[Paper Review] Closures and generating sets related to combinations of structures

S. V. Sudoplatov|arXiv (Cornell University)|Jan 1, 2016

semigroups and automata theory21 citations

TL;DR

This paper investigates closure operators and generating sets in the context of $E$-combinations and $P$-combinations of first-order structures and their theories. It establishes that for $E$-combinations, the existence of a minimal generating set is equivalent to the existence of a least generating set, and provides syntactic and semantic characterizations of this property. The key contribution is a complete characterization of when such least generating sets exist for linearly ordered language uniform theories, in terms of order-theoretic properties of the underlying index set.

ABSTRACT

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories. We prove, for $E$-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set. For the class of linearly ordered language uniform theories we solve the problem of the existence of least generating set with respect to $E$-combinations and characterize that existence in terms of orders.

Motivation & Objective

• To analyze closure operators and generating sets in $E$-combinations and $P$-combinations of first-order structures.
• To determine when a minimal generating set of theories corresponds to a least generating set in $E$-combinations.
• To characterize the existence of a least generating set syntactically and semantically for $E$-combinations.
• To solve the existence problem for least generating sets in the class of linearly ordered language uniform theories.
• To describe $e$-spectra for $E$-combinations of linearly ordered language uniform theories.

Proposed method

• Introduces $E$-operators and $P$-operators to define topologies on combinations of structures.
• Defines $E$-combinations as structures formed by partitioning the universe via an equivalence relation $E$, with each class elementarily equivalent to a given structure.
• Uses closure operators $\mathrm{Cl}_E(\mathcal{T}_0)$ to generate the set of all theories realizable in $E$-classes of models.
• Applies order-theoretic analysis to index sets of theories, particularly focusing on dense and discrete order types.
• Constructs models with specific accumulation point configurations to realize desired $e$-spectra.
• Applies Morleyzation and expansions of languages to preserve or modify generating set properties under theory extensions.

Experimental results

Research questions

• RQ1 When does a minimal generating set of theories for an $E$-combination coincide with a least generating set?
• RQ2 What syntactic and semantic conditions ensure the existence of a least generating set for $E$-combinations?
• RQ3 For which linearly ordered language uniform theories does an $E$-combination admit a least generating set?
• RQ4 How do $e$-spectra of $E$-combinations depend on the order type of the index set of theories?
• RQ5 Which cardinalities can arise as $e$-spectra for $E$-combinations of $\mathrm{IILU}$-theories?

Key findings

• For $E$-combinations, the existence of a minimal generating set is equivalent to the existence of a least generating set.
• The existence of a least generating set is characterized both syntactically and semantically in terms of the theory's model-theoretic structure.
• For linearly ordered language uniform theories, the existence of a least generating set corresponds to the index set of theories being order-isomorphic to a discrete linear order with finitely many or countably many accumulation points.
• When the index set contains a dense interval, the closure operator $\mathrm{Cl}_E(\mathcal{T}_0)$ does not admit a least generating set.
• For any infinite cardinal $\lambda$, there exists an $E$-combination of $\mathrm{IILU}$-theories with $e$-spectrum at least $\max\{2^\omega, \lambda\}$, showing unbounded $e$-spectra in the absence of $e$-least models.
• For any $\mu \leq \omega$, there exists an $E$-combination of $\mathrm{IILU}$-theories with $e$-spectrum exactly $\mu$, and for uncountable $\lambda$, an $E$-combination exists with $e$-spectrum $\lambda$.