Orthogonal adjointness in posets with $0$

Abstract

Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with $0$. A pair of operators $f$, $g$ on a poset with $0$ is called orthogonally adjoint if $f(x)$ is orthogonal to $y$ if and only if $x$ is orthogonal to $g(y)$. We characterize the existence and the uniqueness of $g$ for given $f$ and describe basic properties of orthogonal adjointness. We present constructions of orthogonally adjoint pairs in pseudocomplemented posets. If a given operator $f$ is an order-isomorphism of a pseudocomplemented poset satisfying some natural properties then the corresponding adjoint $g$ can be described explicitly. Moreover, if $f$ and $f^{-1}$ are bijective $\perp$-morphisms then they are orthogonally adjoint, too. Finally we show that a given pair of orthogonally adjoint mappings on a poset $\mathbf{P}$ may not be extendable to the Dedekind-McNeille completion of $\mathbf{P}$ and we present sufficient conditions for the existence of such an extension. We also provide sufficient conditions for the existence of an extension of orthogonally adjoint mappings to the lattice of ideals. Our results are illustrated by numerous examples.