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The homomorphism-safe fragment of Tarski's relation algebra is finitely generated, but the function-preserving and total-function-preserving fragments are not finitely generated. The forward function-preserving and local injective-function-preserving fragments are finitely generated.

Abstract

The paper investigates the question of whether certain semantically-defined fragments of Tarski's relation algebra (TRA) can be generated by a finite set of operations. The main findings are:
The homomorphism-safe fragment of TRA is finitely generated, both over finite and arbitrary structures. This is shown by translating homomorphism-preserved first-order formulas to a finite fragment of TRA.
The function-preserving fragment and the total-function-preserving fragment of TRA are not finitely generated, and in fact, not expressible by any finite set of guarded second-order definable function-preserving operations. This is proven by constructing a counterexample operation that is function-preserving but not definable in any finite fragment.
The forward function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an 'injective union' operation.
The paper also discusses connections to related work on clones of operations on binary relations and expressive completeness results in temporal and interval logics.

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Key Insights Distilled From

by Bart Bogaert... at **arxiv.org** 04-12-2024

Deeper Inquiries

One possible semantic fragment of Tarski's relation algebra that could be investigated for finite generatedness is the injective-function-preserving fragment. This fragment focuses on operations that preserve injective partial functions, which could provide valuable insights into the generative properties of such operations. By studying whether this fragment is finitely generated, researchers can further expand their understanding of the expressibility and complexity of operations on binary relations that maintain injectivity.

Characterizing the ⊆-preserved fragment of Tarski's relation algebra as a finitely generated fragment presents an intriguing challenge. While the preservation of the ⊆ relation is a fundamental property in model theory, the finite generatedness of this specific fragment remains an open question. By exploring whether this fragment can be defined by a finite set of operations, researchers can uncover essential insights into the structural properties and expressibility of operations that maintain the subset relation.

The finite generatedness results in the paper can have significant implications for the axiomatizability and computational complexity of the corresponding fragments of Tarski's relation algebra. Understanding which semantic fragments are finitely generated can provide valuable information about the expressibility and definability of operations on binary relations. Moreover, these results can shed light on the computational properties and efficiency of reasoning tasks involving these fragments, offering insights into the complexity of decision procedures and algorithms for working with relation algebras.

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