Implicational tonoid semilinear logics: Prelinear logics and their relational semantics

This paper deals with semilinear extensions of implicational tonoid and partial Galois logics. To this end, first the class of implicational tonoid prelinear logics is defined and it is verified that these logics are semilinear in an algebraic context, namely an implicational tonoid logic is semilinear if it is complete over linearly ordered matrices. Next, a relational semantics is introduced for finitary implicational tonoid prelinear logics and it is proved that these logics are complete on the semantics. Thirdly the term "semilinear" is generalized to a notion to be applied in a set-theoretic context and it is verified that finitary implicational tonoid prelinear logics are semilinear in this context. Finally some extensions satisfying abstract Galois, dual Galois properties are introduced together with similar relational semantics for them and it is shown that these logics are semilinear in both contexts.