  Our concern is the completeness problem for spi-logics, that is, sets of
implications between strictly positive formulas built from propositional
variables, conjunction and modal diamond operators. Originated in logic,
algebra and computer science, spi-logics have two natural semantics:
meet-semilattices with monotone operators providing Birkhoff-style calculi, and
first-order relational structures (aka Kripke frames) often used as the
intended structures in applications. Here we lay foundations for a completeness
theory that aims to answer the question whether the two semantics define the
same consequence relations for a given spi-logic.
