Operators on complemented lattices

Abstract The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping $$^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>+</mml:mo> </mml:mmultiscripts> </mml:math> assigning to each element a the set $$a^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> of all its complements is investigated as an operator on the given lattice. We can extend the definition of $$a^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> in a natural way from elements to arbitrary subsets. In particular we study the set $$a^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> for complemented modular lattices, and we characterize when the set $$a^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> is a singleton. By means of the operator $$^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>+</mml:mo> </mml:mmultiscripts> </mml:math> we introduce two other operators $$\rightarrow $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>→</mml:mo> </mml:math> and $$\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⊙</mml:mo> </mml:math> which can be considered as implication and conjunction in a certain propositional calculus, respectively. These two logical connectives are “unsharp” which means that they assign to each pair of elements a non-empty subset. However, also these two derived operators share a lot of properties with the corresponding logical connectives in intuitionistic logic or in the logic of quantum mechanics. In particular, they form an adjoint pair. Finally, we define so-called deductive systems and we show their relationship to the mentioned operators as well as to lattice filters.