The structure of residuated lattices. (English) Zbl 1048.06010

Summary: A residuated lattice is an ordered algebraic structure \[ \mathbf L=\langle L,\vee,\wedge,\cdot,e,\setminus,/\rangle \] such that \(\langle L,\vee,\wedge\rangle\) is a lattice, \(\langle L,\cdot,e\rangle\) is a monoid, and \(\setminus\) and \(/\) are binary operations for which the equivalences \[ a\cdot b\leq c \Leftrightarrow a \leq c/b \Leftrightarrow b \leq a\setminus c \] hold for all \(a,b,c \in L\). It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as “dividing” on the right by \(b\) and “dividing” on the left by \(a\). The class of all residuated lattices is denoted by \(\mathcal {RL}\).
The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers and also by Krull. Since that time, there has been substantial research regarding some specific classes of residuated structures, but we believe that this is the first time that a general structural theory has been established for the class \(\mathcal{RL}\) as a whole. In particular, we develop the notion of a normal subalgebra and show that \(\mathcal{RL}\) is an “ideal variety” in the sense that it is an equational class in which congruences correspond to “normal” subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety \(\mathcal{RL}^{\mathcal C}\) that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members.


06F25 Ordered rings, algebras, modules
06B05 Structure theory of lattices
06B20 Varieties of lattices
06B10 Lattice ideals, congruence relations
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