Hoops, Coops and the Algebraic Semantics of Continuous Logic

Rob Arthan; Paulo Oliva

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arXiv · 1212.2887

Hoops, Coops and the Algebraic Semantics of Continuous Logic

Rob Arthan·Paulo Oliva

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Abstract

B\"{u}chi and Owen studied algebraic structures called hoops. Hoops provide a natural algebraic semantics for a class of substructural logics that we think of as intuitionistic analogues of the widely studied {\L}ukasiewicz logics. Ben Yaacov extended {\L}ukasiewicz logic to get what is called continuous logic by adding a halving operator. In this paper, we define the notion of continuous hoop, or coop for short, and show that coops provide a natural algebraic semantics for continuous logic. We characterise the simple and subdirectly irreducible coops and investigate the decision problem for various theories of coops. In passing, we give a new proof that hoops form a variety by giving an algorithm that converts a proof in intuitionistic {\L}ukaseiwicz logic into a chain of equations.

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BibTeX RIS

Rob Arthan, Paulo Oliva. 2012-12-12. Hoops, Coops and the Algebraic Semantics of Continuous Logic. https://arxiv.org/abs/1212.2887

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