Lattice-theoretic properties of algebras of logic

LEDDA, ANTONIO;PAOLI, FRANCESCO;Tsinakis C.

2014-01-01

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Abstract

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a latticetheoretic property for more general classes of algebras of logic. For a class of integralresiduatedlatticesthatincludesHeytingalgebrasandsemi-linearresiduated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.