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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.

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