Exercise 6, § 2

Let $\Sigma$ be the species of structure having a principal base set A and an auxiliary set $k$, whose generic structure (F, H) has as typical characterization

and whose axiom (which can be written more explicitly) is the following : "F is a structure of commutative $k$-algebra with identity element, and H is an irreducible ideal of the algebra A" (we recall that an irreducible ideal in a $k$-algebra A is an ideal $\mathfrak{a}$ such that, for any ideals $\mathfrak{b}$, $\mathfrak{c}$ of A satisfying $\mathfrak{b} \cap \mathfrak{c} = \mathfrak{a}$, either $\mathfrak{b} = \mathfrak{a}$ or $\mathfrak{c} = \mathfrak{a}$). If A, A$'$ are two sets endowed respectively with structures (F, H), (F$'$, H$'$) of species $\Sigma$, we define the $\sigma$-morphisms of A into A$'$ to be homomorphisms $f$ of $k$-algebras, which map identity element to identity element, and satisfy $f(\mathrm{H}) \subset \mathrm{H}'$. Give an example of a family $(\mathscr{S}_\lambda)$ of structures of species $\Sigma$ on a set A, such that there exists a least upper bound for this family of structures (in the ordered set of structures of species $\Sigma$ on A) but such that this least upper bound is not an initial structure for the family $(\mathrm{A}_\lambda, \mathscr{S}_\lambda, \mathrm{Id}_\lambda)$, where $\mathrm{A}_\lambda$ is the set A endowed with the structure $\mathscr{S}_\lambda$, and $\mathrm{Id}_\lambda$ is the identity mapping $\mathrm{A} \to \mathrm{A}_\lambda$. (Consider a polynomial ring $\mathrm{A} = k[\mathrm{T}]$ : the irreducible ideals in A are powers of maximal ideals. If F is the $k$-algebra structure of A, consider the two structures $(\mathrm{F}, \mathfrak{p}_1)$ and $(\mathrm{F}, \mathfrak{p}_2)$ of species $\Sigma$, where $\mathfrak{p}_1$, $\mathfrak{p}_2$ are distinct maximal ideals of A; show that the least upper bound of these two structures is (F, (0)), but that this is not the initial structure for the family in question. To prove this, consider the $k$-subalgebra $\mathrm{B} = k + (\mathfrak{p}_1 \cap \mathfrak{p}_2)$ of A, in which $\mathfrak{p}_1 \cap \mathfrak{p}_2$ is a maximal ideal, and the injection $\mathrm{B} \to \mathrm{A}$.)

¶ Also give an example of a species of structure $\Sigma'$, and a family $(\mathscr{S}'_\lambda)$ of structures of species $\Sigma'$ on a set A$'$, such that the greatest lower bound of this family of structures exists but is not the final structure for the family. (On the dual of A (considered as a vector space) consider the structure of linearly compact $k$-coalgebra deduced from the $k$-algebra structure of A by transposition, and define the species of structures $\Sigma'$ by transposition.)