Narrow Results

Over the past four decades an extensive literature covered the properties of $S$-acts. However, only few studies had generalized some known properties of $S$-acts to the $S$-posets. The reversible, and indecomposable properties in $S$-posets have been addressed previously but connectivity has not been defined in $S$-posets yet. Connectivity property was found to be related to those of reversibility and flatness in the category of $S$-acts. The primary objective of this paper is to define connectivity in the category of $S$-posets for both versions: ordered “poconnected” and unordered “connected”. Examples are presented to show the difference between the two versions. The relationship between connectivity with other properties such as reversibility, and indecomposability had also been investigated. We show that the poconnected in $S$-posets is always indecomposable, but the inverse is not true. We also find that the weakly reversible is always connected and indecomposable. These relations among these properties in $S$-posets are different from their corresponding relations in $S$-acts.