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The concept of operator left residuation has been introduced by the authorsin a previous paper. Modifications of so-called quantum structures, inparticular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean andBoolean posets are investigated here in order to show that they are operatorleft residuated or even operator residuated. In fact they satisfy more generalsufficient conditions for operator residuation assumed for bounded posetsequipped with a unary operation. It is shown that these conditions may be alsonecessary if a generalized version using subsets instead of single elements isconsidered. The above listed posets can serve as an algebraic semantics for thelogic of quantum mechanics in a broad sense. Moreover, our approach showsconnections to substructural logics via the considered residuation.

Left residuated operators induced by posets with a unary operation