Question

Let \(X \subseteq|\mathfrak{A}| .\) Define \(X_{0}=X \cup C\) where \(C\) is the set of constants of \(\mathfrak{A}, X_{n+1}=X_{n} \cup\left\\{f\left(a_{1}, \ldots, a_{m}\right) \mid f\right.\) in \(\left.\mathfrak{A}, a_{1}, \ldots, a_{m} \in X_{n}\right\\}, X_{\omega}=\) \(\bigcup\left\\{X_{n} \mid n \in \mathbb{N}\right\\}\) Show that \(\mathfrak{B}=\left\langle X_{\omega}, R_{1} \cap X_{\omega}^{r_{1}}, \ldots, R_{2} \cap X_{\omega}^{r_{1}}, f_{1}\left|X_{\omega}^{a_{1}}, \ldots, f_{m}\right| X_{\omega}^{a_{m}},\left\\{c_{1} \mid i \in\right.\right.\) \(I\\}\rangle\) is a substructure of \(\mathfrak{A}\). We say that \(\mathfrak{B}\) is the substructure generated by \(X\). Show that \(\mathfrak{B}\) is the smallest substructure of \(\mathfrak{A}\) containing \(X ; \mathfrak{B}\) can also be characterized as the intersection of all substructures containing \(X\).

Short answer

\(\mathfrak{B}\) is a substructure of \(\mathfrak{A}\) and the smallest substructure containing \(X\).

Step-by-step solution

Understanding the components

We begin by analyzing the components of our exercise. We have a structure \(\mathfrak{A}\) with a set of elements \(|\mathfrak{A}|\), a set of constants \(C\), and functions and relations defined over it. The set \(X\) is a subset of \(|\mathfrak{A}|\) and \(X_0 = X \cup C\). This is our starting point to build the sequence \(X_n\).

Constructing the sequence \(X_n\)

For each integer \(n\), we define \(X_{n+1} = X_n \cup \{f(a_1, \ldots, a_m) \mid f\text{ is a function in }\mathfrak{A}, a_1, \ldots, a_m \in X_n\}\). This step ensures that \(X_n\) includes the closure under the functions of \(\mathfrak{A}\). Finally, define \(X_{\omega} = \bigcup \{X_n \mid n \in \mathbb{N}\}\), which includes all the elements generated by iterating the function closure process infinitely.

Defining the substructure \(\mathfrak{B}\)

Define \(\mathfrak{B} = \langle X_{\omega}, R_i \cap X_{\omega}^{r_i}, f_i \mid X_{\omega}^{a_i}, \{c_i\} \rangle\). This means \(\mathfrak{B}\) includes the elements of \(X_{\omega}\), all the relations \(R_i\) restricted to this set, the functions \(f_i\) restricted to \(X_{\omega}\), and constants from \(C\). This ensures that \(\mathfrak{B}\) is closed under the operations of \(\mathfrak{A}\).

Proving \(\mathfrak{B}\) is a substructure

We verify the criteria for a substructure: (1) \(X_\omega\) is non-empty and contains constants of \(\mathfrak{A}\), (2) it is closed under the functions of \(\mathfrak{A}\) as shown in \(X_n\) steps, and (3) relations in \(\mathfrak{A}\) hold for \(\mathfrak{B}\) because they are restricted to \(X_\omega\). Thus, \(\mathfrak{B}\) satisfies being a substructure of \(\mathfrak{A}\).

Showing smallest substructure criterion

To show \(\mathfrak{B}\) is the smallest substructure containing \(X\), suppose there is any substructure \(\mathfrak{C}\) of \(\mathfrak{A}\) that contains \(X\). Since \(\mathfrak{C}\) is closed under the functions of \(\mathfrak{A}\), it must include all elements of \(X_\omega\) (the closure under functions starting from \(X\)). Therefore, \(\mathfrak{B}\) cannot be smaller. Thus, \(\mathfrak{B}\) is the intersection of all such substructures containing \(X\), proving it to be the smallest.

