Forcing of infinity and algebras of distributions of binary semi-isolating formulas for strongly minimal theories

We apply a general approach for distributions of binary isolating and semi-isolating formulas to the class of strongly minimal theories.

containing p and q and having a POSTC R -structure, including all labels that are ✂-dominated by u, the formula θ p,u,q (a, y) has infinitely many solutions.
By the definition a label u ∈ ρ ν(p,q) forces an infinite set of solutions if and only if for any n ∈ ω and some n elements a 1 , . . . , a n in the set of solutions of θ p,u,q (a, y) (in an arbitrary structure), where |= p(a), there exists new element a n+1 , for which |= θ p,u,q (a, a n+1 ) and links between a n+1 and a 1 , . . . , a n are defined by some labels.
Clearly, almost deterministic labels do not force infinite sets of solutions and any label u, ✂-dominating infinitely many labels v i , forces an infinite set of solutions. Moreover, each label u ∧ ¬v i 1 ∧ . . . ∧ ¬v in also forces an infinite set of solutions. Another example with a label, which forces an infinite set of solutions, is presented by theory Th( Q, < ), for which any non-zero label (defining the strict order property) corresponds to formulas having only infinitely many solutions. An infinite set of solutions can be forced by labels in U fin for formulas in stable theories. Such an example is produced by any label corresponding to a special element of an infinite group for an everywhere finitely defined polygonometry [4].
Definition (J. T. Baldwin, A. H. Lachlan [3]). A theory T is called strongly minimal if for any formula ϕ(x,ā) of language obtained by adding parameters ofā (in some model M |= T ) to the language of T , either ϕ(x,ā), or ¬ϕ(x,ā) has finitely many solutions.
An example of strongly minimal theory with the forcing of infinite set is represented by structure M; s with successor function s (having exactly one preimage for any element, and do not having cycles). Since Th( M; s ) has unique 1-type, there is a label u ∈ U cofin for the semi-isolating formula (x ≈ x). This label ✂-dominates infinitely many labels v corresponding to formulas (y ≈ s n (x)), n ∈ Z, and thus, u forces an infinite set of solutions for the formulas θ u (a, y), a ∈ M. Theorem 1. For any strongly minimal theory T , the family R ⇋ S 1 (∅) of 1-types, and a regular family ν(R) of labelling functions for semi-isolating formulas, the following conditions hold: (a) the POSTC R -structure M ν(R) consists of labels belonging to U fin ∪ U cofin ; (b) there is unique type r 0 ∈ R having infinitely many realizations; in particular, any set ρ ν(p,q) is finite, where p, q ∈ R and q = r 0 , and all labels u ∈ ρ ν(p,q) are almost deterministic and belong to U fin ; (c) if R is finite, i. e., all types in R are principal, then all non-zero labels are positive and all labels u, including zero, have complementsū, and for any pair of labels u,ū ∈ ρ ν(p,r 0 ) , exactly one of them is almost deterministic and, in particular, belongs to U fin , and the other label marks a formula θ p,,r 0 (a, y) with infinitely many solutions, where |= p(a), and belongs to U cofin ; (e) only labels in ρ ν(p,r 0 ) with the principal type r 0 can force infinitely many solutions.
Proof. By the definition of strongly minimal theory, each formula ϕ(a, y), where |= p(a), (in particular, witnessing the semi-isolation for a type q(y)) has a finite or a cofinite set ϕ(a, M) of solutions. For the finite set ϕ(a, M), the label u ∈ ρ ν(p,q) , marking the formula ϕ(x, y) with ϕ(a, y) ⊢ q(y) and |= p(a) can ✂-dominate only finitely many labels in ρ ν(p,q) . If ϕ(a, M) is cofinite, then the label u ∈ ρ ν(p,q) for the formula ϕ(x, y) ✂-dominates all labels in ρ ν(p,q) except for a finitely many u 1 , . . . , u k , and u has the complementū, which is obtained from u by disjunctive attachment of labels u i . Thus the condition (a) holds: all labels belong to U fin ∪ U cofin .
Since T is strongly minimal we also have that there is unique type r 0 ∈ R, principal or non-principal, with infinitely many realizations: if there are finitely many 1-types it is implied by the property that models are infinite and there are no two principal formulas with infinitely many realizations, and if there are infinitely many 1-types, then the non-principal type r 0 (x), having infinitely many realizations by Compactness, is isolated by the set of all formulas ¬ϕ(x), where ϕ(x) are principal formulas and none of these formulas can not have infinitely many solutions.
