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Implicational (dual) Residuated Semilinear Gaggle Logics
Eunsuk Yang
Yang and Dunn [28] recently introduced implicational (dual) residuated partial gaggle logics. Here we extend those logics to semilinear ones. More precisely, this paper first defines the class of implicational (dual) residuated prelinear gaggle logics and shows that these logics are semilinear in an algebraic context, that is, they are semilinear in the sense that on linearly ordered matrices they are complete.We next introduce a relational semantics, called Routley–Meyer–style semantics, for finitary those logics and provide completeness for them using the semantics. We finally generalize the term “semilinear” to a notion applicable in both algebraic and set-theoretic contexts and show that finitary those logics are semilinear in this sense.
Keywords: Implicational partial gaggle logic, weakly implicative logic, semilinear logic, (dual) residuation, Routley–Meyer–style semantics
2020 Mathematics Subject Classification. 03B50, 03B52, 03B60.