The regular-injective envelope of S-posets

Abstract

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.