algebras in \(W\) is an elementary class. The paper closes with an example of a variety that is locally finite and therefore contains all the canonical embedding algebras of its finitely generated members, and yet is not canonical.
complex algebra of a relational structure
scientific article; zbMATH DE number 836834