The regular-injective envelope of $$S$$ -posets.

For $$S$$ a partially-ordered monoid (pomonoid), in this paper we consider three variations of the notion of essentiality for regular monomorphisms of $$S$$ -posets, defining regular-essential, mono-essential and weakly regular-essential extensions of $$S$$ -posets. It is shown that, while the last two classes coincide and properly contain the first, they all lead to the same notion of regular-injective envelope. Every $$S$$ -poset has such an envelope, and when $$S$$ is a pogroup, it may be obtained by letting $$S$$ act on the MacNeille completion of the underlying poset. [ABSTRACT FROM AUTHOR]