Key concepts

Substructure Generation

In Mathematical Logic, a "substructure" is essentially a smaller structure within a larger one that maintains certain properties or functions of the original. When we talk about **Substructure Generation**, we refer to creating such a substructure like \(\mathfrak{B}\) from a larger structure \(\mathfrak{A}\).
The process involves choosing a subset \(X\) of the elements from \(|\mathfrak{A}|\). We start by including all constants of \(\mathfrak{A}\) in this set, forming \(X_0 = X \cup C\). This initial step is crucial because constants help establish a base upon which we can build further.
The main goal of substructure generation is to create \(\mathfrak{B}\), which will behave just like \(\mathfrak{A}\) with respect to the operations and relations on the elements of \(X\). Every defined function or relation on \(\mathfrak{B}\) remains consistent within the context of \(\mathfrak{A}\). Thus, \(\mathfrak{B}\) harnesses the essential features of \(\mathfrak{A}\) while respecting the subset \(X\). This process ensures we maintain integrity and consistency with the overall structural properties.

Structure Closure

Closures are pivotal in Mathematical Logic because they help in understanding how new elements can be integrated while maintaining the system's properties. **Structure Closure** refers to forming a closed system within the larger structure \(\mathfrak{A}\) by utilizing the operations and functions defined within it.
For instance, starting with \(X_0\), we continuously apply operations and incorporate all possible elements formed from the functions in \(\mathfrak{A}\). This results in a sequence \(X_1, X_2, \ldots\), where each set is expanded by applying the functions to elements of the preceding set. Finally, this process leads to \(X_\omega\), which encompasses all elements generated from repeatedly applying functions.
Structure closure guarantees that no matter how we operate within the subset \(X\), the resulting elements are always a part of \(X_\omega\). It ensures all operations within \(\mathfrak{B}\) are valid and consistent with those in \(\mathfrak{A}\), thereby allowing us to 'close' \(\mathfrak{B}\) under the operations of the structure.

Function Closure

Within Mathematical Logic, **Function Closure** relates to closing a set under specific functions. The idea is that if we start with set \(X_n\), when we apply functions from \(\mathfrak{A}\), all outcomes are also in \(X_n\). It's a process of including all elements produced by applying the functions continuously.
For each step to form \(X_{n+1}\), apply all applicable functions on every element or combination in \(X_n\). For instance, if \(f(a_1, ..., a_m)\) is a function defined in \(\mathfrak{A}\), then each output for inputs taken from \(X_n\) is added to form \(X_{n+1}\). This logical, recursive expansion is a deliberate way to ensure any functionally derived element also abides by the rules of \(\mathfrak{A}\).
Function closure is vital because it signifies every function can only produce elements still within our growing set \(X_\omega\). Consequently, this closure maintains the integrity of the mathematical structure we are working with, ensuring our substructure \(\mathfrak{B}\) acts as an accurate reflection of \(\mathfrak{A}\).

Relational Closure

Relational Closure in Mathematical Logic ensures that when new elements are added through the closure processes, all original relationships are still respected within the substructure. Relations in \(\mathfrak{A}\) must be consistent when restricted to \(\mathfrak{B}\).
An important part of defining \(\mathfrak{B}\) is the restriction of relations like \(R_i\) to \(X_\omega^{r_i}\). This means that any relation that previously existed on an \(r_i\) tuple within \(\mathfrak{A}\) will still hold true when considering \(\mathfrak{B}\). Such restriction ensures no relation defined in \(\mathfrak{A}\) is violated in the substructure.
Moreover, relational closure confirms that \(\mathfrak{B}\) logically and consistently maintains every relation, as every pair or set of elements in a relation must also be an example within the substructure. This procedural element is crucial for \(\mathfrak{B}\) to act just like \(\mathfrak{A}\) but within a specific context of \(X\). The relational closure thus firmly establishes \(\mathfrak{B}\) as a robust and self-sufficient framework mirroring \(\mathfrak{A}\)'s logical properties.