Hoops, Coops and the Algebraic Semantics of Continuous Logic

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.