Since the type r 0 with infinitely many realizations is unique, then any set ρ ν(p,q) is finite, where p, q ∈ R and q = r 0 . Here all labels u ∈ ρ ν(p,q) are almost deterministic and belong to U fin . Thus, we have the condition (b).
If r 0 is isolated then all 1-types are isolated and by Proposition 1.1 [2] all non-zero labels are positive. Since each isolating formula ϕ(x) has a label, all labels, including zero, have complements. In this case for any pair of labels u,ū ∈ ρ ν(p,r 0 ) , exactly one of these labels is almost deterministic and, in particular, belongs to U fin , and the other label marks a formula θ p,·,r 0 (a, y) with infinitely many solutions, where |= p(a), and belongs to U cofin . Hence, the condition (c) holds.
If r 0 is non-isolated, then all non-zero labels, linking realizations of r 0 are positive, since having a non-positive non-zero label u, linking realizations of r 0 we have the non-symmetric relation SI r 0 and as r 0 is non-isolated there are infinitely many solutions for the formula θ u (x, a), where |= r 0 (a). This contradicts the strong minimality of theory T . By Proposition 1.1 [2], nonzero labels linking realizations of types in R \ {r 0 }, are positive, and labels, linking realizations of r 0 with realizations of types in R\{r 0 }, are negative. In this case, since for non-principal type there are only relative complements, if a label u belongs to ρ ν(p,r 0 ) , then u is positive or zero, almost deterministic and does not have a complement. Moreover, p = r 0 since realizations of principal types cannot semi-isolate realizations of non-principal type r 0 . If the set ρ ν(r 0 ) is finite, then any label in U fin belongs to U cofin and vice versa, i. e., U fin = U cofin , and if ρ ν(r 0 ) is infinite, then all labels are almost deterministic and U cofin = ∅. Thus, the condition (d) holds. The condition (e) is implied by previous items. ✷ If M is a POSTC R -structure and there is a theory T with a family R = S 1 (∅) and a regular family ν(R) of labelling functions for semi-isolating formulas such that M ν(R) = M, then we say that M is represented by T and also say that T represents the POSTC R -structure M. If all types of R are realized in a model N of T , then we say that M is represented by N .
Note that the syntactic representability of POSTC R -structure M (by a theory) is equivalent to the semantic representability of M (by a model).
Notice also that there is a representation T for the POSTC R -structure M such that a label u is almost deterministic if and only if u does not force an infinite set of solutions.
Theorem 2. Let M be a POSTC R -structure satisfying the following conditions: (a) M consists of labels belonging to U fin ∪ U cofin ; (b) there is an element r 0 ∈ R such that any set ρ µ(p,q) is finite, where p, q ∈ R and q = r 0 , and all labels u ∈ ρ µ(p,q) are almost deterministic (in some representation N of M) and belong to U fin ; (c) if R is finite then all non-zero labels are positive and all labels u, including zero, have complementsū, and for any pair of labels u,ū ∈ ρ µ(p,r 0 ) , exactly one of them is almost deterministic and, in particular, belongs to U fin , and the other label marks a formula θ p,·,r 0 (a, y) (for mathcalN) with infinitely many solutions, where |= p(a), and belongs to U cofin ; (d) if R is infinite then all non-zero labels, linking r 0 or elements of R \ {r 0 }, are positive, and labels, linking r 0 with elements in R \ {r 0 }, are negative; in this case, if a label u belongs to ρ µ(p,r 0 ) , then u is positive or zero, almost deterministic, does not have complements and p = r 0 , moreover, U fin = U cofin if ρ µ(r 0 ) is finite, and U cofin = ∅ if ρ µ(r 0 ) is infinite; (e) only labels in ρ µ(p,r 0 ) and with |R| < ω can force infinity. Then there is a strongly minimal theory T representing the POSTC Rstructure M and having unique 1-type r 0 with infinitely many realizations.
Proof. Consider the construction for the proof of Theorems 9.1 [1] and 8.1 [2]. We identify R with the set of 1-types for the required theory T . Now we add to the types describing links between elements with respect to binary relations Q u , u ∈ U, an information for the cardinality of sets of solutions for formulas θ p,u,q (a, y), where |= p(a). The generic construction for the class T 0 of types guarantees that the generic theory of the language {Q u | u ∈ U} is strongly minimal and represents the POSTC R -structure M. ✷