This book contains an algebraic study of semilattice structures. The choice of semilattice structures is motivated by their connections to logic (i.e. the origins and the motivation of axioms for various types of semilattice structures can be found in logic). The authors concentrate on algebraic properties of these structures and do not discuss the impact of their results to logic. They use the usual tools of general algebra. They investigate congruences of algebras, free objects in the corresponding varieties, the subdirectly irreducible algebras in the varieties, etc. After two chapters on algebraic preliminaries, the authors investigate pseudo-complemented semilattices, relatively pseudo-complemented semilattices, implication algebras, sectionally pseudo-complemented semilattices and residuated semilattices. The last chapter on weak systems investigates structures that are not semilattices but where there is still some order in the structure. The topics covered include Hilbert algebras, BCC-algebras and non-associative MV-algebras.