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Some locally tabular logics with contraction and mingle

Hsieh A.

Reports on Mathematical Logic

2010

Vol. 45

143-159

Anderson and Belnap�fs implicational system RMO rightwards arrow can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO* is algebraized by the quasivariety IP of all idem- potent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO* are in one-to-one correspondence with the relative subvarieties of IP. An algebra in IP is called semiconic if it decomposes subdirectly (in IP) into algebras where the iden- tity element t is order-comparable with all other elements. The semiconic algebras in IP are locally finite. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x almost equal to (x rightwards arrow t) rightwards arrow x. It follows that if an axiomatic extension of RMO has ((p rightwards arrow t) rightwards arrow p) rightwards arrow p among its theorems then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized.