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In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function $f$ of one preordered set $X$ into another $Y$ has been called (1) increasingly $g$ -normal, for some function $g$ of $Y$ into $X$ , if for any $x \in X$ and $y \in Y$ we have $f (x) \leq y$ if and only if $x \leq g (y)$ ; (2) increasingly $ϕ$ -regular, for some function $ϕ$ of $X$ into itself,...

Symmetric difference on orthomodular lattices and $Z_{2}$ -valued states

Milan Matoušek, Pavel Pták (2009)

Commentationes Mathematicae Universitatis Carolinae

The investigation of orthocomplemented lattices with a symmetric difference initiated the following question: Which orthomodular lattice can be embedded in an orthomodular lattice that allows for a symmetric difference? In this paper we present a necessary condition for such an embedding to exist. The condition is expressed in terms of $Z_{2}$ -valued states and enables one, as a consequence, to clarify the situation in the important case of the lattice of projections in a Hilbert space.

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