Residuation in finite posets

Abstract When an algebraic logic based on a poset instead of lattice is investigated then there natural problem how to introduce implication be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this for the quantum mechanics which orthomodular or effects so-called effect algebra only partial need not lattice-ordered. For this, we introduced operator residuation where values conjunction elements underlying poset, but certain subsets it. However, approach can generalized posets more general conditions. If these are even finite, focus maximal minimal corresponding formulas mentioned operators essentially simplified. This shown in present paper all theorems explained by examples.