Abstract We are dealing with projective classes (in short $\textrm {PC}$) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$ for any infinite cardinal $\lambda $, and that if $\lambda $ is singular, then $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$. If $\lambda $ is regular, then a class of structures over a $\lambda $-ary vocabulary is $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$-definable iff it is the image of a $\lambda $-continuous functor on a $\lambda $-accessible category; we also provide separating counterexamples for the non $\lambda $-ary case. We prove that many $\textrm {PC}$ classes of structures, previously known not to be closed under elementary equivalence over any $\mathbin {\mathscr {L}}_{\infty \lambda }$, are not even $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. Those classes arise from diverse contexts including convex $\ell $-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K$_0$-theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is $\textrm {PC}$ but not $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a ‘well-behaved’ functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.

Abstract We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, completef-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a completel-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive. The following proposition can be proved by using elementary facts about iterated ultrapowers. Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.