A weak relative pseudocomplementation on a poset is a partial operation \(\varvec{*}\) which associates with every pair \(\varvec{(x,y)}\) of elements such that \(\varvec{x \ge y}\) the greatest \(\varvec{u}\) for which \(\varvec{(u] \cap (x] = (y]}\) . By an extension of \(\varvec{*}\) is meant any total operation \(\varvec{\rightarrow }\) extending it; a relative pseudocomplementation, when it exists, is an example. Such partial operations and their extensions have already been studied on meet semilattices. We reveal basic properties of extended weak relative pseudocomplementations on arbitrary posets and discuss in some detail five particular classes of posets equipped with an operation of this kind. Two of them are new; the three others, actually defined in other ways in the literature, have not earlier been associated with weak relative pseudocomplementation.
