A residuated closure operator $f$ on a lattice $L$ is a closure operator (an idempotent ascending isotone map) whose fixpoints are identically the fixpoints of a coclosure operator (an idempotent descending isotone map) $g$ i.e. a closure operator on the order dual $L^{op}$. These form an adjoint pair. M. F. Janowitz in 1967 gave the following criterion for residuated closure operators. Let $F$ be the fixpoints for a closure operator $f: L \to L$. Then for each $x \in L$, $f(x) = \wedge(\uparrow{}x \cap F)$. If $f$ is a residuated closure operator, then we can define, additionally, $\forall x \in L$, $g(x) = \vee(\downarrow{}x \cap F)$, the corresponding closure operator on $L^{op}$. Then $F$ is the set of fixpoints for both $f$ and $g$ and they form an adjunction.For a complete lattice $L$ there will always always at least one non-identity residuated closure operator: that whose fixpoints are $\{0_{L}, 1_L\}$. It is easy to see that the fixpoints of a residuated closure must always contain these two elements. In fact, for a lattice $L$ and $x \in L$, $\{0_{L},x,1_{L}\}$ is a set of fixpoints of a residuated closure operator.
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