On the structure of rotation-invariant semigroups
A partially ordered set \((M,\leq)\) together with a binary operation \(*\) and an involution \('\) (i.e., an antitone bijection of \(M\) onto itself) is said to have the rotation invariance property with respect to \('\) if \(x*y\leq z\Leftrightarrow y*z'\leq x'\) for all \(x,y,z\in M\). It follows that \((M,*,\leq)\) is right and left residuated. This concept generalizes the notions of Girard monoid and MV-algebra and is studied, in particular, in the case where \((M,*,\leq)\) is a partially ordered semigroup. Several conclusions from the axioms are made and integrality of Girard monoids and divisibility of MV-algebras are deduced. Based on a geometrical characterization several construction methods for rotation invariant semigroups are presented. The motivation for investigating these algebraic structures comes from nonclassical logics such as: full Lambek calculus (with exchange or weakening), fuzzy logics, monoidal propositional calculus, and affine linear logic.
scientific article; zbMATH DE number 1982529