Residuated lattice

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In abstract algebra, a residuated lattice is a lattice with certain simple properties which apply to collections of all two-sided ideals of any ring.

In this context, the study of these objects has stretched as far back as the 1930s. More recently valuations of truth degrees in various multi-valued logics are taken to form such a lattice in order to generalize for Boolean algebras, Heyting algebras, and MV-algebra.

[edit] Definition

A residuated lattice is an algebra L = $\left \langle L, \wedge, \vee, \otimes, \rightarrow, 0, 1 \right \rangle$ with 0 as the infimum, 1 as the supremum, with $\left \langle L, \otimes, 1 \right \rangle$ a commutative monoid which satisfies the adjointness property, that is

$x\le y \rightarrow z\ \iff x \otimes y \le z\qquad\mbox{for all } x, y, z \in L.$

[edit] Comments

Examples of residual lattices are the Łukasiewicz algebra, the standard Gödel algebra, and the standard product algebra. Residual lattices form a variety.