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Cancellative residuated lattices. (English) Zbl 1092.06012
A residuated lattice is a structure \({\mathbf L}=\langle L,\wedge ,\vee ,\cdot ,e,\backslash ,/ \rangle \) such that \(\langle L,\wedge ,\vee \rangle \) is a lattice, \(\langle L,\cdot ,e \rangle \) is a monoid and for all \(a,b,c\in L\), \(a\cdot b\leq c\) iff \(a\leq c/b\) iff \(b\leq a\backslash c\). \(\mathbf L\) is called cancellative if \(\langle L,\cdot ,e\rangle \) is cancellative. Cancellative residuated lattices are a natural generalization of \(l\)-groups. In sharp contrast with \(l\)-groups, the lattice reduct of a cancellative residuated lattice need not be distributive, in fact, any lattice is a subreduct of some simple, cancellative, integral residuated lattice. MV-algebras and BL-algebras are both particular cases of bounded commutative residuated lattices. Generalized MV- and BL-algebras are introduced as residuated lattices satisfying the identities \(x/((x\vee y)\backslash x)=x\vee y=(x/(x\vee y))\backslash x\) and \(((x\wedge y)/y)\cdot y=x\wedge y=y\cdot (y\backslash (x\wedge y))\), respectively. The existence of lattice bounds and the commutativity assumption are dropped. It is shown that \(\mathbf L\) is a cancellative integral GMV-algebra if and only if it is a cancellative integral GBL-algebra if and only if \(\mathbf L\) is the negative cone of an \(l\)-group. Let \(\mathcal L\mathcal G\) be the variety of \(l\)-groups and \(\mathcal L\mathcal G^{-}\) the class of negative cones of \(l\)-groups; similarly, for any subclass \(\mathcal V\) of \(\mathcal L\mathcal G\), \(\mathcal V^{-}\) is the class of negative cones of members of \(\mathcal V\). The authors prove that \(\mathcal V\mapsto \mathcal V^{-}\) is a one-to-one order-preserving correspondence between subvarieties of \(\mathcal L\mathcal G\) and \(\mathcal L\mathcal G^{-}\), and finally they show how the equational basis for \(\mathcal V^{-}\) can be found given the basis for \(\mathcal V\), and vice versa.

06F05 Ordered semigroups and monoids
06D35 MV-algebras
06F15 Ordered groups
08B15 Lattices of varieties
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