Editing Semilattice (section)

==Complete semilattices==

Nowadays, the term "complete semilattice" has no generally accepted meaning, and various mutually inconsistent definitions exist. If completeness is taken to require the existence of all infinite joins, or all infinite meets, whichever the case may be, as well as finite ones, this immediately leads to partial orders that are in fact [[complete lattice]]s. For why the existence of all possible infinite joins entails the existence of all possible infinite meets (and vice versa), see the entry [[completeness (order theory)]].

Nevertheless, the literature on occasion still takes complete join- or meet-semilattices to be complete lattices. In this case, "completeness" denotes a restriction on the scope of the [[homomorphism]]s. Specifically, a complete join-semilattice requires that the homomorphisms preserve all joins, but contrary to the situation we find for completeness properties, this does not require that homomorphisms preserve all meets. On the other hand, we can conclude that every such mapping is the lower adjoint of some [[Galois connection]]. The corresponding (unique) upper adjoint will then be a homomorphism of complete meet-semilattices. This gives rise to a number of useful [[duality (category theory)|categorical dualities]] between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively.

Another usage of "complete meet-semilattice" refers to a [[bounded complete]] [[Complete partial order|cpo]]. A complete meet-semilattice in this sense is arguably the "most complete" meet-semilattice that is not necessarily a complete lattice. Indeed, a complete meet-semilattice has all ''non-empty'' meets (which is equivalent to being bounded complete) and all [[directed set|directed]] joins. If such a structure has also a greatest element (the meet of the empty set), it is also a complete lattice. Thus a complete semilattice turns out to be "a complete lattice possibly lacking a top". This definition is of interest specifically in [[domain theory]], where bounded complete [[algebraic poset|algebraic]] cpos are studied as [[Scott domain]]s. Hence Scott domains have been called ''algebraic semilattices''.

Cardinality-restricted notions of completeness for semilattices have been rarely considered in the literature.<ref>E. G. Manes, ''Algebraic theories'', Graduate Texts in Mathematics Volume 26, Springer 1976, p. 57</ref>