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Modes, modals, and barycentric algebras: a brief survey and an additivity theorem

Smith J. D. H.

Demonstratio Mathematica

2011

Vol. 44, nr 3

571-587

Modes are idempotent and entropic algebras.Modals are both join semi lattices and modes,where the mode structure distributes over the join.Barycentric algebras are equipped with binary operations from the open unit interval,satisfying idempo tence,skew commutativity,and skew associativity.The article aims to give a brief survey of these structures and some of their applications.Special attention is devoted to hierar chical statistical mechanics and the modeling of complex systems.An additivity theorem for the entropy of independent combinations of systems is proved.

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.

Komori identities in algebraic logic

Blok W., La Falce S.

Reports on Mathematical Logic

2000

No. 34

79-106

A variety generated by a class K of BCK-algebras consists of BCK-algebras if and only if it satisfies a certain kind of identity, first discovered by Komori. A similar phenomenon is shown to hold more generally in a certain class of quasivarieties of logic that includes not only the class of BCK-algebras but also such classes as the quasivariety of biresiduation algebras and quasivarieties of algebras with an equivalence operation. We describe a set of identities (which we call Komori identities), and show that the variety generated by a class K of algebras in one of the quasivarieties considered is contained in the quasivariety if and only it it satisfies a Komori identity. We use the result to establish (i) that the subvarieties of any of the quasivarieties studied are congruence 3-permutable and (ii) that the varietal join of two subvarieties of any of the quasivarieties studied is contained in the quasivariety.