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The structure of commutative residuated lattices. (English) Zbl 1011.06006
A commutative residuated lattice is a structure $$\langle L;\cdot,\wedge,\vee,\to,e\rangle$$ where $$\langle L;\cdot,e\rangle$$ is a commutative monoid, $$\langle L;\wedge,\vee\rangle$$ is a lattice, and the operations $$\cdot$$, $$\to$$ are connected by the equivalences: $$a\cdot b\leq c$$ iff $$a\leq b\to c$$ iff $$b\leq c\to a$$, $$a,b,c\in L$$. It is known that $$\mathcal{CRL}$$, the class of all commutative residuated lattices, is a finitely based variety of algebras. Now the theory of $$\mathcal{CRL}$$ is studied from the viewpoint of universal algebra. It is shown that congruences correspond to order-convex subalgebras. Further, the authors present an equational basis for the subvariety $$\mathcal{CRL}^c$$ generated by all commutative residuated chains. Finally, the congruence lattice of any member of $$\mathcal{CRL}^c$$ is described.

##### MSC:
 06B05 Structure theory of lattices 06B20 Varieties of lattices 06B10 Lattice ideals, congruence relations
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##### References:
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