infimum

So, for this diagram, human is the infima species.

In a dual way, one defines properties for the preservation of infima.

The uncertainty is understanding what media, "middle", and infima, "low", mean in this context.

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting.

An interesting situation occurs if a function preserves all suprema (or infima).

The technical name for the lowest species in such a scheme is the "infima species".

Moreover, it has to be a complete lattice so that the suprema and infima always exist.

The rest of the proof is routine manipulation with infima, suprema and inequalities.

In analysis, infima and suprema of subsets S of the real numbers are particularly important.

Thus the central operations of lattices are binary suprema and infima .

Infima and suprema do not necessarily exist.

In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

The strongest form of completeness is the existence of all suprema and all infima.

Dually, upper adjoints preserve all existing infima.

In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima.

Least upper bounds (suprema, ) and greatest lower bounds (infima, )

In this way all infima exist, but not all infima are necessarily interesting.

More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Consequently, bounded completeness is equivalent to the existence of all non-empty lower bounded infima.

More generally, one can capture infima and suprema under the abstract notion of a categorical limit (or colimit, respectively).

Many other types of orders arise when the existence of infima and suprema of certain sets is guaranteed.

In a dual way, the existence of all infima implies the existence of all suprema.

However, it is not true that a function that preserves all suprema would also preserve all infima or vice versa.

Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration.

It suffices to require that all suprema and infima of two elements exist to obtain all non-empty finite ones.