Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].

There are two approaches to lattices used in the Mizar Mathematical Library: on the one hand, these structures are based on the set with two binary operations (with an equational characterization as in [17]). On the other hand, we may look at them as at relational structures (posets – see [12]). As the main result of this article we can state that the Mizar formalization enables us to use both ...

employees are seldom passive observers of the events that occur in the work place. they form impressions of others and the events that affect them. the problem of how to motivate employees has puzzled and frustrated managers for generations however, the topic has received considerable and sustained attention in recent decades different motivational theories are developed and the related researc...