‎‎The investigation of equational compactness was initiated by‎ ‎Banaschewski and Nelson‎. ‎They proved that pure injectivity is‎ ‎equivalent to equational compactness‎. ‎Here we define the so‎ ‎called sequentially compact acts over semigroups and study‎ ‎some of their categorical and homological properties‎. ‎Some‎ ‎Baer conditions for injectivity of S-acts are also presented‎.