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Generalizations of pseudo MV-algebras and generalized pseudo effect algebras. (English) Zbl 1174.06330

Summary: We deal with unbounded dually residuated lattices that generalize pseudo MV-algebras in such a way that every principal order-ideal is a pseudo MV-algebra. We describe the connections of these generalized pseudo MV-algebras with generalized pseudo effect algebras, which allows us to represent every generalized pseudo MV-algebra \(A\) by means of the positive cone of a suitable \(\ell \)-group \(G_A\). We prove that the lattice of all (normal) ideals of \(A\) and the lattice of all (normal) convex \(\ell \)-subgroups of \(G_A\) are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo MV-algebra is commutative.

MSC:

06F05 Ordered semigroups and monoids
06D35 MV-algebras

Software:

Pseudo Hoops
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References:

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