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The Conrad program: from \(l\)-groups to algebras of logic. (English) Zbl 1337.06006
Lattice-ordered groups (\(l\)-groups for short) have a fundamental role in the study of algebras of logic. The term Conrad program refers to Paul Conrad’s approach to \(l\)-groups focusing on lattice theoretic properties of their lattices of convex \(l\)-subgroups.
In the paper under review the authors develop a Conrad type approach to the study of a large class of algebras of logic, showing that a substantial part of the Conrad program can be extended to residuated lattices that satisfy the identity \(x\setminus e\approx e/x\), called \(e\)-cyclic in the present paper. This variety encompasses several varieties of significance in algebraic logic. For any \(e\)-cyclic residuated lattice \(L\) let \(C(L)\) be the lattice of its convex subalgebras. A first result proved in this paper is that \(C(L)\) is an algebraic distributive lattice whose principal convex subalgebras form a sublattice.
A second result is that, in case \(L\) satisfies the left or right prelinearity law, a convex subalgebra \(H\) of \(L\) is prime iff the set of all convex subalgebras exceeding \(H\) is a chain under set-inclusion. Thus the lattice of principal convex subalgebras of \(L\) is a relatively normal lattice. It is also proved that a variety \(V\) of \(e\)-cyclic residuated lattices that satisfy either of the prelinearity laws is semilinear iff for every \(L\) in \(V\) all minimal prime convex subalgebras of \(L\) are normal.
In the final part of the paper the authors consider special classes of residuated lattices, e.g., residuated lattices in which convex subalgebras are normal, and GMV-algebras.

06D35 MV-algebras
06F05 Ordered semigroups and monoids
06F15 Ordered groups
03G10 Logical aspects of lattices and related structures
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
08B15 Lattices of varieties
Full Text: DOI
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