Abstract. A sectionally pseudocomplemented poset $P$ is one which has the top element and in which every principal order filter is a pseudocomplemented poset. The sectional pseudocomplements give rise to an implication-like operation on $P$ which coincides with the relative pseudocomplementation if $P$ is distributive. We characterise this operation and study some elementary properties of upper semilattices, lower semilattices and lattices equipped with this as well as two weaker kinds of implication. We also clarify connections of these algebras with Hilbert algebras and with relatively pseudocomplemented posets and semilattices. Sectionally pseudocomplemented lattices have already been studied in the literature.

AMS Subject Classification (1991): 03G25, 06A12, 06D15, 06F35

Received August 14, 2007, and in final form November 24, 2007. (Registered under 6027/2009.)