Non-Standard Axioms for Various Math Structures 9

In this post, we summarize my previous work of the series in a more readable and concise manner. (Proofs are not included.)

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We are motivated by the following fact: Boolean algebras are exactly the models of the equational theory of $\left\{ 0,1 \right\}$ . Does a similar fact hold for other classes of algebras, or even classes of structures/models? In other words, let $C$ be a class of models, then does there exist a model $c \in C$ such that the members of $C$ are exactly the models of the theory of $c$ ?

We note that this question depends on the logic we are working in, since that determines the form of the statements in the theory. For instance, our motivating example of Boolean algebras is in equational logic, not say first-order logic.

From another point of view, we can remember that for example group theory is typically motivated in education as the “algebra of symmetry,” and this is often motivated by picking some axioms (associativity, identity, and inverse) that are satisfied by all symmetry groups and then defining groups to be the models of these axioms. But is the connection stronger? Can groups be equivalently defined as the models of all axioms satisfied by all symmetry groups? This turns out to indeed be the case, in equational logic (as a result of Cayley’s Theorem.) Thus, we may define groups to be the models of all equations satisfied by all sets of permutations under composition.

To encompass both of these kinds of situations, we can make a general definition: if $C$ is a class of models, then we call a subclass $C^{'} \subseteq C$ a logical generating subclass of $C$ if $C$ is exactly the class of models of the theory satisfied by every member of $C^{'}$ . With this terminology, in equational logic $\left\{ \left\{ 0,1 \right\} \right\}$ is a logical generating subclass of the class of Boolean algebras and the class of permutation groups is a logical generating subclass of the class of groups.

We now mention some general results.

In first-order logic, a class $C$ has a singleton logical generating subclass if and only if $C$ is the class of all models elementarily equivalent to some model $m$ , and in that case, any subclass of $C$ suffices as a logical generating subclass. For example, if $C$ is the class of real-closed fields, then $\left\{ \mathbb{R} \right\}$ is a logical generating subclass, as well as any other subclass of $C$ . More generally, this holds not just in first-order logic but in any logic closed under negation.

Assuming the theory of abstract generations that I have developed in another project, if we extend part of that theory beyond sets to classes, then for any class $C$ of models, there is a natural generation on $C$ , defined as follows: $C^{'} \subseteq C$ generates $c \in C$ if for every statement $\lambda$ true about each member of $C^{'}$ , $\lambda$ is also true about $c$ . Then, logical generating subclasses are exactly abstract generating subclasses.

For the subset of first-order logic containing operations, relations, existential quantifiers, logical-and, and logical-or (in particular excluding negation), every class $C$ of all models of a theory has a singleton logical generating subclass. The element of such a singleton can be constructed in most cases as a free object over an infinite set; specifically, if the theory does not imply the “trivial collapse” statement $\forall a\forall b(a = b)$ , then the free object over an infinite set is well-formed, and it suffices as the element of a singleton logical generating subclass. Otherwise, all members of $C$ are models that each have only one element, so all non-empty models are isomorphic to each other and any one of them can be the element of a singleton logical generating subclass.

However, for general first-order logic, not every class $C$ of all models of a theory has a singleton logical generating subclass. Instead, we conjecture that every such $C$ has a logical generating subclass with at most two models (where for each statement not implied by the theory, one of the models satisfies it and the other one satisfies its negation.)

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EDIT (07-09/2023): Based on an interaction with ChatGPT, it seems like the connection between the free object and the singleton logical generating subclass in equational logic is well-known.

Other results I’ve stated here, like the fact that the class of permutation groups is a logical generating subclass for the class of all groups, are also well-known. It seems that while these specific results may already be familiar to mathematicians, there could be value in the language and point of view of a logical generating subclass. It would be interesting to see whether we can say more in general about this concept (e.g., classifying all logical generating subclasses of a given class), and what applications that could have.

Also, I have learned recently about closure operators, and I thought about it in context of the “closure” I defined in some of the earlier posts of this series. Specifically, I defined the “closure” (let’s call it the “logical generating closure”) of a class $C$ to be the class $\overline{C}$ of all models of the theory satisfied by every member of $C$ . Is this in fact a closure operator?

It clearly satisfies $C\subseteq \overline{C}$ . If $C\subseteq D$ , then if $m\in\overline{C}$ , $m$ would satisfy the theory $T$ satisfied by every member of $C$ . It follows that $m$ would satisfy the theory $T'$ satisfied by every member of $D$ , since for any statement $\lambda\in T'$ , $\lambda$ would be true of every member in $D$ , and thus it would be true of every member in $C$ as $C\subseteq D$ , and $m$ satisfies every statement true about every member of $C$ . This implies that $m\in \overline{D}$ , so $\overline{C}\subseteq\overline{D}$ . Finally, does $\overline{\overline{C}}=\overline{C}$ ? Well, we have that $\overline{\overline{C}}$ is the class of all models of the theory $T$ satisfied by every member of $\overline{C}$ , and $\overline{C}$ is the class of all models of the theory $T'$ satisfied by every member of $C$ . If $\lambda\in T$ , so that $\lambda$ is satisfied by every member of $\overline{C}$ , then since $C\subseteq\overline{C}$ , $\lambda$ is satisfied by every member of $C$ , hence $\lambda\in T'$ and $T\subseteq T'$ . But on the other hand, if $\lambda\in T'$ , so that $\lambda$ is satisfied by every member of $C$ , then by definition of $\overline{C}$ , every member of $\overline{C}$ must satisfy $\lambda$ too, so that $\lambda \in T$ . Thus, $T=T'$ , so $\overline{\overline{C}}=\overline{C}$ . Hence, the logical generating closure is in fact a closure operator.

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